Ronald P. S. Mahler is a senior staff research scientist at Lockheed Martin
Advanced Technology Laboratories in Eagan, MN. He earned his Ph.D. in
mathematics from Brandeis University, Waltham, MA. He is recipient of
the 2005 IEEE AESS Harry Rowe Mimno Award, the 2007 IEEE AESS
M. Barry Carlton Award, and the 2007 JDL-DFG Joseph Mignogna
Data Fusion Award.
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Advances in Statistical Multisource-Multitarget
Contents Overview: Introduction; Elements of Finite-Set Statistics; RFS Filters:
Standard Measurement Model; RFS Filters for Unknown Backgrounds;
RFS Filters for Nonstandard Measurement Models; Sensor, Platform, and
Weapons Management.
Information Fusion
Since 2007, FISST has inspired a considerable amount of research conducted
in more than a dozen nations and reported in nearly a thousand publications.
This sequel addresses the most intriguing practical and theoretical advances
in FISST, for the first time aggregating and systematizing them into a coherent,
integrated, and deep-dive picture. Special emphasis is given to computationally
fast exact closed-form implementation approaches. The book also includes
the first complete and systematic description of RFS-based sensor/platform
management and situation assessment.
Mahler
This is the sequel to the 2007 Artech House best-selling title, Statistical
Multisource-Multitarget Information Fusion. That earlier book was a
comprehensive resource for an in-depth understanding of finite-set statistics
(FISST), a unified, systematic, and Bayesian approach to information fusion.
The cardinalized probability hypothesis density (CPHD) filter, which was first
systematically described in the earlier book, has since become a standard
multitarget detection and tracking technique, especially in research
and development.
Advances in Statistical
Multisource-Multitarget
Information Fusion
Ronald P. S. Mahler
ISBN 13: 978-1-60807-798-4
ISBN: 1-60807-798-5
ARTECH HOUSE
BOSTON I LONDON
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Advances in Statistical
Multisource-Multitarget
Information Fusion
For a complete listing of titles in the
Artech House Electronic Warfare Library,
turn to the back of this book.
Advances in Statistical
Multisource-Multitarget
Information Fusion
Ronald P. S. Mahler
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ISBN 13: 978-1-60807-798-4
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10 9 8 7 6 5 4 3 2 1
Contents
Preface
xxix
Acknowledgments
xxxv
Chapter 1 Introduction to the Book
1.1 Overview of Finite-Set Statistics
1.1.1 The Philosophy of Finite-Set Statistics
1.1.2 Misconceptions About Finite-Set Statistics
1.1.3 The Measurement-to-Track Association Approach
1.1.4 The Random Finite Set (RFS) Approach
1.1.5 Extension to Nontraditional Measurements
1.2 Recent Advances in Finite-Set Statistics
1.2.1 Advances in Conventional PHD and CPHD Filters
1.2.2 Multitarget Smoothers
1.2.3 PHD and CPHD Filters for Unknown Backgrounds
1.2.4 PHD Filters for Nonpoint Targets
1.2.5 Advances in Classical Multi-Bernoulli Filters
1.2.6 RFS Filters for “Raw-Data” Sensors
1.2.7 Theoretical Advances
1.2.8 Advances in Fusing Nontraditional Measurements
1.2.9 Advances Toward Fully Unified Systems
1.3 Organization of the Book
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Advances in Statistical Multisource-Multitarget Information Fusion
Elements of Finite-Set Statistics
41
Chapter 2 Random Finite Sets
2.1 Introduction
2.1.1 Organization of the Chapter
2.2 Single-Sensor, Single-Target Statistics
2.2.1 Basic Notation
2.2.2 State Spaces and Measurement Spaces
2.2.3 Random States and Measurements, Probability-Mass
Functions, and Probability Densities
2.2.4 Target Motion Models and Markov Densities
2.2.5 Measurement Models and Likelihood Functions
2.2.6 Nontraditional Measurements
2.2.7 The Single-Sensor, Single-Target Bayes Filter
2.3 Random Finite Sets (RFSs)
2.3.1 RFSs and Point Processes
2.3.2 Examples of RFSs
2.3.3 Algebraic Properties of RFSs
2.4 Multiobject Statistics in a Nutshell
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Chapter 3 Multiobject Calculus
3.1 Introduction
3.2 Basic Concepts
3.2.1 Set Functions
3.2.2 Functionals
3.2.3 Functional Transformations
3.2.4 Multiobject Density Functions
3.3 Set Integrals
3.4 Multiobject Differential Calculus
3.4.1 Gâteaux Directional Derivatives
3.4.2 Volterra Functional Derivatives
3.4.3 Set Derivatives
3.5 Key Formulas of Multiobject Calculus
3.5.1 Fundamental Theorem of Multiobject Calculus
3.5.2 Change of Variables Formula for Set Integrals
3.5.3 Set Integrals on Joint Spaces
3.5.4 Constant Rule
3.5.5 Sum Rule
3.5.6 Linear Rule
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3.5.7
3.5.8
3.5.9
3.5.10
3.5.11
3.5.12
3.5.13
3.5.14
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Monomial Rule
Power Rule
Product Rules
First Chain Rule
Second Chain Rule
Third Chain Rule
Fourth Chain Rule
Clark’s General Chain Rule
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Chapter 4 Multiobject Statistics
4.1 Introduction
4.2 Basic Multiobject Statistical Descriptors
4.2.1 Belief-Mass Functions
4.2.2 Multiobject Probability Density Functions
4.2.3 Convolution and Deconvolution
4.2.4 Probability Generating Functionals (p.g.fl.’s)
4.2.5 Multivariate p.g.fl.’s
4.2.6 Cardinality Distributions
4.2.7 Probability Generating Functions (p.g.f.’s)
4.2.8 Probability Hypothesis Densities (PHDs)
4.2.9 Factorial Moment Density
4.2.10 Equivalence of the Fundamental Descriptors
4.2.11 Radon-Nikodým Formulas
4.2.12 Campbell’s Theorems
4.3 Important Multiobject Processes
4.3.1 Poisson RFSs
4.3.2 Identical, Independently Distributed Cluster (i.i.d.c.)
RFSs
4.3.3 Bernoulli RFSs
4.3.4 Multi-Bernoulli RFSs
4.4 Basic Derived RFSs
4.4.1 Censored RFSs
4.4.2 Cluster RFSs
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Chapter 5 Multiobject Modeling and Filtering
5.1 Introduction
5.2 The Multisensor-Multitarget Bayes Filter
5.3 Multitarget Bayes Optimality
5.4 RFS Multitarget Motion Models
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5.5
5.6
5.7
5.8
RFS Multitarget Measurement Models
Multitarget Markov Densities
Multisensor-Multitarget Likelihood Functions
The Multitarget Bayes Filter in p.g.fl. Form
5.8.1 The p.g.fl. Time Update Equation
5.8.2 The p.g.fl. Measurement Update Equation
5.9 The Factored Multitarget Bayes Filter
5.10 Approximate Multitarget Filters
5.10.1 The p.g.fl. Time Update for Independent Targets
5.10.2 The p.g.fl. Measurement Update for Independent
Measurements
5.10.3 A Principled Approximation Methodology
5.10.4 Poisson Approximation: PHD Filters
5.10.5 i.i.d.c. Approximation: CPHD Filters
5.10.6 Multi-Bernoulli Approximation: Multi-Bernoulli
Filters
5.10.7 Bernoulli Approximation: Bernoulli Filters
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Chapter 6 Multiobject Metrology
139
6.1 Introduction
139
6.2 Multiobject Miss Distance
140
6.2.1 Multiobject Miss Distance: A History
141
6.2.2 The Optimal Sub-Pattern Assignment (OSPA) Metric 144
6.2.3 Extension of OSPA to Covariance (COSPA)
147
6.2.4 OSPA for Labeled Tracks (LOSPA)
149
6.2.5 Temporal OSPA (TOSPA)
152
6.3 Multiobject Information Functionals
153
6.3.1 Csiszár Information Functionals
154
6.3.2 Csiszár Functionals for Poisson Processes
157
6.3.3 Csiszár Functionals for i.i.d.c. Processes
158
II
RFS Filters: Standard Measurement Model
Chapter 7 Introduction to Part II
7.1 Summary of Major Lessons Learned
7.2 Standard Multitarget Measurement Model
7.2.1 Standard Multitarget Measurement Submodels
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7.2.2
7.3
7.4
7.5
7.6
Standard Multitarget Measurement Model: p.g.fl.
and Likelihood
167
7.2.3 Standard Multitarget Measurement Model: Special
Cases
168
7.2.4 Measurement-to-Track Association (MTA)
169
7.2.5 Relationship Between the MTA and RFS Approaches 173
An Approximate Standard Likelihood Function
173
Standard Multitarget Motion Model
174
Standard Motion Model with Target Spawning
178
Organization of Part II
178
Chapter 8 Classical PHD and CPHD Filters
8.1 Introduction
8.1.1 Summary of Major Lessons Learned
8.1.2 Organization of the Chapter
8.2 A General PHD Filter
8.2.1 General PHD Filter: Motion Modeling
8.2.2 General PHD Filter: Predictor
8.2.3 General PHD Filter: Measurement Modeling
8.2.4 General PHD Filter: Corrector
8.3 Arbitrary-Clutter PHD Filter
8.3.1 Time Update Equations for the Arbitrary-Clutter
Classical PHD Filter
8.3.2 Measurement Modeling for the Arbitrary-Clutter
Classical PHD Filter
8.3.3 Arbitrary-Clutter PHD Filter: Corrector
8.4 Classical PHD Filter
8.4.1 Classical PHD Filter: Predictor
8.4.2 Classical PHD Filter: Measurement Modeling
8.4.3 Classical PHD Filter: Corrector
8.4.4 Classical PHD Filter: State Estimation
8.4.5 Classical PHD Filter: Uncertainty Estimation
8.4.6 Classical PHD Filter: Characteristics
8.5 Classical Cardinalized PHD (CPHD) Filter
8.5.1 Classical CPHD Filter Motion Modeling
8.5.2 Classical CPHD Filter: Predictor
8.5.3 Classical CPHD Filter: Measurement Modeling
8.5.4 Classical CPHD Filter: Corrector
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8.6
8.7
8.5.5 Classical CPHD Filter: State Estimation
8.5.6 Classical CPHD Filter: Characteristics
8.5.7 Approximate Classical CPHD Filter
Zero False Alarms (ZFA) CPHD Filter
8.6.1 Comparison of the PHD and ZFA-CPHD Filters
PHD Filter for State-Dependent Poisson Clutter
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Chapter 9 Implementing Classical PHD/CPHD Filters
217
9.1 Introduction
217
9.1.1 Summary of Major Lessons Learned
217
9.1.2 Organization of the Chapter
218
9.2 “Spooky Action at a Distance”
219
9.3 Merging and Splitting for PHD Filters
221
9.3.1 Merging for PHD Filters
221
9.3.2 Splitting for PHD Filters
222
9.4 Merging and Splitting for CPHD Filters
223
9.4.1 Merging for CPHD Filters
223
9.4.2 Splitting for CPHD Filters
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9.5 Gaussian Mixture (GM) Implementation
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9.5.1 Standard GM Implementation
227
9.5.2 Pruning Gaussian Components
228
9.5.3 Merging Gaussian Components
229
9.5.4 GM-PHD Filter
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9.5.5 GM-CPHD Filter
244
9.5.6 Implementation with Nonconstant pD
250
9.5.7 Implementation with Partially Uniform Target Births 251
9.5.8 Implementation with Target Identity
257
9.6 Sequential Monte Carlo (SMC) Implementation
261
9.6.1 SMC Approximation
262
9.6.2 SMC-PHD Filter
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9.6.3 SMC-CPHD Filter
267
9.6.4 Using Measurements to Choose New Particles
269
9.6.5 Implementation with Target Identity
275
Chapter 10Multisensor PHD and CPHD Filters
10.1 Introduction
10.1.1 Summary of Major Lessons Learned
10.1.2 Organization of the Chapter
10.2 The Multisensor-Multitarget Bayes Filter
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10.3 The General Multisensor PHD Filter
10.3.1 General Multisensor PHD Filter: Modeling
10.3.2 General Multisensor PHD Filter: Update
10.4 The Multisensor Classical PHD Filter
10.4.1 Implementations of the Exact Classical Multisensor
PHD Filter
10.5 Iterated-Corrector Multisensor PHD/CPHD Filters
10.5.1 Limitations of the Iterated-Corrector Approach
10.6 Parallel Combination Multisensor PHD and CPHD Filters
10.6.1 Parallel Combination Multisensor CPHD Filter
10.6.2 Parallel Combination Multisensor PHD Filter
10.6.3 Simplified PCAM-PHD Filter
10.7 An Erroneous “Averaged” Multisensor PHD Filter
10.8 Performance Comparisons
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Chapter 11Jump-Markov PHD/CPHD Filters
11.1 Introduction
11.1.1 Summary of Major Lessons Learned
11.1.2 Organization of the Chapter
11.2 Jump-Markov Filters: A Review
11.2.1 The Jump-Markov Bayes Recursive Filter
11.2.2 State Estimation for Jump-Markov Filters
11.3 Multitarget Jump-Markov Systems
11.3.1 What Is a Multitarget Jump-Markov System?
11.3.2 The Multitarget Jump-Markov Filter
11.4 Jump-Markov PHD Filter
11.4.1 Jump-Markov PHD Filter: Models
11.4.2 Jump-Markov PHD Filter: Time Update
11.4.3 Jump–Markov PHD Filter: Measurement Update
11.4.4 Jump-Markov PHD Filter: State Estimation
11.5 Jump-Markov CPHD Filter
11.5.1 Jump-Markov CPHD Filter: Modeling
11.5.2 Jump-Markov CPHD Filter: Time Update
11.5.3 Jump-Markov CPHD Filter: Measurement Update
11.5.4 Jump-Markov CPHD Filter: State Estimation
11.6 Variable State Space Jump-Markov CPHD Filters
11.6.1 Variable State Space CPHD Filters: Modeling
11.6.2 Variable State Space CPHD Filters: Time Update
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11.6.3 Variable State Space CPHD Filters: Measurement
Update
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11.6.4 Variable State Space CPHD Filters: State Estimation 338
11.7 Implementing Jump-Markov PHD/CPHD Filters
340
11.7.1 Gaussian Mixture Jump-Markov PHD/CPHD Filters 340
11.7.2 Particle Implementation of Jump-Markov PHD and
CPHD Filters
346
11.8 Implemented Jump-Markov PHD/CPHD Filters
346
11.8.1 Jump-Markov PHD Filter of Pasha et al.
347
11.8.2 IMM-Type JM-PHD Filter of Punithakumar et al.
347
11.8.3 Best-Fitting-Gaussian PHD Filter of Wenling Li and
Yingmin Jia
348
11.8.4 JM-CPHD Filter of Georgescu et al.
349
11.8.5 Current Statistical Model (CSM) PHD Filter of
Mengjun et al.
349
11.8.6 The Variable State Space CPHD Filter of Chen et al. 350
Chapter 12Joint Tracking and Sensor-Bias Estimation
12.1 Introduction
12.1.1 Example: “Gridlocking” of Sensor Platforms
12.1.2 Gridlocking in General
12.1.3 Summary of Major Lessons Learned
12.1.4 Organization of the Chapter
12.2 Modeling Sensor Biases
12.3 Optimal Joint Tracking and Registration
12.3.1 Optimal BURT Filter: Single-Filter Version
12.3.2 Optimal BURT Filter: Two-Filter Version
12.3.3 Optimal BURT Procedure
12.4 The BURT-PHD Filter
12.4.1 BURT-PHD Filter: Single-Sensor Case
12.4.2 BURT-PHD Filter: Multisensor Case Using Iterated
Corrector
12.4.3 BURT-PHD Filter: Multisensor Case Using Parallel
Combination
12.5 Single-Filter BURT-PHD Filters
12.5.1 Single-Filter BURT-PHD Filter for Static Biases
12.5.2 A Heuristic Single-Filter BURT-PHD Filter
12.6 Implemented BURT-PHD Filters
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Contents
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12.6.1 The BURT-PHD Filter of Ristic and Clark
12.6.2 The BURT-PHD Filter of Lian et al.
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Chapter 13Multi-Bernoulli Filters
13.1 Introduction
13.1.1 Summary of Major Lessons Learned
13.1.2 Organization of the Chapter
13.2 The Bernoulli Filter
13.2.1 Bernoulli Filter: Modeling
13.2.2 Bernoulli Filter: Time-Update
13.2.3 Bernoulli Filter: Measurement Update
13.2.4 Bernoulli Filter: State Estimation
13.2.5 Bernoulli Filter: Error Estimation
13.2.6 The Bernoulli Filter as an Exact PHD Filter
13.2.7 Bernoulli Filter: Practical Implementation
13.2.8 Bernoulli Filter: Implementations
13.3 The Multisensor Bernoulli Filter
13.4 The CBMeMBer Filter
13.4.1 CBMeMBer Filter: Modeling
13.4.2 CBMeMBer Filter: Predictor
13.4.3 CBMeMBer Filter: Corrector
13.4.4 CBMeMBer Filter: Merging and Pruning
13.4.5 CBMeMBer Filter: State and Error Estimation
13.4.6 CBMeMBer Filter: Track Management
13.4.7 CBMeMBer Filter: Gaussian-Mixture and Particle
Implementation
13.4.8 CBMeMBer Filter: Performance
13.5 Jump-Markov CBMeMBer Filter
13.5.1 Jump-Markov CBMeMBer Filter: Modeling
13.5.2 Jump-Markov CBMeMBer Filter: Predictor
13.5.3 Jump-Markov CBMeMBer Filter: Corrector
13.5.4 Jump-Markov CBMeMBer Filter: Performance
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Chapter 14RFS Multitarget Smoothers
14.1 Introduction
14.1.1 Summary of Major Lessons Learned
14.1.2 Organization of the Chapter
14.2 Single-Target Forward-Backward Smoother
14.2.1 Derivation of Forward-Backward Smoother
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14.2.2 Vo-Vo Alternative Form of the Forward-Backward
Smoother
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14.2.3 Vo-Vo Exact Closed-Form GM Forward-Backward
Smoother
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14.3 General Multitarget Forward-Backward Smoother
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14.4 Bernoulli Forward-Backward Smoother
416
14.4.1 Bernoulli Forward-Backward Smoother: Modeling 417
14.4.2 Bernoulli Forward-Backward Smoother: Equations 417
14.4.3 Bernoulli Forward-Backward Smoother: Exact GM
Implementation
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14.4.4 Bernoulli Forward-Backward Smoother: Results
421
14.5 PHD Forward-Backward Smoother
421
14.5.1 PHD Forward-Backward Smoother Equation
422
14.5.2 Derivation of the PHD Forward-Backward Smoother 424
14.5.3 Fast Particle-PHD Forward-Backward Smoother
426
14.5.4 Alternative PHD Forward-Backward Smoother
429
14.5.5 Gaussian-Mixture PHD Smoother
430
14.5.6 Implementations of the PHD Forward-Backward
Smoother
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14.6 ZTA-CPHD Smoother
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Chapter 15Exact Closed-Form Multitarget Filter
435
15.1 Introduction
435
15.1.1 Exact Closed-Form Solution of the Single-Target
Bayes Filter
437
15.1.2 Exact Closed-Form Solution of the Multitarget Bayes
Filter
440
15.1.3 Overview of the Vo-Vo Filter Approach
442
15.1.4 Summary of Major Lessons Learned
444
15.1.5 Organization of the Chapter
445
15.2 Labeled RFSs
445
15.2.1 Target Labels
446
15.2.2 Labeled Multitarget State Sets
447
15.2.3 Set Integrals for Labeled Multitarget States
448
15.3 Examples of Labeled RFSs
449
15.3.1 Labeled i.i.d.c. RFSs
449
15.3.2 Labeled Poisson RFSs
453
15.3.3 Labeled Multi-Bernoulli (LMB) RFSs
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15.3.4 Generalized Labeled Multi-Bernoulli (GLMB) RFSs 458
15.4 Modeling for the Vo-Vo Filter
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15.4.1 Labeling Conventions
465
15.4.2 Overview of the Vo-Vo Filter
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15.4.3 Basic Motion and Measurement Models
472
15.4.4 Motion and Measurement Models with Target ID
473
15.4.5 The Labeled Multitarget Likelihood Function
474
15.4.6 The Labeled Multitarget Markov Density—Standard
Version
476
15.4.7 Labeled Multitarget Markov Density—Modified
480
15.5 Closure of Multitarget Bayes Filter
481
15.5.1 A “Road Map” for the Derivations
482
15.5.2 Closure Under Measurement Update with Respect
to Vo-Vo Priors
485
15.5.3 Closure Under Time Update with Respect to Vo-Vo
Priors
489
15.6 Implementation of the Vo-Vo Filter: Sketch
496
15.6.1 δ-GLMB Distributions
496
15.6.2 δ-GLMB Version of the Vo-Vo Filter
498
15.6.3 Characterization of Pruning
498
15.7 Performance Results
499
15.7.1 Gaussian Mixture Implementation of Vo-Vo Filter
499
15.7.2 Particle Implementation of the Vo-Vo Filter
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III
RFS Filters for Unknown Backgrounds
501
Chapter 16Introduction to Part III
16.1 Introduction
16.2 Overview of the Approach
16.3 Models for Unknown Backgrounds
16.3.1 A Model for Unknown Detection Profile
16.3.2 A General Model for Unknown Clutter
16.3.3 Unknown-Clutter Models: Poisson-Mixture
16.3.4 Unknown-Clutter Models: General Bernoulli
16.3.5 Unknown-Clutter Models: Simplified Bernoulli
16.4 Organization of Part III
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Chapter 17RFS Filters for Unknown pD
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17.1 Introduction
17.1.1 Converting RFS Filters into pD -Agnostic Filters
17.1.2 A Motion Model for Probability of Detection
17.1.3 Summary of Major Lessons Learned
17.1.4 Organization of the Chapter
17.2 The pD -CPHD Filter
17.2.1 pD -CPHD Filter Models
17.2.2 pD -CPHD Filter Time Update
17.2.3 pD -CPHD Filter Measurement Update
17.2.4 pD -CPHD Filter Multitarget State Estimation
17.3 Beta-Gaussian Mixture (BGM) Approximation
17.3.1 Overview of the BGM Approach
17.3.2 Beta-Gaussian Mixtures (BGMs)
17.3.3 Pruning BGM Components
17.3.4 Merging BGM Components
17.4 BGM Implementation of the pD -PHD Filter
17.4.1 BGM pD -PHD Filter Modeling Assumptions
17.4.2 BGM pD -PHD Filter Time Update
17.4.3 BGM pD -PHD Filter Measurement Update
17.4.4 BGM pD -PHD Filter Multitarget State Estimation
17.5 BGM Implementation of the pD -CPHD Filter
17.5.1 BGM pD -CPHD Filter Modeling Assumptions
17.5.2 BGM pD -CPHD Filter Time Update
17.5.3 BGM pD -CPHD Filter Measurement Update
17.5.4 BGM pD -CPHD Filter Multitarget State Estimation
17.6 The pD -CBMeMBer Filter
17.7 Implementations of pD -Agnostic RFS Filters
Chapter 18RFS Filters for Unknown Clutter
18.1 Introduction
18.1.1 Summary of Major Lessons Learned
18.1.2 Organization of the Chapter
18.2 A General Model for Unknown Bernoulli Clutter
18.2.1 The General Joint Target-Clutter Model
18.2.2 Phenomenology-Nonintermixing Motion Model
18.2.3 Phenomenology-Intermixing Motion Model
18.3 CPHD Filter for General Bernoulli Clutter
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Contents
18.3.1 General Bernoulli Clutter-Generator Model: CPHD
Filter Time Update
18.3.2 General Bernoulli Clutter Model: CPHD Filter
Measurement Update
18.3.3 General Bernoulli Clutter-Generator Model: PHD
Filter Special Case
18.3.4 General Bernoulli Clutter Model: Multitarget State
Estimation
18.3.5 General Bernoulli Clutter-Generator Model: Clutter
Estimation
18.4 The λ-CPHD Filter
18.4.1 λ-CPHD Filter: Models
18.4.2 λ-CPHD Filter: Time Update
18.4.3 λ-CPHD Filter: Measurement Update
18.4.4 λ-CPHD Filter: Multitarget State Estimation
18.4.5 λ-CPHD Filter: Clutter Estimation
18.4.6 Special Case: The λ-PHD Filter
18.4.7 λ-CPHD Filter Implementation: Gaussian Mixtures
18.5 The κ-CPHD Filter
18.5.1 κ-CPHD Filter: Models
18.5.2 κ-CPHD Filter: Time Update
18.5.3 κ-CPHD Filter: Measurement Update
18.5.4 κ-CPHD Filter: Multitarget State Estimation
18.5.5 κ-CPHD Filter: Clutter Estimation
18.5.6 Special Case: The κ-PHD Filter
18.5.7 κ-CPHD Filter: Beta-Gaussian Mixtures
18.5.8 κ-CPHD Filter Implementation: Normal-Wishart
Mixtures
18.6 Multisensor κ-CPHD Filters
18.6.1 Iterated-Corrector κ-CPHD Filter
18.6.2 Parallel-Combination κ-CPHD Filter
18.7 The κ-CBMeMBer Filter
18.7.1 κ-CBMeMBer Filter: Modeling
18.7.2 κ-CBMeMBer Filter: Time Update
18.7.3 κ-CBMeMBer Filter: Measurement Update
18.7.4 κ-CBMeMBer Filter: Multitarget State Estimation
18.7.5 κ-CBMeMBer Filter: Clutter Estimation
18.8 Implemented Clutter-Agnostic RFS Filters
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18.8.1 Implemented λ-CPHD Filter
18.8.2 “Bootstrap” λ-CPHD Filter
18.8.3 Implemented λ-CBMeMBer Filter
18.8.4 Implemented NWM-PHD Filter
18.9 Clutter-Agnostic Pseudofilters
18.9.1 The λ-PHD Pseudofilter
18.9.2 Pathological Behavior of the λ-PHD Pseudofilter
18.10CPHD/PHD Filters with Poisson-Mixture Clutter
18.10.1 Poisson-Mixture Clutter-Agnostic CPHD Filter
18.10.2 Poisson-Mixture Clutter-Agnostic PHD Filter
18.11Related Work
18.11.1 Decoupled Target-Clutter PHD Filter
18.11.2 The “Dual PHD” Filter
18.11.3 The “iFilter ”
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Chapter 19RFS Filters for Superpositional Sensors
19.1 Introduction
19.1.1 Examples of Superpositional Sensor Models
19.1.2 Summary of Major Lessons Learned
19.1.3 Organization of the Chapter
19.2 Exact Superpositional CPHD Filter
19.3 Hauschildt’s Approximation
19.3.1 Hauschildt Σ-CPHD Filter: Overview
19.3.2 Hauschildt Σ-CPHD Filter: Models
19.3.3 Hauschildt Σ-CPHD Filter: Measurement Update
19.3.4 Hauschildt Σ-CPHD Filter: Implementations
19.4 Thouin-Nannuru-Coates (TNC) Approximation
19.4.1 Generalized TNC Approximation: Overview
19.4.2 TNC Σ-CPHD Filter: Models
19.4.3 TNC Σ-CPHD Filter: Measurement Update
19.4.4 TNC Σ-CPHD Filter: Implementations
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Chapter 20RFS Filters for Pixelized Images
20.1 Introduction
20.1.1 Summary of Major Lessons Learned
20.1.2 Organization of the Chapter
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20.2 The IO Multitarget Measurement Model
20.3 IO Motion Model
20.4 IO-CPHD Filter
20.5 IO-MeMBer Filter
20.5.1 IO-MeMBer Filter: Measurement Update
20.5.2 IO-MeMBer Filter: Track Merging
20.5.3 IO-MeMBer Filter: Multitarget State Estimation
20.5.4 IO-MeMBer Filter: Track Management
20.6 Implementations of IO-MeMBer Filters
20.6.1 Track-Before-Detect (TBD) in Image Data
20.6.2 Tracking in Color Videos
20.6.3 Tracking Road-Constrained Targets
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Chapter 21RFS Filters for Cluster-Type Targets
21.1 Introduction
21.1.1 Summary of Major Lessons Learned
21.1.2 Organization of the Chapter
21.2 Extended-Target Measurement Models
21.2.1 The Statistics of Extended Targets
21.2.2 Exact Rigid-Body (ERB) Model
21.2.3 Approximate Rigid-Body (ARB) Model
21.2.4 Approximate Poisson-Body (APB) Model
21.3 Extended-Target Bernoulli Filters
21.3.1 Extended-Target Bernoulli Filters: Performance
21.4 Extended-Target PHD/CPHD Filters
21.4.1 General Extended-Target PHD Filter
21.4.2 PHD Filter for Extended Targets: ERB Model
21.4.3 PHD Filter for Extended Targets: APB Model
21.5 Extended-Target CPHD Filter: APB Model
21.5.1 APB-CPHD Filter: Theory
21.5.2 Gaussian Mixture APB-CPHD Filter: Performance
21.5.3 Gamma Gaussian Inverse-Wishart APB-CPHD Filter: Performance
21.5.4 APB-CPHD Filter of Lian et al.: Performance
21.6 Cluster-Target Measurement Model
21.6.1 Likelihood Function for Cluster Targets
21.6.2 Estimation of Soft Clusters
21.7 Cluster-Target PHD and CPHD Filters
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21.7.1 Cluster-Target CPHD Filter
21.7.2 Cluster-Target PHD Filter
21.8 Measurement Models for Level-1 Group Targets
21.8.1 “Natural” State Representation of Single Level-1
Group Targets
21.8.2 “Natural” State Representation of Multiple Level-1
Group Targets
21.8.3 Simplified State Representation of Multiple Level-1
Group Targets
21.8.4 Multiple Level-1 Group Targets with the Standard
Measurement Model
21.9 PHD/CPHD Filters for Level-1 Group Targets
21.9.1 PHD Filter for Level-1 Group Targets: Standard
Model
21.9.2 CPHD Filter for Level-1 Group Targets: Standard
Model
21.9.3 PHD Filter for Single Level-1 Group Targets: Standard Measurement Model
21.9.4 CPHD Filter for Single Level-1 Group Targets:
Standard Model
21.10Measurement Models for General Group Targets
21.10.1 Simplified State Representation of Level-ℓ Group
Targets
21.10.2 Standard Measurement Model for Level-ℓ Group
Targets
21.11PHD/CPHD Filters for Level-ℓ Group Targets
21.12A Model for Unresolved Targets
21.13Motion Model for Unresolved Targets
21.14The Unresolved-Target PHD Filter
21.15Approximate Unresolved-Target PHD Filter
21.16Approximate Unresolved-Target CPHD Filter
Chapter 22RFS Filters for Ambiguous Measurements
22.1 Introduction
22.1.1 Motivation: Quantized Measurements
22.1.2 Generalized Measurements, Measurement Models,
and Likelihoods
22.1.3 Summary of Major Lessons Learned
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22.1.4 Organization of the Chapter
22.2 Random Set Models of Ambiguous Measurements
22.2.1 Imprecise Measurements
22.2.2 Vague Measurements
22.2.3 Uncertain Measurements
22.2.4 Contingent Measurements (Inference Rules)
22.2.5 Generalized Fuzzy Measurements
22.3 Generalized Likelihood Functions (GLFs)
22.3.1 GLFs for Nonnoisy Nontraditional Measurements
22.3.2 GLFs for Noisy Nontraditional Measurements
22.3.3 Bayesian Processing of Generalized Measurements
22.3.4 Bayes Optimality of the GLF Approach
22.4 Unification of Expert-System Theories
22.4.1 Bayesian Unification of Measurement Fusion
22.4.2 Dempster’s Rule Arises as a Particular Instance of
Bayes’ Rule
22.4.3 Bayes-Optimal Measurement Conversion
22.5 GLFs for Imperfectly Characterized Targets
22.5.1 Example: Imperfectly Characterized Target Types
22.5.2 Example: Received Signal Strength (RSS)
22.5.3 Modeling Imperfectly Characterized Targets
22.5.4 GLFs for Imperfectly Characterized Targets
22.5.5 Bayes Filtering with Imperfectly Characterized Targets
22.6 GLFs for Unknown Target Types
22.6.1 Unmodeled Target Type
22.6.2 Unmodeled Target Types—Imperfectly Characterized Measurement Function
22.7 GLFs for Information with Unknown Correlations
22.8 GLFs for Unreliable Information Sources
22.9 Using GLFs in Multitarget Filters
22.10GLFs in RFS Multitarget Filters
22.10.1 Using GLFs in PHD Filters
22.10.2 Using GLFs in CPHD Filters
22.10.3 Using GLFs in CBMeMBer Filters
22.10.4 Using GLFs in Bernoulli Filters
22.10.5 Implementations of RFS Filters for Nontraditional
Measurements
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22.11Using GLFs with Conventional Multitarget Filters
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22.11.1 Measurement-to-Track Association (MTA) with Nontraditional Measurements
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22.11.2 A Closed-Form Example: Fuzzy Measurements
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22.11.3 MTA with Joint Kinematic and Nonkinematic Measurements
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V
Sensor, Platform, and Weapons Management
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Chapter 23Introduction to Part V
23.1 Basic Issues in Sensor Management
23.1.1 Top-Down or Bottom-Up?
23.1.2 Single-Step or Multistep?
23.1.3 Information-Theoretic or Mission-Oriented?
23.2 Information Theory and Intuition: An Example
23.2.1 PENT for “Cookie Cutter” Sensor Fields of View
(FoVs)
23.2.2 PENT for General Sensor Fields of View
23.2.3 Characteristics of PENT
23.2.4 The Cardinality-Covariance Objective Function
23.2.5 The Cauchy-Schwartz Objective Function
23.3 Summary of RFS Sensor Control
23.3.1 RFS Control Summary: General Approach (SingleStep)
23.3.2 RFS Control Summary: Ideal Sensor Dynamics
23.3.3 RFS Control Summary: Simplified Nonideal Sensor Dynamics
23.3.4 RFS Control Summary: Control with PHD and
CPHD Filters
23.3.5 RFS Control Summary: “Pseudosensor” Approximation for Multisensor Control
23.3.6 RFS Control Summary: General Approach (Multistep)
23.4 Organization of Part V
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Chapter 24Single-Target Sensor Management
24.1 Introduction
24.1.1 Summary of Major Lessons Learned
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24.1.2 Organization of the Chapter
24.2 Example: Missile-Tracking Cameras
24.2.1 Single-Camera Missile Tracking
24.2.2 Two-Camera Missile Tracking
24.3 Single-Sensor, Single-Target Control: Modeling
24.4 Single-Sensor, Single-Target Control: Single-Step
24.5 Single-Sensor, Single-Target Control: Objective
Functions
24.5.1 Kullback-Leibler Information Gain
24.5.2 Csiszár Information Gain
24.5.3 Cauchy-Schwartz Information Gain
24.6 Single-Sensor, Single-Target Control: Hedging
24.6.1 Expected-Value Hedging
24.6.2 Minimum-Value Hedging
24.6.3 Multisample Approximate Hedging
24.6.4 Single-Sample Approximate Hedging
24.6.5 Mixed Expected-Value and PM Hedging
24.7 Single–Sensor, Single-Target Control: Optimization
24.8 Special Case 1: Ideal Sensor Dynamics
24.9 Simple Example: Linear-Gaussian Case
24.10Special Case 2: Simplified Nonideal Dynamics
24.10.1 Simplified Nonideal Single-Sensor Dynamics: Modeling
24.10.2 Simplified Nonideal Single-Sensor Dynamics: Filtering Equations
24.10.3 Simplified Nonideal Single-Sensor Dynamics: Optimization
Chapter 25Multitarget Sensor Management
25.1 Introduction
25.1.1 Summary of Major Lessons Learned
25.1.2 Organization of the Chapter
25.2 Multitarget Control: Target and Sensor State Spaces
25.2.1 Target State Spaces
25.2.2 Sensor State Spaces
25.2.3 Joint Multisensor-Multitarget State Space
25.2.4 Integrals and Set Integrals on State Spaces
25.2.5 p.g.fl.’s on Target/Sensor State Spaces
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25.3 Multitarget Control: Control Spaces
25.4 Multitarget Control: Measurement Spaces
25.4.1 Sensor Measurements
25.4.2 Actuator-Sensor Measurements
25.4.3 Joint Multisensor-Multitarget Measurements
25.4.4 Integrals and Set Integrals on Measurement Spaces
25.4.5 p.g.fl.’s on Measurement Spaces
25.5 Multitarget Control: Motion Models
25.5.1 Single-Target and Multitarget Motion Models
25.5.2 Single-Sensor Motion and Multisensor Motion with
Sensor Controls
25.5.3 Joint Multisensor-Multitarget Motion
25.6 Multitarget Control: Measurement Models
25.6.1 Measurements: Assumptions
25.6.2 Measurements: Sensor Noise
25.6.3 Measurements: Fields of View (FoVs) and Clutter
25.6.4 Measurements: Actuator Sensors and Transmission
Failure
25.6.5 Measurements: Multitarget Likelihood Functions
25.6.6 Measurements: Joint Multitarget Likelihood Functions
25.7 Multitarget Control: Summary of Notation
25.7.1 Notation for Spaces of Interest
25.7.2 Notation for Motion Models
25.7.3 Notation for Measurement Models
25.8 Multitarget Control: Single Step
25.9 Multitarget Control: Objective Functions
25.9.1 Information-Theoretic Objective Functions
25.9.2 The PENT Objective Function
25.9.3 The Cardinality-Variance Objective Function
25.9.4 PENT as an Approximate Information-Theoretic
Objective Function
25.10Multisensor-Multitarget Control: Hedging
25.10.1 Hedging Using Predicted Measurement Set (PMS)?
25.10.2 Predicted Ideal Measurement Set (PIMS): A General Approach
25.10.3 Predicted Ideal Measurement Set (PIMS): Special
Cases
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25.10.4 Predicted Ideal Measurement Set (PIMS): Derivation of General Approach
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25.11Multisensor-Multitarget Control: Optimization
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25.12Sensor Management with Ideal Sensor Dynamics
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25.13Simplified Nonideal Multisensor Dynamics
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25.13.1 Simplified Nonideal Multisensor Dynamics: Assumptions
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25.13.2 Simplified Nonideal Multisensor Dynamics: Filtering Equations
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25.13.3 Simplified Nonideal Single-Sensor Dynamics: Hedgingand Optimization
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25.14Target Prioritization
25.14.1 The Concept of Tactical Significance
25.14.2 Tactical Importance Functions (TIFs) and HigherLevel Fusion
25.14.3 Characteristics of TIFs
25.14.4 The Multitarget Statistics of TIFs
25.14.5 Posterior Expected Number of Targets of Interest
(PENTI)
25.14.6 Biasing the Cardinality Variance to Targets of Interest (ToIs)
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Chapter 26Approximate Sensor Management
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26.1.2 Organization of the Chapter
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26.2 Sensor Management with Bernoulli Filters
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26.2.1 Sensor Management with Bernoulli Filters: Filtering Equations
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26.2.2 Sensor Management with Bernoulli Filters: Objective Functions
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26.2.3 Bernoulli Filter Control: Hedging
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26.2.4 Bernoulli Filter Control: Multisensor
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26.3 Sensor Management with PHD Filters
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26.3.1 Single-Sensor, Single-Step PHD Filter Control
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26.3.2 PHD Filter Sensor Management: Multisensor SingleStep
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26.4 Sensor Management with CPHD Filters
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26.4.1 Single-Sensor, Single-Step CPHD Filter Control
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26.4.2 Multisensor, Single-Step CPHD Filter Control
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26.5 Sensor Management with CBMeMBer Filters
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26.5.2 Multisensor, Single-Step CBMeMBer Control
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26.6 RFS Sensor Management Implementations
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26.6.1 RFS Control Implementations: Multitarget Bayes
Filter
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26.6.2 RFS Control Implementations: Bernoulli Filters
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26.6.3 RFS Control Implementations: PHD Filters
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26.6.4 RFS Control Implementations: CBMeMBer Filters 1030
Appendix A Glossary of Notation and Terminology
A.1 Transparent Notational System
A.2 General Mathematics
A.3 Set Theory
A.4 Fuzzy Logic and Dempster-Shafer Theory
A.5 Probability and Statistics
A.6 Random Sets
A.7 Multitarget Calculus
A.8 Finite-Set Statistics
A.9 Generalized Measurements
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Appendix B Bayesian Analysis of Dynamic Systems
B.1 Formal Bayes Modeling in General
B.2 The Bayes Filter in General
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Appendix C Rigorous Functional Derivatives
C.1 Nonconstructive Definition of the Functional Derivative
C.2 The Constructive Radon-Nikodým Derivative
C.3 Constructive Definition of the Functional Derivative
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Appendix D Partitions of Finite Sets
D.1 Counting Partitions
D.2 Recursive Construction of Partitions
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Appendix E Beta Distributions
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Appendix F Markov Time Update of Beta Distributions
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Appendix G Normal-Wishart Distributions
G.1 Proof of (G.8)
G.2 Proof of (G.22)
G.3 Proof of (G.23)
G.4 Proof of (G.29)
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Appendix H Complex-Number Gaussian Distributions
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Appendix I Statistics of Level-1 Group Targets
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Appendix J FISST Calculus and Moyal’s Calculus
J.1 A “Point Process ”Functional Calculus
J.2 Volterra Functional Derivatives
J.3 Moyal’s Functional Calculus of p.g.fl.’s
J.3.1 Moyal’s p.g.fl.
J.3.2 Moyal’s Functional Calculus
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Appendix K Mathematical Derivations
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References
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About the Author
1109
Index
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Preface
This book is a sequel to my 2007 book, Statistical Multisource-Multitarget Information Fusion [179]. That earlier book was a textbook-style introduction to finiteset statistics (also known as random set information fusion), a fundamentally new,
seamlessly unified, and fully probabilistic approach to multisource-multitarget detection, tracking, classification, and information fusion.
This sequel provides a comprehensive description of the state of the art in
random set information fusion since 2007—a description not otherwise available.
Its intended audience is signal processing graduate students, researchers, and engineers, as well as mathematicians and statisticians interested in tracking, information
fusion, robotics, and related subjects.
Finite-set statistics has five major elements:
• A general theory of measurements, based on a stochastic-geometry formulation of random set theory.
• A general theory of stochastic multiobject systems, based on a stochasticgeometry formulation of point process theory or, equivalently, random finite
set theory.
• A general approach to multisource-multitarget modeling based on multiobject
integro-differential calculus.
• A general optimal approach to multisource-multitarget processing based on
these models and Bayesian filter theory.
• A general approach to multisource-multitarget algorithmic approximation,
also based on multiobject integro-differential calculus.
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Tutorial introductions to the subject can be found in [198], [181], and [311];
and extended summaries in the book chapters [177] and [311]. An authorized
Chinese-language edition of Statistical Multisource-Multitarget Information Fusion
is also available [180]. Other information sources are:
• The “Random Finite Set Filtering” web site, with U.K. and Australian mirrors:
– http://randomsets.eps.hw.ac.UK/index.html.
– http://randomsets.ee.unimelb.edu.au/index.html.
• Professor Ba-Tuong Vo’s web sites:
– http://ba-tuong-vo-au.com.
– http://ba-tuong-vo-au.com/codes/html.
– Various MATLAB algorithm codes are available at the second link:
probability hypothesis density (PHD) filter; cardinalized PHD (CPHD)
filter; cardinality-balanced multi-Bernoulli (CBMeMBer) filter; the singletarget RFS filter described in [309]; and (eventually) the λ-CPHD filter
(described in Chapter 18) and the track-before-detect multi-Bernoulli
filter (described in Chapter 20).
Since 2007, the approach has inspired a considerable amount of research,
conducted by many dozens of researchers in at least a dozen nations, reported
in many hundreds of research publications. As a result, progress in random set
information fusion has been rapid and has proceeded in diverse and sometimes
unexpected directions, propelled by many clever new ideas. Indeed, the rapidity
and extent of progress has itself been somewhat unexpected, especially given
my cautious disclaimer in Statistical Multisource-Multitarget Information Fusion
([179], p. 566):
“...preliminary research has suggested that the PHD and CPHD filters
may be more effective than [multihypothesis correlator-type] filters
in some conventional multitarget detection and tracking problems.
Whether such claims hold up is for future research to determine. Here
we emphasize that the PHD and CPHD approaches were originally
devised to address non-traditional tracking problems such as those just
described [that is, tracking of target clusters].”
Preface
xxxi
A summary of advances in the field will be given in Section 1.2. In brief,
the progression of research emphasis has been roughly as follows: from PHD filter
to CPHD filter; from CPHD filter to multi-Bernoulli filters and, in particular, to
the CBMeMBer filter; and most recently, to “background-agnosic” CPHD and CBMeMBer filters and the Vo-Vo exact closed-form multitarget detection and tracking
filter. Ancillary advances have occurred in regard to joint tracking and sensor registration; superpositional sensors and track-before-detect (TBD); distributed fusion;
sensor management; and robotics.
As time progressed, it became increasingly clear to me that the most intriguing aspects of the new research should be aggregated and systematized, in a single
place, into a coherent and integrated picture. That is the purpose of this book. Thus
one of my primary goals is to provide a deep-dive overview of the state of the art in
the field.
However, this overview is not intended to be comprehensive. Many rather
intricate implementation strategies for PHD and CPHD filters have been neglected,
for example. I have placed a greater emphasis on exact closed-form implementation
approaches, rather than particle implementation approaches. I have not included
research whose veracity I have been unable to verify in detail. In particular I have,
albeit with a few exceptions, excluded mention of research that is mathematically
erroneous or that I have been unable to understand. Also, as the book neared
finalization at the end of 2013, it became impossible to include later-breaking
developments.
Certain advances are not described here, or are described at only a broad
conceptual level, because they are already addressed in book-length or book-chapter
form elsewhere. The reader’s attention is directed to the following publications:
• B. Ristic, Particle Filters for Random Set Models [250].
– As the title indicates, this book emphasizes techniques for the sequential
Monte Carlo (SMC) implementation of RFS multitarget detection and
tracking filters.
– An unusual aspect of the book is its application of particle-RFS filters to
nontraditional measurements, such as natural-language statements and
inference rules.
• J. Mullane, B.-N. Vo, M. Adams, and B.-T. Vo, Random Finite Sets in Robotic
Map Building and SLAM [210].
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– This is a concise introduction to the new random finite set approach
to the robotics field known as Simultaneous Localization and Mapping
(SLAM). Its topics include an introduction to RFSs, estimation with
RFSs, a “brute force” PHD filter approach to SLAM, and a RaoBlackwellized PHD filter approach to SLAM.
– The RFS-SLAM techniques reported in this book provide the first
provably Bayes-optimal approach to SLAM.
– In high-clutter environments, PHD filter-based SLAM algorithms described in the book have been shown to significantly outperform conventional approaches such as EKF-SLAM and MH-FastSLAM.
– A tutorial overview of the approach can be found in [1].
– Another source is Professor Martin Adams’ web site:
∗ http://www.cec.uchile.cl/˜martin/Martin research 18 8 11.html.
– The RFS-SLAM paper [141] by Lee, Clark, and Salvi is also pertinent
in this regard, as is the June 2014 special issue of the IEEE Robotics &
Automation Magazine on “Applications of Stochastic Geometry.”
• R. Mahler, “Toward a Theoretical Foundation for Distributed Fusion” [183].
– This book chapter is essentially a summary and elaboration of work
by D. Clark and his associates. This work—for example, the paper
[294] by Uney, Clark, and Julier—leverages the multitarget covarianceintersection concept of [172], which is immune to unknown doublecounting of measurements.
– Of special interest in this regard is the potentially breakthrough-level
paper [15] by Battistelli, Chisci, Fantacci, Farina, and Graziano.
– Additional papers of interest are [242], [10].
For those who may wish to employ this publication as a university textbook,
I suggest the following. It is divided into five parts, devoted to increasingly
more specialized and increasingly more research-oriented topics. Part I provides
a fairly condensed summary of the basic elements of finite-set statistics. When
used in conjunction with Statistical Multisource-Multitarget Information Fusion, it
would be suitable for the first part of an introductory one-semester course. The
beginning of Part II is devoted to topics that have become standard in finite-set
statistics (PHD, CPHD, Bernoulli, and multi-Bernoulli filters). This material is
Preface
xxxiii
suitable for the second part of an introductory one-semester course. Chapter 22
is appropriate for a course oriented more towards expert systems and higher-level
information fusion. The remaining chapters of Part II, as well as Parts III, IV, and
V, address more specialized topics and leading-edge research. When selected with
a particular research focus in mind, a subset of these chapters could be the basis of
a graduate seminar or advanced one-semester course, with the purpose of leading
students to the threshold of dissertation-level research. For pedagogical reasons
and ease of reference, many well-known concepts are reviewed—for example,
unscented Kalman filters, jump-Markov filters, complex Gaussian distributions,
beta distributions, and Wishart and inverse-Wishart distributions.
As with Statistical Multisource-Multitarget Information Fusion, it is my
sincere hope that the reader will find the book informative, useful, stimulating,
thought-provoking, occasionally provocative, and possibly even a bit exciting.
Acknowledgments
The contents of this book do not necessarily reflect the position or policy of
Lockheed Martin Corporation. No official endorsement should be inferred.
I gratefully acknowledge the ongoing original research and correspondences
of Professors Ba-Ngu Vo and Ba-Tuong Vo (Curtin University, Perth, Australia);
Professor Daniel Clark (Heriot-Watt University, Edinburgh, U.K.); and Dr. Branko
Ristic (Defence Science and Technology Organization, Melbourne, Australia).
Readers will also quickly discover the importance of the research of the following individuals and their students: Professor Thia Kirubarajan (McMaster University, Hamilton, Ontario, Canada); Professor Peter Willett (University of Connecticut, Storrs, Connecticut, United States of America); Professor Lennart Svensson (Chalmers University of Technology, Göteborg, Sweden); and Professor Mark
Coates (McGill University, Montreal, Canada).
I also wish to express my gratitude for the long-standing assistance and
researches of my collaborators at Scientific Systems Company, Inc.—especially,
Dr. Adel El-Fallah and Dr. Alexsander Zatezalo. I am particularly grateful to Dr.
El-Fallah for his invaluable help in preparing the final camera-ready copy of the
book.
The work of dozens of researchers is described in this book. I have endeavored to describe their work as accurately and as clearly as possible. In many cases,
I found it necessary to directly contact them for clarification of various technical
issues. I wish to thank them for their time, assistance, and patience.
Finally, and as I did in Statistical Multisource-Multitarget Information Fusion, I acknowledge my profound debt to the groundbreaking research of Dr. I.R.
Goodman (U.S. Navy SPAWAR Systems Center, retired). It was Dr. Goodman who
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first made me aware of the potentially revolutionary implications of random set
techniques in information fusion.
The manuscript for this book was produced using Version 5.50 of MacKichan
Software’s Scientific WorkPlace. The task of writing it would have been vastly more
difficult without it. Figures were prepared using Scientific WorkPlace and Microsoft
PowerPoint.
Chapter 1
Introduction to the Book
The subject of the earlier book, Statistical Multisource-Multitarget Information Fusion [179], was finite-set statistics (FISST), a form of random finite set (RFS) theory
specialized for application to information fusion Finite-set statistics unifies much
of information fusion under a single probabilistic—in fact, Bayesian—paradigm. It
does so by directly generalizing, to the multisource-multitarget realm, the “statistics
101” formalism that most signal processing practitioners learn as undergraduates.
Increasingly more detailed tutorial descriptions of the approach can be found in the
following publications:
• R. Mahler, “‘Statistics 101’ for multisensor, multitarget data fusion” [198] (a
tutorial introduction at a very elementary level);
• R. Mahler, “‘Statistics 102’ for multisensor-multitarget tracking” [181] (a
more detailed tutorial);
• R. Mahler, “Random set theory for target tracking and identification” [177]
(an extended summary);
• B.-N.Vo, B.-T. Vo, and D. Clark, “Bayesian multiple target filtering using
random finite sets” [311] (a tutorial with emphasis on implementation of the
PHD, CPHD, and multi-Bernoulli filters).
Finite-set statistics was introduced in 1994 in its basic form (set integrals, set
derivatives, RFS motion and measurement models, multitarget Bayes filter) [162],
[94]; and, in its current refined form (probability generating functionals, functional
derivatives) in 2001 [168]. From the beginning it has been conceived as a systematic
1
2
Advances in Statistical Multisource-Multitarget Information Fusion
Figure 1.1 The finite-set statistics research program. Unification of expert systems, and of generalized measurements, sets the stage for a unification of multisource integration. This sets the stage for a unification of sensor and platform management. This is intended to set the stage for a unification of all of the information
fusion levels.
Introduction to the Book
3
research program, as indicated in Figure 1.1. In the United States at least, information fusion has been conceptualized in terms of a taxonomy of “fusion levels” [25].
In the original and simplest form of this taxonomy, there are four fusion levels.
Level 1 fusion (also known as “multisource integration”) algorithms address basic
issues: the detection, tracking, localization, and identification of one or more targets
using one or more information sources. Levels 2 and 3 fusion (“higher-level fusion,”
“threat assessment,” “situation assessment”) algorithms address more complex and
amorphous issues such as degree of threat and adversarial intent. Level 4 fusion
(“information refinement” or “resource allocation”) algorithms enable sensing and
other assets to collect better information about poorly understood targets of interest.
Statistical Multisource-Multitarget Information Fusion was primarily concerned with the top two layers of Figure 1.1: unification of expert-systems theory,
and unification of Level 1 fusion. The purpose of this book is therefore threefold:
• To provide detailed and integrated descriptions of many of the most interesting and significant advances in Level 1 information fusion.
• To place these advances in the context of the finite-set statistics research
program, and with respect to each other.
• To systematically address, for the first time, the bottom two layers in Figure
1.1—unification of sensor and platform management (see Part V), and the
elements of a foundation for Levels 2 and 3 information fusion (Section 25.14
).
In consequence of the first goal, the reader will discover that much of the
book is devoted to the innovations of other researchers. While I have endeavored
to describe these as accurately as possible, any errors in their description should be
attributed to me.
The remainder of this chapter is structured as follows:
1. Section 1.1: An overview of the philosophy, methods, and techniques of
finite-set statistics.
2. Section 1.2: A summary of recent advances in finite-set statistics.
3. Section 1.3: The organization of the book.
4
Advances in Statistical Multisource-Multitarget Information Fusion
1.1
OVERVIEW OF FINITE-SET STATISTICS
The purpose of this section is to provide a brief introduction to finite-set statistics
and random finite set (RFS) methods for those readers who may not yet be familiar
with them. An extended and detailed summary will be provided in Chapters 2
through 6 and Chapter 22. The section is organized as follows:
1. Section 1.1.1: The philosophy of finite-set statistics.
2. Section 1.1.2: Misconceptions about finite-set statistics.
3. Section 1.1.3: Measurement-to-track association (MTA)—the conventional
approach to multisensor-multitarget information fusion.
4. Section 1.1.4: The random finite-set (RFS) approach, compared to MTA.
5. Section 1.1.5: Extension of the RFS approach to nontraditional measurements, using general random set theory.
1.1.1
The Philosophy of Finite-Set Statistics
Finite-set statistics has attracted a considerable amount of interest in a relatively
short period of time. The lesson that a few seem to have drawn is that renown
will follow if one skims a few insights from finite-set statistics while changing its
notation and terminology; strips off some or all of the mathematical tools that make
these insights rigorous, general, and useful; and then proclaims the resulting halfcopy to be an advance over finite-set statistics. All such imitator-critics have not
only completely missed the point, but have embraced the same fallacy: the belief
that mere changes of notation and terminology add technical substance.
The point of finite-set statistics is not that multitarget problems can be
formulated in terms of random finite sets, random counting measures, or anything
else. The choice of a particular mathematical formalism is of limited practical
interest in and of itself. The point is, rather, that random set techniques provide
a carefully constructed practitioner’s toolbox of explicit, rigorous, systematic, and
general procedures. This toolbox cannot be supplanted by extemporized, ad hoc
reasoning that (as we shall see) actually facilitates the commission of error.
The purpose of this section is to summarize the finite-set statistics toolbox
and the philosophy that underlies it.
Finite-set statistics is based on Bayesian probability and filtering theory,
which is reviewed in Appendix B. Challa, Evans, and Musicki have succinctly
Introduction to the Book
5
summarized one of the primary viewpoints that motivate finite-set statistics [31, p.
437]:
In practice, an intelligent combination of probability theory using
Bayes’ theorem and ad hoc logic is used to solve tracking problems.
The ad hoc aspects of practical tracking algorithms are there to limit
the complexity of the probability theory based solutions. One of the
key things that is worth noting is that the success of Bayesian solution
depends on the models used. Hence, before applying Bayes’ theorem
to practical problems, a significant effort must be spent on modeling.
In pursuit of this goal, finite-set statistics addresses multisource-multitarget
information fusion problems using the following systematic methodology:
• Step 1: Approach information fusion problems in a unified, statistically topdown fashion, by constructing comprehensive statistically accurate models of
multitarget-multisensor-multiplatform systems, including:
– Top-down, comprehensive statistically accurate (as opposed to extemporized, ad hoc) models of multitarget sensing—encompassing phenomena such as sensor-platform dynamics, sensor slew rates, sensor
noise, sensor fields of view (FoVs), missed-detection processes, clutter
processes, obscurations, and transmission dropouts.
– Top-down, comprehensive statistically accurate (as opposed to extemporized, ad hoc) models of multitarget motion—encompassing phenomena such as individual target motion, target disappearance, and target
appearance.
– Top-down, comprehensive statistically accurate models of “nontraditional measurements”—encompassing attributes, features, naturallanguage statements, and inference rules, as well as certain expertsystem uncertainty representations such as fuzzy sets, the DempsterShafer theory, and rule-based inference.
• Step 2: Use these statistically accurate models to construct the optimal
solution to the problem at hand—typically, some kind of multitarget recursive
Bayes filter. This necessitates the explicit construction of:
– “True” multitarget Markov transition densities—meaning that:
∗ These densities faithfully reflect the underlying multitarget motion
model and thus are not heuristic or ad hoc.
6
Advances in Statistical Multisource-Multitarget Information Fusion
∗ No extraneous information has inadvertently been introduced.
– “True” multitarget likelihood functions—meaning that
∗ These functions faithfully reflect the underlying multitarget measurement model and thus are not heuristic or ad hoc.
∗ No extraneous information has inadvertently been introduced.
• Step 3: Since the optimal solution will usually be computationally intractable
in general, use principled approximation techniques to “trim down” the
optimal solution to an approximate one that is tractable and yet preserves, as
faithfully as possible, the underlying models and their interrelationships. In
particular, it is presumed that:
– A principled approximation must be statistically top-down, in the sense
that it has been directly constructed from the optimal multitarget solution.
However, a statistically top-down approach forces us into unfamiliar theoretical territory:
• Multisensor, multitarget systems are comprised of randomly varying numbers
of randomly varying objects of various kinds: varying numbers of targets;
varying numbers of sensors with varying number of sensor measurements
collected by each sensor; and varying numbers of sensor-carrying platforms.
A rigorous mathematical foundation for stochastic multiobject problems—
point process theory [55], [278]—has been in existence for a half-century. However, this theory has traditionally been formulated with the requirements of mathematicians rather than tracking and information fusion practitioners in mind. The
formulation usually preferred by mathematicians, random counting-measures, is
inherently abstract and complex (especially in regard to probabilistic foundations)
and not easily assimilable with practical physical intuition (see Section 2.3.1).
The fundamental motivation underlying the finite-set statistics treatment of
point process theory is this:
• Tracking and information fusion R&D researchers and practitioners should
not have to be virtuoso experts in point process theory to produce meaningful
practical innovations.
As was emphasized in [198], engineering statistics is a tool and not an end in
itself. It must have two qualities:
Introduction to the Book
7
• Trustworthiness: Constructed upon a systematic, reliable mathematical foundation, to which we can appeal when the going gets rough.
• Fire and forget: This foundation can be safely neglected in most situations,
leaving a serviceable mathematical machinery in its place.
These two qualities are inherently in conflict. If foundations are so mathematically complex that they cannot be taken for granted in most practical situations,
then they are shackles and not foundations. If they are so simple that they repeatedly
result in practical blunders, then they are simplistic rather than simple.
This inherent gap between mathematical trustworthiness and practical pragmatism is what finite-set statistics attempts to bridge. Four objectives are paramount:
• Directly generalize familiar single-sensor, single-target Bayesian “Statistics
101” concepts to the multisource-multitarget realm.
• Avoid all avoidable abstractions.
• As much as possible, replace theorem-proving with “mechanical,” “turn-thecrank,” purely algebraic procedures.
• Nevertheless retain all mathematical power necessary for effective practical
problem-solving.
The following are specific illustrations of the second point.
1.1.1.1
Illustration 1: Avoid Avoidable Concepts
Consider the concepts of “thinning” and “marking.” These are basic to purely
mathematical treatments of point process theory. But in multitarget detection
and tracking, they appear only in a few, concrete contexts that can be adequately
addressed at a practitioner level of complexity. Missed detections and disappearing
targets can both be described as forms of thinning; and target identity as a form
of marking. But does the imposition of such concepts represent an increase of
practically actionable understanding—or of pedantry?
1.1.1.2
Illustration 2: Avoid Abstract Point Process Theory
Finite-set statistics is based on a specific formulation of point process theory—the
stochastic-geometry version of the theory of random finite sets (RFSs) [134], [278].
The reason for this is that the stochastic geometry formulation offers the following
advantages:
8
Advances in Statistical Multisource-Multitarget Information Fusion
1. It is more “practitioner friendly” than the random counting measure and other
formulations of point processes. A finite set {x1 , ..., xn } is easily visualizable as a point pattern—for example, in the plane or in three dimensions.
Similarly, an RFS is easily visualizable as a random point pattern. The following is an everyday example of an RFS regarded as a random point pattern:
the stars in a night sky, with many stars winking in and out of visibility, and/or
slightly varying in their apparent position ([179], pp. 349-356).
2. It usually permits us to avoid abstractions such as topologies, measurable
mappings, and the “randomness” of point processes in the formal mathematical sense. The topology presumed in the stochastic-geometry formulation
of RFS theory—the Fell-Matheron topology—permits a major theoretical
simplification. Abstract probability measures, defined on “hyperspaces” of
finite sets, can be equivalently replaced by belief-mass functions (b.m.f.’s,
also known as belief measures) defined on ordinary (that is, nonhyper) spaces.
Just as the probability measure
pk|k (S) = Pr(Xk|k ∈ S)
(1.1)
completely characterizes the probability law of the random target-state Xk|k ,
so the b.m.f.
βk|k (S) = Pr(Ξk|k ⊆ S)
(1.2)
completely characterizes the probability law of a random finite set Ξk|k .
Formulas for the probability distribution fΞk|k (X) of Ξk|k can be derived
from βk|k (S) via set differentiation and “turn-the-crank” differentiation
rules.
3. It results in a multitarget mathematical formalism that is nearly identical to
the single-target “Statistics 101” formalism with which tracking practitioners
are already familiar.
4. It provides a systematic probabilistic foundation for expert systems theory
(fuzzy logic, Dempster-Shafer theory, rule-based inference) in addition to
multisensor-multitarget estimation and filtering. This permits a systematic
and mathematically rigorous unification of these two quite different aspects
of information fusion (see Chapter 22).
Introduction to the Book
1.1.1.3
9
Illustration 3: Avoid Abstract Measure Theory
In finite-set statistics density functions are systematically used in place of measures,
except when this is not possible. Thus the Dirac delta function is employed
even though it produces practitioner-heuristic abbreviations of rigorous measuretheoretic expressions.
In particular, in finite-set statistics probability functionals and their functional
derivatives are addressed using a constructive, rather than an abstract measuretheoretic, approach. The reasons for this choice, explained in greater detail in
Appendix J, are as follows.
1. The purely measure-theoretic treatments preferred by pure mathematicians,
such as Moyal [207], are ill-suited for practical purposes because:
(a) Measure-theoretic definitions of probability functionals are based on abstract probability measures on abstract hyperspaces—concrete formulas
for which are extremely cumbersome and difficult to determine.
(b) Measure-theoretic treatments of functional differentiation are nonconstructive. They lead to formulas for abstract measures on product
spaces (or, equivalently, multilinear functionals) rather than ordinary
density functions. Even if we succeed in proving that these very abstract
measures are absolutely continuous, the Radon-Nikodým theorem tells
us only that their densities exist. It does not provide what is required for
practical application: concrete formulas for these densities themselves.
2. By way of contrast, finite-set statistics has, from its inception in 1996, been
based on belief-mass functions on (nonhyper) spaces and their set derivatives.
In particular:
(a) Probability functionals, such as the probability generating functional
(p.g.fl.), can be defined in terms of belief-mass functions on ordinary
(nonhyper) spaces.
(b) A functional derivative is a particular kind of set derivative, and set
derivatives are constructive. That is, they produce concrete formulas
for density functions—as opposed to abstract formulas for abstract measures on product spaces (equivalently, abstract multilinear functionals).
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Advances in Statistical Multisource-Multitarget Information Fusion
1.1.2
Misconceptions About Finite-Set Statistics
As might be expected to occur with any new technical specialty, a few misconceptions about finite-set statistics have arisen and sometimes dogmatically asserted as
fact. The purpose of this section is to address the more common of these.
• Misconception 1: PHD filters, like all RFS filters, are defined only for
Euclidean state and measurement spaces.
– RFS filters are based on the stochastic-geometry formulation of random
set theory—in particular, on the Fell-Matheron topology on the hyperspace of closed subsets of an underlying space Y. The topology on the
subhyperspace of finite sets is the restriction of the Fell-Matheron topology to that subhyperspace. Consequently, a state space or measurement
space can be any topological space that is Hausdorff, locally compact,
and completely separable [278]. This space is further assumed to be
endowed with a measure-theoretic integration concept.
– In applications, the underlying space Y commonly has the “hybrid”
continuous-discrete form L × S, where S ⊆ RN is some region of a
Euclidean space and L is a finite set.
• Misconception 2: Because finite sets are order-independent, RFS filters are
inherently incapable of constructing time sequences of labeled tracks—and
therefore are not true tracking filters.
– This misconception is a consequence of the first one. Target states can
have the non-Euclidean form
x = (ℓ, u),
(1.3)
where ℓ ∈ L is an identifying label unique to each track and where
u ∈ S is the kinematic part of the state. Given this, the multitarget
Bayes filter—as well as any RFS approximation of it, including PHD
and CPHD filters—are in principle inherently capable of maintaining
temporally-connected tracks. (See pp. 505-508 of [179].) They are
capable of track maintenance also in practice, as much recent research
has shown. (The fact that the first such algorithms did not appear until
2004 reflects the fact that implementers did not begin addressing the
track labeling issue until then.)
Introduction to the Book
11
– In particular and as is explained in detail in Chapter 15, Vo and Vo have
devised an exact, closed-form, computationally tractable solution of the
multitarget recursive Bayes filter. This results in what appears to be the
first provably Bayes-optimal, implementable, multitarget detection and
tracking algorithm. The track management approach inherent to this
filter is, therefore, also provably Bayes-optimal.
• Misconception 3: Target identifiability is lost in RFS models.
– This misconception is a second consequence of the first one. An
identifying label ℓ can include target-identity information as well as
track label information—meaning that target identifiability is not lost in
RFS models.
• Misconception 4: Because target identifiability is lost in RFS models, PHD
and other RFS filters require that motion models and likelihood functions are
the same for all targets.
– This misconception is a third consequence of the first one. Since singletarget likelihood functions are allowed to have the form
fk+1 (z|ℓ, u),
(1.4)
one can specify a different measurement model for each choice of ℓ.
Similarly, because single-target Markov densities are allowed to have
the form
fk+1|k (ℓ, u|ℓ′ , u′ ) = fk+1|k (ℓ|ℓ′ , u′ ) · fk+1|k (u|ℓ, ℓ′ , u′ ),
(1.5)
one can specify a different single-target Markov density
fk+1|k (u|ℓ, ℓ′ , u′ )
(1.6)
for each choice of ℓ, ℓ′ .
• Misconception 5: The CPHD filter cannot address target-spawning because—
unlike the PHD filter—it does not have a target-spawning model.
– Consider the following analogy. A jump-Markov motion model is not
necessarily required for a single-target Bayes filter to successfully track
maneuvering targets. In the same manner, a spawning model is not
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Advances in Statistical Multisource-Multitarget Information Fusion
necessarily required for an RFS filter to track spawning targets. The
CPHD filter can potentially detect and track spawning targets (though
with a time-delay, as the filter accounts for measurements generated by
the spawned targets).
– Furthermore, the CPHD filter’s model for (nonspawned) target appearance can be used to address spawning targets.
– Moreover: a spawning model can actually cause tracking performance
to deteriorate. Consider, for example, the staging events of a multistage
missile. If the missile type is unknown, these events will occur at unpredictable times. If the spawning model is active at all times, it will almost
always mismodel the missile’s actual spawning behavior—since spawning occurs only at one, two, or possibly three isolated instants. Because
the spawning model is thereby usually quite inaccurate, a tracking filter
with such a model will be forced to “waste” measurements to overcome
the ongoing mismatch between reality (no spawning is occurring) and
the spawning model (spawning is occurring).
• Misconception 6: The previous misconception derives from a deeper
misconception—that models are always applicable in all situations. As the
target-spawning example illustrates, this is not true:
– Detailed statistical models can be counterproductive if inappropriately
applied.
– Some models are appropriate in some circumstances and not in others.
• Misconception 7: Both the time-update Dk|k (x|Z (k) ) → Dk+1|k (x|Z (k) )
and the measurement-update Dk+1|k (x|Z (k) ) → Dk+1|k+1 (x|Z (k+1) ) for
the (classical) PHD filter require the following assumptions: for the timeupdate, fk|k (X|Z (k) ) is approximately Poisson; and for the measurementupdate, fk+1|k (X|Z (k) ) is approximately Poisson.
– In actuality, only the second assumption is required. The formula for
the time-update does not require the Poisson approximation and, in
this sense, is exact (given the underlying models and independence
assumptions). An exact derivation is possible because of the product
and chain rules of the multitarget differential calculus.
Introduction to the Book
13
• Misconception 8: The (classical) PHD filter is obsolete, because it has been
superseded by better RFS filters such as the (classical) CPHD filter and the
CBMeMBer filter.
– Practical real-time application usually involves a trade-off between algorithm performance and algorithm computational complexity. While
the (classical) CPHD filter has significantly better performance than the
(classical) PHD filter, it is also significantly more computationally intensive. Consequently, there will be applications in which the CPHD
filter cannot be employed, but the PHD filter can. Likewise, the CBMeMBer filter has different limitations that sometimes hinder its use.
– In any case, the (classical) PHD filter often exhibits surprisingly good
performance, especially in dense-clutter situations that tax the capabilities of conventional combinatorial algorithms.
• Misconception 9: The RFS model of the multiple target state is an approximation, because the Bayes posterior RFS is not exact, but is an approximation
based on the earlier invocations of the PHD approximation used to close
the Bayesian recursion. The Bayes posterior RFS is an approximation even
before the PHD approximation is invoked.
– This misconception appears to be due to an extremely superficial reading of the finite-set statistics literature. In this reading, the entire RFS
approach is misunderstood to be synonymous with one particular aspect
of the RFS approach: the PHD filter. That is, it is mistakenly believed
that the multitarget RFS is always presumed to be Poisson (“PHD approximation”).
– In actuality, in the RFS approach a random multitarget state at time
tk is mathematically represented as an RFS Ξk|k , and the statistical behavior of Ξk|k is represented by its RFS probability distribution fk|k (X|Z (k) ). This distribution is statistically exact—not approximate. In particular, the RFS approach includes a “multitarget calculus”
methodology for explicitly constructing fk|k (X|Z (k) ) from explicitlydefined RFS multitarget motion and measurement models.
– For purposes of computation, in the RFS approach multitarget distributions are approximated in various ways. To arrive at the (classical)
PHD filter, the predicted multitarget distribution fk+1|k (X|Z (k) ) is
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Advances in Statistical Multisource-Multitarget Information Fusion
approximated as a Poisson multitarget distribution:
fk+1|k (X|Z (k) ) ∼
= e−Nk+1|k
∏
Dk+1|k (x|Z (k) ).
(1.7)
x∈X
However, this is not the only approximation used in the RFS approach. For example, fk+1|k (X|Z (k) ) can also be approximated as
an i.i.d.c. process, a multi-Bernoulli process, or a generalized labeled
multi-Bernoulli process—none of which are necessarily Poisson.
– The Poisson approximation is the simplest and least accurate RFS approximation of the random multitarget state. It is not a “representation”
of it.
• Misconception 10: The right model of the multitarget state is that used in
the multihypothesis tracker (MHT) paradigm, not the RFS paradigm.
– The MHT “representation” of the multitarget state is—no less than the
Poisson, i.i.d.c., multi-Bernoulli, or generalized labeled multi-Bernoulli
“representations” used in the RFS paradigm—an approximation of
fk|k (X|Z (k) ).
– Moreover, if the MHT “representation” is the “right” one, then it
should be—as is the case with the RFS representation—provably Bayesoptimal. But it is apparently the case that no proof exists showing that
MHT is Bayes-optimal—or even approximately Bayes-optimal.
– An algorithm is not Bayes-optimal simply because it employs Bayes’
rule in some fashion. The term “Bayes optimal” has a specific mathematical meaning. In the multitarget case, it requires the minimization
of the multitarget Bayes risk corresponding to some multitarget cost
function (see Section 5.3).
– By way of contrast, the approximation of fk|k (X|Z (k) ) as a generalized labeled multi-Bernoulli distribution is provably Bayes-optimal—
because it leads to a exact closed-form solution of the multitarget Bayes
filter (see Chapter 15).
• Misconception 11: The RFS approach is questionable because it is computationally intractable, and requires extreme approximations to make it
tractable.
Introduction to the Book
15
– This statement appears to reflect the existence of a double standard. The “ideal” MHT is inherently computationally intractable. Its
combinatorics can be “beat down” only by resort to rather extreme
approximations—approximations that can severely degrade its performance in heavy-clutter and other scenarios.
• Misconception 12: The RFS approach has not “panned out”—that is,
RFS algorithms, such as PHD and CPHD filters, have failed to demonstrate
significant improvement over more conventional approaches.
– Provided that we eschew less magnanimous interpretations of this statement, this appears to be another misconception that is attributable to a
superficial reading of the finite-set statistics literature. While this issue
will be more fully addressed in Section 1.2, the following examples will
suffice here:
– Vo, Vo, and Cantoni have reported a single-target RFS filter that significantly outperforms traditional approaches such as the probabilistic
data association (PDA) filter [309]. (This RFS filter can also address
state-dependent clutter, such as multipath returns.)
– In conventional multitarget detection and tracking, CPHD filters can
successfully perform in clutter with clutter rates of 70 measurements per
frame and higher. In such circumstances, conventional algorithms such
as MHTs tend to suffer combinatorial breakdown. This is especially the
case when there are newly appearing targets, in which case such targets
tend to be overlooked because of measurement-gating.
– Another example is the TNC Σ-CPHD filter for superpositional sensors
(Chapter 19). It has been shown to significantly outperform conventional Markov Chain Monte Carlo (MCMC) methods, while also being
30 to 87 times faster (depending on the specific application).
– Still another example is the IO-MeMBer track-before-detect (TBD) filter for tracking in images (Chapter 20). It has been shown to significantly outperform the previously best TBD algorithm, the histogramPMHT.
– As a final example, consider the RFS-SLAM algorithms [210], [208],
[1]. In high-clutter environments, these have been shown to significantly outperform conventional simultaneous localization and mapping
(SLAM) algorithms such as MH-FastSLAM.
16
Advances in Statistical Multisource-Multitarget Information Fusion
1.1.3
The Measurement-to-Track Association Approach
The purpose of this and the following subsections is to provide an overview of
the finite-set statistics approach by contrasting it with the ubiquitous conventional
approach, measurement-to-track association (MTA). A more complete and precise
discussion will be provided in Sections 7.2.2, 7.2.4, and 7.2.5.
It should also be pointed out that the following discussion is conceptual. It is
not intended to be a description of the internal logic of any particular conventional
multitarget tracking algorithm. (For an encyclopedic treatment of conventional
tracking methods, see the book by Blackman and Popoli [24].)
The most familiar tracking algorithms presume the following measurement
model—hereafter referred to as the “standard multitarget measurement model.” A
detection process (for example, a threshold) is applied to a sensor signature (for
example, a radar scan or an image). Signal-to-noise ratio (SNR) is relatively small,
meaning that the number m of measurements will typically outnumber the number
n of targets. The result is a set Z = {z1 , ...., zm } of measurements (for example,
measured positions). For each zi there are three possibilities:
• zi was generated by a target (a “target detection”).
• zi was caused by sensor noise (a “false detection”).
• zi was generated by some real entity in the background environment that is
momentarily target-like but not an actual target (a “clutter detection”).
There is also a fourth possibility:
• A target was present but did not generate a measurement (a “missed detection”).
For mathematical convenience, false and clutter detections are usually grouped
together as a single statistical process, which is referred to as “clutter,” and which
is usually assumed to be Poisson.
For the target-generated detections, the “small-target” model is presumed.
That is:
• Targets are distant enough (relative to the sensor’s resolution capability) that
a single target generates at most a single detection.
• They are also near enough that any given detection is generated by at most a
single target.
Introduction to the Book
17
Because of the small-target assumption, a bottom-up, “divide and conquer”
strategy can be applied to the multitarget detection and tracking problem ([179].
pp. 321-335). Because of this strategy, the multitarget tracking problem can be
decomposed into multiple single-target tracking problems.
Suppose that, at time tk , a multitarget tracking algorithm has produced n
hypothesized targets, called “tracks.” These tracks have the form
k|k
k|k
k|k
k|k
k|k
(ℓ1 , x1 , P1 ), ..., (ℓk|k
n , xn , Pn )
where xi is a state-vector (for example, position, velocity), Pi a error covariance
k|k
matrix (modeling the uncertainty in xi ), and ℓi
is a “track label” (to clearly
distinguish the tracks from one another as they evolve through time). The Gaussian
k|k
distribution fi (x) = NP k|k (x − xi ) is called the “track density” or “track
i
distribution” or “spatial density” of the ith track.
Next, suppose that at time tk+1 the sensor collects m detections Zk+1 =
{z1 , ..., zm } with |Zk+1 | = m where, in more difficult tracking scenarios, m > n
because of clutter. The prediction step of some single-target filter—most typically,
an extended Kalman filter (EKF)—is used to construct predicted tracks
k+1|k
(ℓ1
k+1|k
, x1
k+1|k
, P1
), ..., (ℓk+1|k
, xnk+1|k , Pnk+1|k ).
n
Let I be a (possibly empty) subset of {1, ..., n}. From it, construct the following
hypothesis HI,τ about how the predicted tracks are related to the new measurements at time tk+1 :
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k
• The tracks (ℓi
, xi
, Pi
) with i ∈ I generated the detections
zτ (i) for some selection τ (i) ∈ {1, ..., m} of indices, so that ZI,τ =
{zτ (i) | i ∈ I} is the set of target-generated measurements.
• The tracks (ℓi
, xi
, Pi
) with i ∈
/ I were not detected.
• The excess measurements Zk+1 − ZI,τ were generated by clutter.1
The hypothesis HI,τ is a measurement-to-track association (MTA) or
k+1|k+1
association hypothesis. We end up with a list HI,τ
of MTAs, one for each
k+1|k+1
I ⊆ {1, ..., n} and τ . For each HI,τ
, we can apply the corrector step of the
single-target filter—most typically, an EKF—to use the measurements in ZI,τ to
1
The possibility of newly appearing targets has been ignored for the sake of conceptual clarity.
18
Advances in Statistical Multisource-Multitarget Information Fusion
construct measurement-updated tracks for i ∈ I:
k+1|k+1
(ℓi,τ (i)
k+1|k+1
, xi,τ (i)
k+1|k+1
, Pi,τ (i)
).
This process is repeated indefinitely.
Multihypothesis trackers (MHTs) are currently the dominant MTA-based
tracking algorithms [245], [23].
1.1.4
The Random Finite Set (RFS) Approach
In contrast to MTA, finite-set statistics employs a top-down paradigm grounded in a
particular version of point process theory—the stochastic-geometry formulation of
the theory of random finite sets. The basic ideas are as follows.
1.1.4.1
Multitarget Density Functions
k+1|k+1
In the place of the list {HI,τ
}I,τ of measurement-updated association
hypotheses, the RFS approach employs a multitarget probability density function
(m.p.d.f.)
fk+1|k+1 (X|Z (k+1) )
defined on the finite-set variable X = {x1 , ..., xn } with n ≥ 0, where
Z (k+1) : Z1 , ..., Zk+1
is the time history (sample path) of measurement sets at time tk+1 . The quantity
fk+1|k+1 (X|Z (k+1) ) is the probability (density) that the targets have state set
X = {x1 , ..., xn } with n ≥ 0, given the measurement history Z (k+1) . Because
the number n of targets can be variable, we can have:





∅
{x1 }
X=
{x1 , x2 }



..

.
if
if
if
..
.
no targets are present
one target with state x1 is present
two targets with states x1 ̸= x2 are present .
..
.
(1.8)
The units of measurement of fk+1|k+1 (X|Z (k+1) ) are u−|X| , where u denotes
the units of measurement of the single-target state x.
Introduction to the Book
1.1.4.2
19
RFS Measurement Models
In the place of the standard multitarget measurement model, one constructs from
this model an RFS measurement model of the form
all measurements
measurements, 1st target
measurements, nth target
clutter
Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1
(1.9)
where Υk+1 (x) is the RFS of the measurement generated by a target with state
x; where Ck+1 is the RFS describing the clutter-measurement process; and where
Σk+1 is the RFS of all measurements.
For the standard multitarget measurement model, Ck+1 is assumed Poisson,
and either Υk+1 (x) = ∅ (the target x was not detected) or Υk+1 (x) = {Zk+1 }
(the target was detected and the random measurement Zk+1 was collected). For a
nonlinear-additive single-target measurement model, for example, if Υk+1 (x) ̸= ∅
then
Υk+1 (x) = {ηk+1 (x) + Vk+1 }.
(1.10)
1.1.4.3
Multitarget Likelihood Functions
The RFS measurement model is transformed into an equivalent multitarget likelihood function
LZ (X) abbr.
= fk+1 (Z|X).
(1.11)
The value LZ (X) is the likelihood that, at time tk+1 , the measurement set
Z = {z1 , ..., zm } with m ≥ 0 will be generated, if targets with state set
X = {x1 , ..., xn } with n ≥ 0 are present. The transformation of the RFS model
to a multitarget likelihood function is accomplished using multiobject differential
calculus:
[
]
δ
fk+1 (Z|X) =
βk+1 (T |X)
(1.12)
δZ
T =∅
where
βk+1 (T |X) = Pr(Σk+1 ⊆ T |Ξk+1|k = X)
(1.13)
is the belief-mass function (belief measure) of the multitarget measurement model
Σk+1 ; and where δ/δZ denotes a set derivative. This belief-mass function is the
multiobject analog of the probability-mass function
pk+1 (T |x) = Pr(Zk+1 ∈ T |Xk+1|k = x)
of a single-target measurement model Zk+1 = ηk+1 (x) + Vk+1 .
(1.14)
20
Advances in Statistical Multisource-Multitarget Information Fusion
1.1.4.4
Multitarget Bayes’ Rule
Instead of constructing the list of measurement-updated hypotheses, we instead
apply the multitarget analog of Bayes’ rule:
fk+1|k+1 (X|Z (k+1) ) = ∫
fk+1 (Z|X) · fk+1|k (X|Z (k) )
.
fk+1 (Z|Y ) · fk+1|k (Y |Z (k) )δY
(1.15)
∫
Here, ·δY is a set integral that accounts for the fact that both the elements and
the number of elements of the finite-set variable Y are variable. The set integral
operation is inverse to the set derivative operation.
1.1.4.5
RFS Multitarget Motion Models
The standard multitarget motion model is an analog of the standard multitarget
measurement model. It is based on the following presumptions:
• A target may disappear from the scene (this is the motion-model analog of a
missed detection).
• New targets may appear in the scene (this is the motion-model analog of
clutter).
• Target motions are independent of each other and of those of newly appearing
targets.
From this standard model, one constructs an RFS motion model,
all targets
transition for target 1
transition for target n′
new targets
Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n′ ) ∪ Bk+1|k
(1.16)
where Tk+1|k (x′ ) is the RFS at time tk+1 of a target that had state x′ at time tk ;
Bk+1|k is the RFS of all appearing targets; and Ξk+1|k is the RFS of all predicted
tracks at time tk+1 .
For the standard multitarget motion model, Bk+1|k is assumed Poisson, and
either Tk+1|k (x′ ) = ∅ (the target x′ did not persist) or Tk+1|k (x′ ) = {Xk+1|k }
(the target persisted and transitioned to Xk+1|k ). For a nonlinear-additive singletarget motion model, for example, if Tk+1|k (x′ ) ̸= ∅ then
Tk+1|k (x′ ) = {φk (x′ ) + Wk }.
(1.17)
Introduction to the Book
1.1.4.6
21
Multitarget Markov Densities
The RFS motion model is transformed into a multitarget Markov transition density
MX (X ′ ) abbr.
= fk+1|k (X|X ′ ).
The value MX (X ′ ) is the likelihood that, at time tk+1 , the targets will have state
set X = {x1 , ..., xn } with n ≥ 0 if, at time tk , the state set of the targets
was X ′ = {x′1 , ..., x′n′ } with n′ ≥ 0. The transformation of an RFS model to a
multitarget Markov density is accomplished using, once again, a set derivative:
[
]
δ
′
′
fk+1|k (X|X ) =
βk+1|k (S|X )
(1.18)
δX
S=∅
where
βk+1|k (S|X ′ ) = Pr(Ξk+1|k ⊆ S|Ξk|k = X ′ )
(1.19)
is the belief-mass function of the RFS motion model Ξk+1|k . This belief-mass
function is the multiobject analog of the probability-mass function
pk+1 (S|x′ ) = Pr(Xk+1|k ∈ S|Xk|k = x′ )
(1.20)
of a single-target motion model Xk+1|k = φk (x′ ) + Wk .
1.1.4.7
Multitarget Prediction Integral
Instead of constructing the predicted hypothesis list, in finite-set statistics one
instead applies a multitarget analog of the prediction integral:
∫
(k)
fk+1|k (X|Z ) = fk+1|k (X|X ′ ) · fk|k (X ′ |Z (k) )δX ′
(1.21)
where
1.1.4.8
∫
·X ′ is a set integral.
The Multitarget Recursive Bayes Filter
The time-update and measurement-update steps just summarized result in the
multitarget recursive Bayes filter:
... →
fk|k (X|Z (k) )
→
fk+1|k (X|Z (k) )
→
fk+1|k+1 (X|Z (k+1) )
→ ...
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Advances in Statistical Multisource-Multitarget Information Fusion
This is the optimal solution to the problem of detecting, tracking, and classifying
an unknown number of unknown targets. It is optimal because of the existence
of Bayes-optimal multitarget state estimators—for example, the joint multitarget
(JoM) estimator ([179], p. 500):
JOM
Xk+1|k+1
= arg sup
X
c|X|
· fk+1|k+1 (X|Z (k+1) ).
|X|!
(1.22)
Here, c is a constant that has the same units of measurement as the single-target
state x; and its magnitude should be the size of the desired localization accuracy
([179], pp. 498-499).
1.1.4.9
Approximate RFS Filters
The multitarget Bayes filter is computationally intractable in most situations of practical interest. Consequently, approximations are necessary. These approximations
should be principled, in the sense that they are statistically top-down in form, and
thus preserve the essential nature of the original motion and measurements models,
as well as the relationships between these models.
The type of principled approximation employed in finite-set statistics is that of
assuming that the multitarget distributions fk|k (X|Z (k) ) and/or fk+1|k (X|Z (k) )
have some simplified form—one that permits approximate closed-form solution
of the multitarget Bayes filter. This is accomplished using probability generating
functionals (p.g.fl.’s) and multiobject calculus.
The p.g.fl. of a multitarget distribution is defined as
∫
(k)
Gk|k [X|Z ] =
hX · fk|k (X|Z (k) )δX
(1.23)
∫
Gk+1|k [X|Z (k) ] =
hX · fk+1|k (X|Z (k) )δX
(1.24)
where the power functional is defined as
hX =
{
∏
1
x∈X h(x)
if
if
X=∅
X ̸= ∅
(1.25)
and where h(x) is a “test function” such that 0 ≤ h(x) ≤ 1 identically.
Three types of approximation have been extensively investigated in the literature thus far, and are most simply expressed using p.g.fl.’s:
Introduction to the Book
23
1. fk+1|k (X|Z (k) ) and/or fk|k (X|Z (k) ) are the distributions of Poisson
RFSs. The various probability hypothesis density (PHD) filters are the result:
... → Dk|k (x|Z (k) ) → Dk+1|k (x|Z (k) ) → Dk+1|k+1 (x|Z (k+1) ) → ...
where Dk|k (x|Z (k) ) is a probability hypothesis density (PHD).
2. fk|k (X|Z (k) ) and/or fk+1|k (X|Z (k) ) are the distributions of independent
identically distributed cluster (i.i.d.c.) processes. The various cardinalized
PHD (CPHD) filters are the result:
... →
pk|k (n|Z (k) )
→
pk+1|k (n|Z (k) )
... →
sk|k (x|Z (k) )
→
sk+1|k (x|Z (k) )
→
↑↓
→
pk+1|k+1 (n|Z (k+1) )
→ ...
sk+1|k+1 (x|Z (k+1) )
→ ...
where sk|k (x|Z (k) ) is a spatial distribution (that is, the normalized PHD)
and where pk|k (n|Z (k) ) is the probability distribution on the number n of
targets (also known as the “cardinality distribution”).
3. fk|k (X|Z (k) ) and/or fk+1|k (X|Z (k) ) are the distributions of multiBernoulli processes. The various multi-Bernoulli filters are the result:
k|k v
k+1|k vk+1|k
}i=1
k|k
{qi }i=1
→ {qi
... →
k|k
v
k+1|k
k|k
... → {si (x)}i=1
→ {si
{qi
→
{si
v
k+1|k
(x)}i=1
k+1|k+1 vk+1|k+1
}i=1
→ ...
→
k+1|k+1
↑↓
vk+1|k+1
(x)}i=1
→ ...
k|k
k|k
where s1 (x), ..., sνk|k (x) are the track distributions (spatial densities) of
k|k
k|k
νk|k target tracks; and where q1 , ..., qνk|k are the probabilities that these
tracks are actual targets.
1.1.4.10
Derivation of Approximate RFS Filters
These filters are derived using the following methodology. It can be shown that the
multitarget Bayes filter can be equivalently expressed as a filter on p.g.fl.’s:
... →
Gk|k [h|Z (k) ]
→
Gk+1|k [h|Z (k) ]
→
Gk+1|k+1 [h|Z (k+1) ]
→ ...
Given one of the above approximations, the formulas for Gk+1|k [h|Z (k) ] and
Gk+1|k+1 [h|Z (k+1) ] become algebraically simplified. The formulas for items such
24
Advances in Statistical Multisource-Multitarget Information Fusion
k+1|k+1
k+1|k+1
as pk+1|k+1 (n|Z (k+1) ), Dk+1|k+1 (x|Z (k+1) ), qi
, and si
then be derived using multiobject calculus. For example,
[
]
δ
(k+1)
(k+1)
Dk+1|k+1 (x|Z
)=
Gk+1|k+1 [h|Z
]
.
δx
h=1
(x) can
(1.26)
Here, δG/δx is a generalization of a set derivative called a functional derivative.
It is defined by
δ
G[h + ε · δx ] − G[h]
G[h] = lim
(1.27)
ε→0
δx
ε
where δx denotes the Dirac delta function concentrated at x. (Note that this is a
heuristic, intuitive definition since δx is not a valid test function. See Appendix C
or [181] for a more rigorous definition.)
1.1.5
Extension to Nontraditional Measurements
In this book, the terminology “traditional measurement” refers to a measurement
that is produced by a conventional sensor—whether this measurement be a signature
or a detection extracted from a signature. Other forms of information, referred to as
“nontraditional,” usually (but not always) involve human mediation. These include:
• Attribute—an identifying characteristic of a target. An example is an identifying characteristic of a target inferred by a human operator while examining
an image.
• Feature—typically, an identifying characteristic extracted from a signature
by a digital signal processing (DSP) algorithm. Examples include intuitively
understandable features such as “blobs” extracted from images; but also
mathematically abstract features such as principal components or wavelet
coefficients.
• Natural-language statement. These consist of verbal or written texts generated by a human information source.
• Inference rule. This is contingent information, in which a consequent statement is held to be true in the event that an antecedent statement is true.
Besides its employment of RFS theory to model multiobject systems, finiteset statistics includes:
• A general theory of measurements—called “generalized measurements”—
based on the concept of a random (but not necessarily finite) set.
Introduction to the Book
25
• Single-target and multitarget Bayes-optimal processing of general measurements via generalized likelihood functions (GLFs), which are employed in
Bayes’ rule in the same way as conventional likelihood functions.
1.2
RECENT ADVANCES IN FINITE-SET STATISTICS
As has already been stated, this book is a consequence of the many new developments that have arisen in RFS-based multisource-multitarget information fusion.
The purpose of this section is to briefly describe some of these advances. Although
most will be described in technical detail in later chapters, in some cases the reader
will be directed to other publications.
The section is organized as follows:
1. Section 1.2.1: Advances in conventional PHD and CPHD filters: multisensor
PHD and CPHD filters, and multiple-model PHD and CPHD filters.
2. Section 1.2.2: PHD multitarget smoothers.
3. Section 1.2.3: PHD and CPHD filters for unknown detection profiles and
unknown clutter backgrounds.
4. Section 1.2.4: PHD filters for extended targets, cluster targets, group targets,
and unresolved targets.
5. Section 1.2.5: Advances in conventional multi-Bernoulli filters.
6. Section 1.2.6: RFS filters for “raw data” sensors.
7. Section 1.2.7: Theoretical advances: Clark’s general chain rule for functional
derivatives; the general PHD filter; and an exact closed-form solution of the
multitarget Bayes filter.
8. Section 1.2.8: Fusing nontraditional information sources.
9. Section 1.2.9: Advances in the development of unified processing systems: unified simultaneous localization and mapping (SLAM); unified sensor/platform management; unified multisensor-multitarget tracking and sensor registration; and unified track-to-track fusion.
26
1.2.1
Advances in Statistical Multisource-Multitarget Information Fusion
Advances in Conventional PHD and CPHD Filters
Chapters 10 and 11 report advances in, respectively, multisensor PHD/CPHD filters
and jump-Markov PHD/CPHD filters.
1.2.1.1
Principled Approximate Classical Multisensor PHD/CPHD Filters
The classical PHD and CPHD filters are single-sensor filters. The corresponding
multisensor filters are computationally intractable in general. The most common
approximate approach for implementing multisensor PHD and CPHD filters, the
heuristic “iterated corrector” approach, involves repeating the single-sensor corrector formulas, once for each sensor. However, this approach produces different
answers, depending on the order of the sensors. In particular, sensors with larger
probabilities of detection should be processed first.
Section 10.6 reports a significant advance: multisensor PHD and CPHD
filters that are principled, computationally tractable, and do not depend on sensor
order.
1.2.1.2
Multiple Motion Model PHD and CPHD Filters
The PHD and CPHD filters presume the existence of an a priori single-target motion
model, in the form of a single-target Markov density fk+1|k (x|x′ ). Since in
general x = (u, c) where u is the kinematic state and c is a target-ID variable,
this model can have different motion models for targets of different type:
fk+1|k (x|x′ )
=
=
fk+1|k (u, c|u′ , c′ )
′
′
(1.28)
′
′
fk+1|k (c|u , c ) · fk+1|k (u|u , c , c)
(1.29)
where the final equation is a consequence of Bayes’ rule. However, because the
motion model fk+1|k (u|u′ , c′ , c) is specified a priori, the filter may fail to
adequately track evasive, rapidly maneuvering targets.
Later chapters report significant advances, due to several researchers, that
extend jump-Markov (multiple motion model) techniques to PHD and CPHD filters
(see Chapter 11) and multi-Bernoulli filters (see Section 13.5).
1.2.2
Multitarget Smoothers
The single-target Bayes filter is an online algorithm. That is, it propagates a track
density fk|k (x|Z k ) that describes the state x of the target at time tk , given
Introduction to the Book
27
the measurement-stream Z k : z1 , ..., zk up through time tk . If we are willing
to employ time-late “batch” processing, it is possible—for purposes such as track
reconstruction—to produce more accurate tracking results. This can be done by
using the data in the entire time-window Z k of measurements to estimate the
target states at time tℓ for all 1 ≤ ℓ ≤ k. An algorithm that computes the track
distributions fℓ|k (x|Z k ) for 1 ≤ ℓ ≤ k is called a smoother.
Single-target smoothers can be directly extended to multitarget smoothers, but
these smoothers will in general be computationally intractable. Chapter 14 reports
four significant advances. The first is a forward-backward Bernoulli smoother that
provides an optimal approach when at most a single target is present. The second is a
principled and computationally tractable PHD smoother. The third is a closed-form
Gaussian-mixture implementation of this smoother. The fourth, generated as a byproduct, is what appears to be the first-ever closed-form Gaussian mixture solution
to the conventional single-sensor, single-target smoother (Section 14.2.3).
The PHD smoother “works,” but its performance turns out to be less satisfactory than one might hope—only about a 30% improvement. However, the PHD
smoother may point the way to more effective approaches in the future.
1.2.3
PHD and CPHD Filters for Unknown Backgrounds
The conventional PHD and CPHD filters require two a priori sensing models. First,
a state-dependent probability of detection pD (x) that describes the detection
profile of the sensor. Second, a model of the clutter process, in the form of an
independent, identically distributed cluster (i.i.d.c.) process model. This consists of
a clutter spatial distribution ck+1 (z) and a distribution pκk+1 (m) on the number
m of clutter measurements. (For the PHD filter, pκk+1 (m) = e−λk+1 λm
k+1 /m! is
assumed to be Poisson.)
In practical application, one or both of these models is typically unknown.
Chapters 17 and 18 report significant advances: PHD and CPHD filters that do not
require these a priori models. These can be summarized as follows.
1.2.3.1
PHD/CPHD Filters for Unknown Detection Profiles
As is reported in Chapter 17, any RFS filter can be converted to a filter that does not
require a priori knowledge of pD (x). The approach is simple: append a new state
variable 0 ≤ a ≤ 1 to the single-target state x, which represents the unknown
probability of detection. These RFS filters are capable in principle of estimating
the probability of detection at any given track. The closed-form implementation
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Advances in Statistical Multisource-Multitarget Information Fusion
of such filters—using “beta-Gaussian mixtures (BGMs)”—is reported in Section
17.3.2.
1.2.3.2
PHD/CPHD Filters for Unknown Clutter
As is reported in Chapter 18, it is possible to derive PHD and CPHD filters that
do not require the i.i.d.c. clutter model ck+1 (z), pκk+1 (m). This is accomplished
by assuming a multi-Bernoulli clutter model instead of an i.i.d.c. clutter model.
That is, clutter generators are assumed to be target-like, in that they generate single
measurements rather than a cluster of measurements. The usual single-target state
space X is replaced by a joint target-clutter state space X ⊎ C where C is the
space of clutter generators.
Mahler, Vo, and Vo have shown that the resulting PHD and CPHD filters can
be implemented in exact closed form using BGMs (see Section 18.5.7). Chen Xin,
Tharmarasa, Kirubarajan, and Pelletier have shown that they can be implemented in
exact closed form using normal-Wishart mixtures (NWMs, see Section 18.5.8).
1.2.4
PHD Filters for Nonpoint Targets
The small-target model, which was discussed in Section 1.1.3, is based on the
assumption that targets are neither too near nor too far from the sensor. This model
can be violated in two ways. First, if a target is sufficiently near the sensor, it will
generate multiple detections rather than a single detection. Such a target is called an
extended target. Second, some unknown phenomenon may generate time-evolving
measurement-clusters. Such a target is called a cluster target. Third, measurementclusters may be produced by a coordinated system of conventional targets. Such a
target is called a group target. Finally, if a group of targets are sufficiently distant
then they will generate a single detection rather than multiple detections. Such
targets are called unresolved targets. Several advances have been made in the
use of RFS filters to address these kinds of targets—especially, for extended and
unresolved targets.
1.2.4.1
PHD and CPHD Filters for Extended Targets
As is reported in Chapter 21, it is possible to derive filtering equations for a
PHD filter designed to track multiple extended targets. The measurement-update
equation for this filter is combinatorial and thus computationally impractical in
general. However, recent research by Lundquist, Granström, and Orguner has
Introduction to the Book
29
shown how certain approximations can be employed to render the filter potentially
practical.
1.2.4.2
PHD and CPHD Filters for Unresolved Targets
As is reported in Chapter 21, it is possible to derive filtering equations for a
PHD filter designed to track multiple unresolved targets. The basic approach is to
represent an unresolved target as a point target cluster—that is, multiple targets that
have the same (single-target) state, but with a continuously variable target number.
Given this, it is possible to derive filtering equations for an unresolved-target PHD
filter.
As with the extended-target PHD filter, the measurement-update equation
is combinatorial. However, approximations similar to those being used for the
extended-target case can potentially be applied here.
1.2.5
Advances in Classical Multi-Bernoulli Filters
Three significant advances in the theory and application multi-Bernoulli filters have
been proposed since their introduction in 2007. These are the cardinality balanced
multi-Bernoulli (CBMeMBer) filter; multiple-model versions of the CBMeMBer
filter; and “background agnostic” versions of the CBMeMBer filter.
1.2.5.1
Cardinality-Balanced Multi-Bernoulli (CBMeMBer) Filters
The first multi-Bernoulli filter, the “multitarget multi-Bernoulli (MeMBer)” filter,
was proposed in Chapter 17 of [179]. Because of an ill-conceived linear approximation, it was subsequently shown to exhibit a pronounced bias in the estimate of
target number. Vo and Vo corrected this deficiency with their “cardinality balanced”
MeMBer (CBMeMBer) filter. This advance is reported in Chapter 13.
1.2.5.2
Multiple Motion Model CBMeMBer Filters
As with PHD and CPHD filters, the CBMeMBer filter employs a single-target
motion model, in the form of a single-target Markov density fk+1|k (x|x′ ) that
is specified a priori. Dunne and Kirubarajan have devised a jump-Markov version
of the CBMeMBer filter that more easily tracks evasive, maneuvering targets. This
advance is reported in Section 13.5.
30
1.2.5.3
Advances in Statistical Multisource-Multitarget Information Fusion
CBMeMBer Filters with Unknown Backgrounds
Again like the PHD and CPHD filters, the CBMeMBer filter employs a priori
models of the detection profile and the clutter background. Vo and Vo have
generalized the CBMeMBer filter so that these models are no longer necessary.
This advance is reported in Section 18.7.
1.2.6
RFS Filters for “Raw-Data” Sensors
Until recently, all RFS filters (including the ones just described) have been based on
the presumption that measurements are detections. That is, the measurements are
points extracted from a sensor signature using some detection technique. However,
detection-based approaches inherently discard information, since the extracted detections only approximate the original signatures. This is especially the case when
signal-to-noise ratio (SNR) is small. It is therefore desirable to devise PHD and
CPHD filters that can be used directly with the original “raw” signature measurement. Such filters are often called track-before-detect (TBD) filters.
1.2.6.1
Exact Multi-Bernoulli Filters for Pixelized Image Data
In modern optical, infrared, and other imaging systems, the measurement is usually
an image-signature in the form of a two-dimensional matrix of gray-scale pixels or
red-green-blue (RGB) color pixels.
As is reported in Chapter 20, Vo and Vo have shown that it is possible to
derive exact closed-form multi-Bernoulli filters for the optimal TBD processing of
pixelized images. In this approach, targets are assumed to have physical extent, and
thus cannot occlude each other. These filters have been applied successfully to real
video data, and have been shown to outperform the previously best TBD algorithm
for image data, the histogram-PMHT filter.
1.2.6.2
PHD/CPHD Filters for Superpositional Sensors
The most familiar sensors are based on electromagnetic-wave or acoustic-wave
signatures. Such sensors are superpositional, in that the received measurements are
summations of the (usually complex-valued) signatures generated by each target.
Mahler has shown that it is possible to derive exact RFS filters for such signatures
(Section 19.2) , but these filters are computationally intractable in general.
As is reported in Chapter 19, approximation techniques make it possible to
derive computationally tractable RFS filters for superpositional signatures. The
Introduction to the Book
31
Hauschildt technique (Section 19.3) employs a direct approximation of the general
approach. A second approach, due to Thouin, Nannuru, and Coates and subsequently generalized by Mahler (Section 19.4), employs Campbell’s theorem. A
CPHD filter based on this approximation has been shown to outperform a conventional Markov Chain Monte Carlo (MCMC) algorithm for multitarget tracking in
both radio-frequency (RF) tomography data and passive-acoustic data.
1.2.7
Theoretical Advances
Three advances of a more theoretical nature have been achieved. The first two
are a general chain rule for functional derivatives, due to D. Clark; and one of its
consequences, a general PHD filter based on this chain rule. The third is an exact
closed-form solution of the multitarget Bayes filter.
1.2.7.1
Clark’s General Chain Rule for Functional Derivatives
The derivation of approximate RFS filters, as summarized in Section 1.1.4.10,
frequently results in expressions of the general form G[T [h]] where G[h] is
a functional and where T : h ?→ T [h] is a functional transformation—that is, a
function that transforms functions into functions. Consequently, these derivations
require the determination of functional derivatives of the form
δ
G[T [h]].
δX
That is, they require a chain rule. Various chain rules were given in [179], but
only for special cases. As a result, the derivation of new results typically required
a complicated induction proof.
Thanks to a powerful general chain rule due to D. Clark, this is no longer
required. Intuitively speaking, the previously required case-by-case induction proof
is now built into the general chain rule. This advance is reported in Section 3.5.14.
1.2.7.2
The Generalized Classical PHD Filter
An immediate consequence of Clark’s general chain rule is the derivation of
the measurement-update formula for a generalization of the classical PHD filter.
This filter is general in that both the clutter process and the target measurementgeneration process can be general. This advance is reported in Section 8.2.
32
1.2.7.3
Advances in Statistical Multisource-Multitarget Information Fusion
Exact Closed-Form Solution of the Multitarget Bayes Filter
PHD, CPHD, and multi-Bernoulli filters are based on approximations that attempt
to approximate the actual multitarget density functions fk|k (X|Z (k) ) with increasingly greater accuracy. The filtering steps for these filters are approximate (with the
exception of the PHD filter and CBMeMBer filter time-updates). Chapter 15 reports
a major theoretical advance, due to B.-T. Vo and B.-N. Vo: an exact closed-form
solution of the multitarget Bayes filter.
This filter is based on the assumption that the targets in a single continuous
target-track all share a distinctive identifying track label. Given this, Vo and Vo
construct a class of multitarget distributions—“generalized labeled multi-Bernoulli”
distributions. They then show that the class of these distributions is algebraically
closed with respect to the multitarget prediction integral, and with respect to the
multitarget version of Bayes’ rule.
One consequence is that this new filter appears to be the first provably
Bayes-optimal, implementable multitarget detection and tracking algorithm. A
second consequence is that it apparently has the first provably Bayes-optimal trackmanagement scheme.
A Gaussian-mixture implementation of this filter is computationally tractable,
and resembles track-oriented multihypothesis tracker (track-oriented MHT) algorithms.
1.2.8
Advances in Fusing Nontraditional Measurements
Two advances in the fusion of nontraditional measurements are reported in Chapter 22: the Bayes optimality of the random set “generalized likelihood function”
approach in the single-target case (when the measurement function is precisely
known); and the Bayes-optimal extension of this approach to both RFS and conventional multitarget filters.
1.2.8.1
Bayes Optimality of the Generalized Likelihood Function (GLF)
Approach
GLFs, as introduced in Section 1.1.5, differ from conventional likelihood functions
in that they are unitless probabilities rather than probability densities. It has been
shown that, despite this difference, GLFs are provably Bayes-optimal in the singletarget case. (This result is true, however, only when the underlying measurement
Introduction to the Book
33
function ηk+1 (x) is precisely known—that is, for “UGA measurements.”) This
advance is reported in Section 22.3.4.
1.2.8.2
Extension of the GLF Approach to Multitarget Filtering
It has also been shown that:
• The GLF approach can be naturally generalized for use with both RFS and
conventional multitarget filters.
• The generalization to RFS multitarget filters is Bayes-optimal (given that the
measurement function is precisely known)—though the RFS filters themselves may not be Bayes-optimal.
The Bayes-optimal generalization of the GLF approach to RFS filters—the
PHD filter, CPHD filter, and multi-Bernoulli filter—is reported in Section 22.10. A
paper by Bishop and Ristic is particularly interesting in this regard—see [22] and
Section 22.10.5.3.
The extension of GLF techniques to conventional measurement-to-track association (MTA), and thereby to conventional multitarget filters, is described in
Section 22.11.
1.2.9
Advances Toward Fully Unified Systems
Existing information fusion systems are typically patched together in an ad hoc
fashion from various subsystems, many of which are themselves based on ad hoc
heuristics. The following are some of the information fusion functionalities that
must be integrated after being addressed separately: sensor registration; target
detection; target tracking; target classification; measurement-to-track fusion; trackto-track fusion; sensor management; platform management; and so on.
ad hoc design and integration reduces algorithmic efficiency by introducing
unknown errors. Every heuristic, whether in the integration approach or in the
individual components, introduces hidden assumptions. Not only does each such
assumption potentially introduce a hidden error and/or statistical bias, but the
inefficiencies due to such errors and/or biases are typically magnified as they are
propagated through a heuristically designed system.
A recent book and book chapter, and Chapters 12, 24, 25, and 26, report
recent advances towards the goal of constructing fully unified information fusion
systems, based on principled RFS statistical reasoning. These advances include:
unified simultaneous localization and mapping (SLAM) for robotics; unified sensor
34
Advances in Statistical Multisource-Multitarget Information Fusion
and platform management; unified multisensor-multitarget tracking and sensor
registration; and unified multisensor-multitarget track-to-track fusion.
1.2.9.1
Unified Simultaneous Localization and Mapping (RFS-SLAM)
In SLAM, one or more robots are inserted into an unknown environment. They
must construct a map of the environment based on detected landmarks, and then
situate themselves within this map. The book Random Finite Sets in Robotic
Map Building and SLAM, by Mullane, Vo, Adams, and Vo, describes an RFS
approach to SLAM—indeed, the first provably Bayes-optimal approach to SLAM
[210]. An approximate SLAM algorithm, based on the PHD filter, has been
shown to significantly outperform traditional methods such as EKF-SLAM and
MH-FastSLAM in high-clutter environments. See also the papers [208], [209], [1].
1.2.9.2
Unified Sensor and Platform Management
The performance of multitarget detection and tracking algorithms can be greatly improved if allocatable sensor resources (including the platforms that carry them) can
be efficiently and adaptively redirected to preferentially collect measurements from
under-collected targets. This process is also known as sensor/platform management
or Level 4 information fusion.
Chapters 24 through 26 report a significant advance in the development of
a unified but potentially tractable RFS-based, control-theoretic and informationtheoretic approach to multisensor-multitarget sensor and platform management.
However, the use of purely abstract measures of information, such as KullbackLeibler cross-entropy or Rényi α-divergence, involves a certain amount of risk: one
is “flying blind” insofar as practical intuition is concerned. Thus the emphasis of
this work is on the development of information-theoretic objective functions that
are not only computationally tractable but physically intuitive (rather than purely
abstract). Examples of such objective functions are the posterior expected number
of targets (PENT), the cardinality variance (that is, the variance of PENT), and
the Cauchy-Schwartz divergence. The work regarding the cardinality variance and
Cauchy-Schwartz divergence is quite new but promising—see Section 26.6.4.3.
1.2.9.3
Unified Multisensor-Multitarget Tracking and Sensor Registration
Existing RFS algorithms, like essentially all current multitarget tracking algorithms,
are based on the presumption that the sensors are temporally and spatially registered
Introduction to the Book
35
with perfect accuracy. In practice, sensor measurements are often contaminated
by one or more unknown biases (also known as misregistrations). One example
is a terrain map that has an unknown translational offset. Another is drift error
in the inertial navigation system (INS) of a sensor-carrying platform. In such
circumstances, the performance of multitarget detection and tracking algorithms
will typically be seriously degraded unless such sensor biases can be estimated and
removed.
Conventional approaches attempt to estimate registration errors prior to target
detection and tracking. However, recent research has demonstrated that, at least in
certain circumstances, sensor registration and multitarget tracking can be accomplished simultaneously. These advances are reported in Chapter 12.
1.2.9.4
Unified Multisensor-Multitarget Track-to-Track Fusion
Conventional information fusion is based on the concept of collecting measurements from sensors in order to detect and track targets. Such an approach is often
not viable for distributed information fusion systems. This is because raw measurements are often too large to be effectively transmitted through bandwidth-limited
communications channels. This difficulty is typically addressed by transmitting
track estimates rather than measurements. However, this approach introduces two
new difficulties: temporal correlation and double-counting (spatial correlation).
Measurements can usually be assumed to be statistically independent. This
is the not the case with tracks which, being the output of some filtering process, are
time-correlated. They therefore cannot be fused in the same manner as measurements.
The other difficulty arises from the fact that, in ad hoc networks, seemingly
independent data arriving at a fusion node may have actually originated with
the same information source. Consider a simple network in which node X
communicates data D about a target to the nodes A and B, each of which
then transmits D to node Y as data D ′ and D ′′ . If Y fuses D ′ and
D ′′ presuming that they are independent, the result will be a spuriously accurate
localization of the target. This phenomenon is known as double-counting.
A unified RFS approach to track-to-track fusion, much of it based on the work
of Clark and his associates, has been described in Chapter 8 of the book Distributed
Data Fusion for Network-Centric Operations [183].
36
Advances in Statistical Multisource-Multitarget Information Fusion
1.2.9.5
Unified Situation Assessment
The term situation assessment (also known as Levels 2 and 3 information fusion)
refers to the process of determining the current and/or predicted levels of threat in
a given environment. Section 25.14.2 describes the elements of a unified approach
to the problem, based on the concept of a tactical importance function (TIF).
1.3
ORGANIZATION OF THE BOOK
The purpose of this section is to summarize the contents of the book. The emphasis
of the book is on the derivation of concrete techniques and formulas that can be
applied to concrete problems. Consequently, the introduction to every chapter
includes a “Major Lessons Learned” section detailing the most significant concepts,
formulas, and results of that chapter.
Mathematical proofs and other extended mathematical derivations have been
relegated to the appendices, when it has not been possible or desirable to direct the
more theoretically engaged reader to other publications.
Because of a “transparent” system of notation (see Appendix A.1), the reader
will usually be able to infer the meaning of mathematical symbols at a glance. A
crawl-walk-run style of exposition is also employed. We will begin with more
familiar concepts and techniques and then build upon them to introduce more
complex ones.
The book is organized as follows and as indicated in Figure 1.2:
Part I: Elements of Finite-Set Statistics
1. Chapter 2: Basic concepts of random finite sets.
2. Chapter 3: Basic concepts of multiobject calculus.
3. Chapter 4: Basic concepts of multiobject statistics.
4. Chapter 5: Basic concepts of RFS modeling and multitarget filtering.
5. Chapter 6: Basic concepts of multitarget metrology: multitarget missdistances, and measures of multitarget information.
Part II: RFS Filters for Standard Measurement Models
1. Chapter 7: Introduction to Part II.
Introduction to the Book
37
Figure 1.2 The structure of the book at a glance. It is divided into five major
parts: Elements of FISST, Standard Measurement Models, Unknown Backgrounds,
Nonstandard Measurement Models, and Sensor/Platform Management.
38
Advances in Statistical Multisource-Multitarget Information Fusion
2. Chapter 8: The “classical” PHD and CPHD filters; the zero false-alarm (ZFA)
CPHD filter; the state-dependent-clutter PHD filter; and the generalized
classical PHD filter.
3. Chapter 9: Implementation of the classical PHD and CPHD filters.
4. Chapter 10: Extension of the classical PHD and CPHD filters to multiple
sensors.
5. Chapter 11: Jump-Markov versions of the classical PHD and CPHD filters
(for rapidly maneuvering, “noncooperative” targets).
6. Chapter 12: Joint sensor registration and multitarget detection and tracking.
7. Chapter 13: Multi-Bernoulli filters and the CBMeMBer filter.
8. Chapter 14: PHD smoothers and the ZTA-CPHD smoother.
9. Chapter 15: The Vo-Vo exact closed-form solution of the multitarget Bayes
filter.
Part III: RFS Filters for Unknown Backgrounds
1. Chapter 16: Introduction to Part III.
2. Chapter 17: Unknown probability of detection.
3. Chapter 18: Unknown clutter backgrounds.
Part IV: RFS Filters for Nonstandard Measurement Models
1. Chapter 19: PHD and CPHD filters for superpositional sensors.
2. Chapter 20: Track-before-detect multi-Bernoulli filters for pixelized images.
3. Chapter 21: PHD filters for extended, cluster, group, and unresolved targets.
4. Chapter 22: The theory and RFS filtering of nontraditional measurements.
Part V: RFS Sensor and Platform Management
1. Chapter 23: Introduction to Part V.
2. Chapter 24: Sensor management for single-sensor, single-target systems.
3. Chapter 25: RFS sensor management for multisensor-multitarget systems.
Introduction to the Book
39
4. Chapter 26: Sensor management using approximate RFS filters: Bernoulli,
PHD, CPHD, and CBMeMBer.
Appendices
Various mathematical details and tangential matters have been relegated to
the appendices, as follows:
• Appendix A: A glossary of notation.
• Appendix B: The core Bayesian approach.
• Appendix C: Functional derivatives.
• Appendix D: The theory of partitions.
• Appendix E: Beta distributions.
• Appendix F: Markov time-update of beta distributions.
• Appendix G: Wishart and normal-Wishart distributions.
• Appendix H: Complex-valued Gaussian random variables.
• Appendix I: The statistics of level-1 group targets.
• Appendix J: Comparing the functional calculi of FISST and Moyal.
• Appendix K: Mathematical derivations: (This appendix can be found
online. See the citations in the text.)
Part I
Elements of Finite-Set Statistics
Chapter 2
Random Finite Sets
2.1
INTRODUCTION
The purpose of this chapter is to provide a summary of the basic concepts, formulas,
and methodologies of finite-set statistics. It begins with an introduction to conventional single-sensor, single-target statistics and ends with a conceptual sketch of
multisource-multitarget statistics. The latter has three basic aspects:
• A formulation of multisource-multitarget detection, tracking, identification,1
and information fusion in terms of the theory of random finite sets (RFSs).
• A general theory of measurements, which encompasses many forms of nontraditional, human-mediated information.
• A set of practitioner-oriented mathematical tools, based on multitarget integral and differential calculus, that facilitates problem-solving.
2.1.1
Organization of the Chapter
The chapter is organized as follows:
1. Section 2.2: A review of single-sensor, single-target Bayes statistics.
2. Section 2.3: An introduction to the theory of random finite sets (RFSs).
1
In this book the term “target identification” will be used flexibly. Depending on application, it can
refer to determining (1) the broad class or type of a target (for example, jet fighter versus commercial
jet); (2) the narrow class or type of a target (for example, F-16 versus F-35); or even (3) a specific
identity (for example, an aircraft tail number).
43
44
Advances in Statistical Multisource-Multitarget Information Fusion
3. Section 2.4: Multitarget statistics in a nutshell.
2.2
SINGLE-SENSOR, SINGLE-TARGET STATISTICS
For the sake of conceptual and notational clarity, let us begin with a summary of
the basic concepts of single-sensor, single-target Bayesian statistics. The following
topics are covered:
1. Section 2.2.1: Basic notation.
2. Section 2.2.2: Single-target state spaces and single-sensor measurement
spaces.
3. Section 2.2.3: Random vectors, probability-mass functions, and probability
density functions.
4. Section 2.2.4: Target motion models and Markov state transition densities.
5. Section 2.2.5: Sensor measurement models and likelihood functions.
6. Section 2.2.6: Generalized measurement models and generalized likelihood
functions for nontraditional information sources.
7. Section 2.2.7: The single-sensor, single-target recursive Bayes filter.
2.2.1
Basic Notation
A glossary of notation can be found in Appendix A. In addition, the following
notation will be employed throughout the book:
• Binomial (also known as combinatorial) coefficient:
Cn,i =
{
n!
i!·(n−i)!
0
if
if
0≤i≤n
.
otherwise
(2.1)
• Multidimensional Gaussian distribution: Let x be an N -dimensional
column vector and C be an N × N covariance matrix. Then this is
defined by
(
)
1
1
NC (x) = √
· exp − xT C −1 x .
(2.2)
2
det 2πC
Random Finite Sets
45
• Fundamental identity for multidimensional Gaussian distributions: Let x be
an N -dimensional column vector, z be an M -dimensional column vector
with M ≤ N , P be an N × N covariance matrix, C be an M × M
covariance matrix, and H be an M × N matrix. Then
NR (z − Hx) · NP (x − x0 ) = NR+HP H T (z − Hx0 ) · NC (x − c) (2.3)
where
C −1
C −1 c
P −1 + H T R−1 H
P −1 x0 + H T R−1 z
(2.4)
(2.5)
x0 + K(z − Hx0 )
(I − KH)P
(
)−1
P H T HP H T + R
.
(2.6)
(2.7)
=
=
or, equivalently, where
2.2.2
c
C
=
=
K
=
(2.8)
State Spaces and Measurement Spaces
The state of a single-target contains the primary information about the target that
we want to know. It is, most commonly, a column vector
x = (x1 , ..., xN )T
(2.9)
in some Euclidean space X = RN or in some region S ⊆ RN . Less commonly,
it has the hybrid discrete-continuous form
x = (c, x1 , ..., xN )T ∈ X = C × S
(2.10)
where C is a finite set of discrete state-variables—most typically, a set of track
labels and/or target-identity classes. But, as indicated in Appendix B, in general
X can be any Hausdorff, locally compact, and completely separable topological
space. For example—and as will be the case in Chapter 18—it could have the form
X = RN1 ⊎ RN2 where ‘⊎’ indicates a disjoint union.
The measurement generated by a target, and collected by a single-sensor, is
the information from the target that is actually observable. It is, most commonly, a
column vector
z = (z1 , ..., zM )T
(2.11)
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Advances in Statistical Multisource-Multitarget Information Fusion
in some Euclidean space Z = RM or in some region T ⊆ RM . Less commonly,
it is hybrid discrete-continuous:
z = (b, z1 , ..., zM )T ∈ Z = B × T
(2.12)
where B is a finite set of discrete state-variables—most typically, a set of attributes
associated with target-identity classes. In general, a measurement space can be any
Hausdorff, locally compact, and completely separable topological space.
State and measurement spaces are both presumed to come equipped with a
measure-theoretic integration concept. For X = RN and Z = RM this is
typically the Lebesgue integral. For X = C × S and Z = B × T it is the product
discrete-Lebesgue integral:
∫
∑∫
h(x)dx =
f (c, x1 , ..., xN )dx1 · · · dxN
(2.13)
c∈C
∫
g(z)dz
=
b∈B
2.2.3
S
∑∫
g(b, z1 , ..., zM )dz1 · · · dzM .
(2.14)
T
Random States and Measurements, Probability-Mass Functions, and
Probability Densities
A random state X is a random element of the state space X, and a random
measurement is a random element Z of the measurement space Z. They both have
corresponding probability-mass functions (also known as probability measures),
defined as
pX (S) = Pr(X ∈ S),
pZ (T ) = Pr(Z ∈ T ).
(2.15)
If pX (S) = 0 whenever S is of measure zero (with respect to the baseline
measure), then the Radon-Nikodým theorem tells us that
∫
pX (S) =
fX (x)dx
(2.16)
S
where fX (x) is an almost-everywhere unique function called the probability
density function (p.d.f.) of X. Similarly,
∫
pZ (T ) =
fZ (z)dz.
(2.17)
T
Random Finite Sets
2.2.4
47
Target Motion Models and Markov Densities
A single-target target motion model has the general form
Xk+1|k = φk (x′ , Wk )
(2.18)
where Wk is a random noise vector and x′ is the target state at time tk , and φk
is the state transition function. More commonly, it has the additive form
Xk+1|k = φk (x′ ) + Wk
(2.19)
where Wk is zero-mean and φk (x) is the deterministic motion model (state
transition function). For a linear-Gaussian sensor, Wk is Gaussian and φk (x′ ) =
Fk x′ , where Fk is the state transition matrix.
The probability-mass function corresponding to Xk+1|k , conditioned on the
event Xk|k = x′ , is
pk+1|k (S|x′ ) = Pr(Xk+1|k ∈ S|x′ ) =
∫
fk+1|k (x|x′ )dx
(2.20)
S
where Mx (x′ ) = fk+1|k (x|x′ ) is the Markov state transition density function. It
is the probability (density) that the target will have state x at time tk+1 , if it had
state x′ at time tk .
2.2.5
Measurement Models and Likelihood Functions
A single-sensor, single-target measurement model has the general form
Zk+1 = ηk+1 (x, Vk+1 )
(2.21)
where Vk+1 is a random noise vector and ηk+1 is the measurement function.
More commonly, it has the additive form
Zk+1 = ηk+1 (x) + Vk+1
(2.22)
where Vk+1 is zero-mean and ηk+1 (x) is the deterministic measurement model
(measurement function). For a linear-Gaussian sensor, Vk+1 is Gaussian and
ηk+1 (x) = Hk+1 x, where Hk+1 is the measurement matrix.
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Advances in Statistical Multisource-Multitarget Information Fusion
The probability-mass function corresponding to Zk+1 , conditioned on the
event Xk+1|k = x, is
pk+1 (T |x) = Pr(Zk+1 ∈ T |x) =
∫
fk+1 (z|x)dz
(2.23)
T
where
Lz (x) abbr.
= fk+1 (z|x)
(2.24)
is the sensor likelihood function. It is the probability (density) that the measurement
z will be collected, given that there is a target with state x in the scene at time
tk+1 .
2.2.6
Nontraditional Measurements
“Traditional measurements” are those produced by conventional sensor information
sources, such as radars. However, much information originates with intermediary
sources such as digital signal processors and human observers. As was indicated
in Section 1.1.5, such “nontraditional” information includes attributes, features,
natural-language statements, and inference rules.
A fundamental feature of finite-set statistics is that information of this sort
can, from a rigorous Bayesian point of view, be processed in the same way as
traditional information. The basic concepts here are the generalized measurement,
the generalized measurement model, and the generalized likelihood function (GLF).
Detailed discussion of this aspect of finite-set statistics is deferred until Chapter 22.
2.2.7
The Single-Sensor, Single-Target Bayes Filter
The single-sensor, single-target recursive Bayes filter is the theoretical foundation
for single-sensor, single-target tracking and identification ([179], Chapter 2). Given
a time sequence Z k : z1 , ..., zk of measurements collected by the sensor through
time tk , this filter propagates a posterior distribution fk|k (x|Z k )—commonly
Random Finite Sets
49
known as the “track distribution”—through time:2
... →
fk|k (x|Z k )
→
fk+1|k (x|Z k )
→
fk+1|k+1 (x|Z k+1 )
It is defined by the time-update and measurement-update equations
∫
fk+1|k (x|Z k ) =
fk+1|k (x|x′ ) · fk|k (x′ |Z k )dx′
fk+1|k+1 (x|Z k+1 )
=
fk+1 (zk+1 |x) · fk+1|k (x|Z k )
fk+1 (zk+1 |Z k )
where the second equation is Bayes’ rule and where
∫
k
fk+1 (zk+1 |Z ) = fk+1 (zk+1 |x) · fk+1|k (x|Z k )dx
→ ...
(2.25)
(2.26)
(2.27)
is the Bayes normalization factor. When motion and measurement models are
linear-Gaussian and the initial distribution f0|0 (x|Z 0 ) = f0|0 (x) is linearGaussian, then the Bayes filter reduces to the Kalman filter.
Information of interest—target position, velocity, type, and so on—can be
extracted from fk|k (x|Z k ) using a Bayes-optimal state estimator, such as the
maximum a posteriori (MAP) estimator3
AP
k+1
x̂M
)
k+1|k+1 = arg sup fk+1|k+1 (x|Z
(2.28)
x∈X
or the expected a posteriori (EAP) estimator:
∫
x̂EAP
=
x · fk+1|k+1 (x|Z k+1 )dx.
k+1|k+1
2
(2.29)
Note: In two other common systems of notation, the probability distribution fk+1|k (x|Z k ) is
written as
fk+1|k (x|Z k ) = f (xk+1 |z1:k )
or as
fk+1|k (x|Z (k) ) = fXk+1|k |Z1 ,....,Zk (x|z1 , ..., zk )
3
where Xk+1|k is the predicted-target process at time tk+1 and Zj is the measurement process
at time tj .
Caution: It is commonly asserted that the MAP estimator is Bayes-optimal. In actuality, when the
state space is continuously infinite it is only approximately Bayes-optimal (although it is Bayesoptimal to within an arbitrarily small degree of accuracy). This is due to the fact that the associated
cost function is a “finite notch” C(x, y) = 1E (x − y), where E is some arbitrarily small
neighborhood of 0.
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2.3
Advances in Statistical Multisource-Multitarget Information Fusion
RANDOM FINITE SETS (RFSs)
Let Y be an underlying space, such as a state space X or a measurement space
Z. Then Y∞ denotes the hyperspace of all finite subsets of Y, the empty set
included.4 A random finite set (RFS) is a random variable Ψ on Y∞ .5 In this
book, we will deal primarily with two types of RFSs:
• Y = X is the space of single-target states, Y∞ = X∞ is the hyperspace of
all finite subsets of X, and Ψ = Ξ is a random target state set.
• Y = Z is the space of single-target, single-sensor measurements, Y∞ = Z∞
is the hyperspace of all finite subsets of Z, and Ψ = Σ is a random
measurement set.
Thus a random state set Ξ will have the instantiations
X
X
=
=
∅
{x1 }
X
=
{x1 , x2 }
..
.
(no targets are present)
(a single target with state x1 is present)
(two targets with states x1 ̸= x2 are present)
(2.30)
(2.31)
(2.32)
Similarly, a random measurement set Σ will have the instantiations
Z
Z
Z
4
5
=
=
=
∅
(no measurement has been collected)
(2.33)
{z1 }
(a single measurement z1 has been collected)
(2.34)
{z1 , z2 } (two measurements z1 ̸= z2 have been collected) (2.35)
..
.
In the mathematical literature, a “hyperspace” is any space whose points are subsets of another
space.
Formally, an RFS Ψ is a measurable mapping Ψ : Ω → Y∞ from an underlying probability
space Ω to Y∞ . In turn, the definition of a measurable mapping requires us to first define a
topology on Y∞ . In finite-set statistics, this topology is the restriction to Y of the Fell-Matheron
“hit and miss” topology (defined on the class of all closed subsets of Y0 ). For more details, see
[179], Appendix F, pp. 711-716.
Random Finite Sets
2.3.1
51
RFSs and Point Processes
As was noted earlier in Section 1.1.1, RFS theory is the mathematically simplest
version of point process theory. Kingman’s book on Poisson point processes is a
classic introduction [134].6
However, point process theory was introduced fifty years ago by Moyal.
He formulated two equivalent versions, one based on random unordered finite
sequences ([207], pp. 2-3, 5) and the other—the one that is now dominant among
pure mathematicians—based on counting measures ([207], p. 6).
˜∞
It will be instructive to briefly examine Moyal’s first formulation. Let Y
7
be the set of unordered finite sequences θ = [y1 ...yn ] for any n. Then a point
˜ ∞ . If it
process is a random variable P whose instantiations are elements of Y
is always the case that the y1 , ..., yn are distinct, then the point process is called
simple. A finite sequence of distinct, unordered elements is the same thing as a
finite set: [y1 ...yn ] = {y1 ...yn }. Therefore, a simple point process is the same
thing as an RFS.
The statistics of P are described by its probability measure pP (O) =
˜ ∞ . A basic result of
Pr(P ∈ O), where O is a measurable subset of Y
point process theory is the following: the probability density function fP (θ)
corresponding to pP (O) exists (that is, is finite-valued and integrable) if and
only if P is simple—that is, if it is an RFS ([55], p. 138, Prop. 5.4.V). In this case
the functions
n! · jn,P (x1 , ..., xn ) = fP ([y1 ...yn ]) = fP ({y1 , ..., yn })
(2.36)
are known as “Janossy densities”—or, as described in Section 4.2.2, as the multiobject probability distribution of the RFS P. The intuitive meaning of Prop. 5.4.V of
[55] is this: if there are repeated elements in [y1 ...yn ] (i..e, if yi = yj for some
i ̸= j) then fP ([y1 ...yn ]) = ∞ which cannot be used in formulas of practical
interest. Stated differently:
• To be of practical interest, a point process must be an RFS.
6
7
More precisely, RFSs in Kingman are local RFSs—that is, their intersection with any bounded
closed set is finite.
Caution: Contrary to modern notation, Moyal used the notation {y1 , ..., yn } to denote an
˜ ∞ = ⊎n≥0 (Yn /Rn )
unordered finite sequence rather than a finite set. Technically speaking, Y
where (Yn /Rn ) denotes the space of equivalence classes of Yn with respect to to equivalence
Rn
relation (y1 , ..., yn ) ≡ (y1 , ..., yn ) if and only if (y1 , ..., yn ) = (yπ1 , ..., yπn ) for some
permutation π on 1, ..., n.
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Advances in Statistical Multisource-Multitarget Information Fusion
As was argued in [179], Appendix E.3, pp. 708-710, for the purposes of information fusion and multitarget detection and tracking, the non-RFS formulations
of point process theory:
• Unnecessarily increase notational and theoretical complexity.
• Add no new substance, especially from a practitioner’s point of view.
• Lose the simple and geometrically intuitive tools of ordinary set theory.
• Are the same thing as RFS theory, the minute that one applies them to
practical problems.
Despite this fact, the increasing notoriety of finite-set statistics has inspired
the creation of supposed “point process” alternatives to RFS-based information
fusion. As a typical example, the authors of [281] cite Kingman as an authority
on Poisson point processes.8 Rather than adopting Kingman’s RFS formulation,
however, they define an instantiation of a point process on Y as follows. Rather
than a finite set {y1 , ..., yn }, it is an (n + 1)-tuple ξ = (n, y1 , ..., yn ) for any
n ≥ 0.9 In this case the probability distribution f (ξ) of ξ must have the form
f (n, y1 , ..., yn ) with f (n, y1 , ..., yn ) = f (n, yπ1 , ..., yπn ) for any permutation
π on 1, ..., n. To be of practical use, f (ξ) must be finite-valued—in which case
the point process must be an RFS. Thus: what in [281] is called a “point process”
is actually an RFS with different notation and terminology.
Another difficulty with all such “point process” alternatives is that they are
technically vacuous. They imitate a single insight from finite-set statistics, while
stripping away the mathematical tools and systematic methodologies that give it
its problem-solving power. The most visible of these tools, all introduced in 2001
[168], are the probability generating functional (p.g.fl.), the functional derivative,
and their employment to derive PHD and CPHD filters. The vacuousness of the
original “point process” imitations appears to have inspired the discovery—a full
decade after the fact—of a supposed alternative “point process” formulation of this
aspect of finite-set statistics [279], [282].
The following question must be raised every time that a newly-minted “point
process” alternative arrives on the scene: notation and terminology aside, is there
any actual difference between it and finite-set statistics? As will be shown in
Appendix J:
8
9
To wit: “For further background on PPP’s from a multidimensional perspective, see Kingman [6]”
[281], first paragraph of Section 2.
See [281], discussion following Eq. (1).
Random Finite Sets
53
• Except for changes in notation and terminology, the “point process” formulation in [279], [282] is identical to a truncated version of finite-set statistics.
• Its treatment of functional derivatives is mathematically erroneous in two
different respects—but is identical to the engineering-heuristic version of the
functional derivative of finite-set statistics.
As was pointed out in [198] and the beginning of Section 1.1.1, the mere
fact that multitarget tracking can be formulated in terms of RFSs—or any other
mathematical formalism—is of little practical interest in and of itself. A mere
change of notation or terminology does not add new substance. In particular:
• “Point process” or “PP” means the same thing as “RFS.”
• “Poisson point process” or “PPP” means the same thing as “Poisson RFS.”
• “Superposition of point processes” means the same thing as “set-theoretic
union of RFSs.”
• “Intensity” or “intensity function” means the same thing as “first-moment
density” or “probability hypothesis density” or “PHD.”
• “Intensity filters,” in the general sense of the term, means the same thing as
“PHD filters.”
• “Cardinal number density”10 is a rather transparent renaming of “cardinality
distribution” (as well as being incorrect terminology, since the term “density”
is reserved for continuously infinite spaces). ”Canonical number distribution”
is yet another transparent renaming.
There are many more such examples. It is the tools of finite-set statistics—
as summarized in this and the following chapters—that render the RFS formalism
useful for information fusion from a practical point of view.
2.3.2
Examples of RFSs
Many specific examples of RFSs are presented in Section 11.2, pp. 348-356, of
[179]. Here are a few simple instances.
• Random singleton:
Ψ = {Y}
10 [282], p. 45, paragraph following Eq. (14).
(2.37)
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Advances in Statistical Multisource-Multitarget Information Fusion
where Y is a random element of Y.
• Union of random singletons:
Ψ = {Y1 , ..., Yn }
(2.38)
where n > 1 is a fixed integer and Y1 , ..., Yn are random elements of
Y. If Y1 , ..., Yn are independent and Y is continuously infinite, then
|Ψ| = n with probability 1.
• “Twinkling” random singleton:
Ψ = {Y} ∩ ∅q
(2.39)
where ∅q is the discrete random subset of Y defined by
Pr(∅q = T ) =


q
1−q

0
if
if
if
T =Y
T =∅ .
otherwise
(2.40)
That is, Ψ ̸= ∅ with probability q; and in this case it is a singleton. As an
example, the RFS Ψ could be employed as a simple model of a twinkling
star in the night sky.
2.3.3
Algebraic Properties of RFSs
• Union: Suppose that Ψ1 , ..., Ψn ⊆ Y are RFSs. Then their set-theoretic
union—also called “superposition”—is an RFS:
Ψ = Ψ1 ∪ ... ∪ Ψn .
(2.41)
• Intersection: Suppose that Ψ1 , Ψ2 ⊆ Y are RFSs. Then their set-theoretic
intersection
Ψ = Ψ1 ∩ Ψ2
(2.42)
is an RFS. If Ψ1 , Ψ2 are independent and Y is continuously infinite, then
they are almost always disjoint:
Pr( Ψ1 ∩ Ψ2 = ∅) = 1.
(2.43)
Random Finite Sets
55
As a special case, let Ψ ⊆ Y be an RFS and y ∈ Y. If {y} is regarded as
a constant RFS, then Ψ, {y} are independent and so
Pr(y ∈ Ψ) = Pr({y} ∩ Ψ ̸= ∅) = 0.
(2.44)
That is, y ∈ Ψ is a zero-probability event. As we shall see, the PHD DΨ (y) of
Ψ provides a mathematically rigorous description of the event y ∈ Ψ, in the same
way that a conventional probability density function fY (y) provides a rigorous
description of the zero-probability event Y = y.
More generally, let Θ ⊆ Y be any random closed subset and Ψ ⊆ Y an
RFS. Then
Ψ′ = Ψ ∩ Θ
(2.45)
is an RFS. In particular, let Θ = T be a constant subset. Then
Ψ′ = Ψ ∩ T
(2.46)
is an RFS, called a censored RFS. It will be further considered in Section 4.4.1,
and will play an important role throughout the book.
2.4
MULTIOBJECT STATISTICS IN A NUTSHELL
Finite-set statistics draws on a general, Bayesian formulation of the dynamic state
space model. This general formulation is described in detail in Appendix B. The
finite-set statistics special case of this formulation consists of the following basic
elements:
1. A practitioner-oriented multisource-multitarget integral and differential calculus ([179], Chapter 11). Just as “turn-the-crank” rules exist for ordinary
differential and integral calculus, so similar “turn-the-crank” rules exist for
the multitarget differential and integral calculus—specifically, for the set integral, set derivative, and functional derivative.
2. Formal statistical multisouce-multitarget measurement models ([179], Chapter 12). Just as single-sensor, single-target data can be modeled using a measurement model
Zk+1 = ηk+1 (x, Vk+1 ),
(2.47)
so multitarget, multisensor data can be modeled using a multisensor and
multitarget measurement model—a randomly varying finite set
Σk+1 = Υk+1 (X) ∪ Ck+1 (X)
(2.48)
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Advances in Statistical Multisource-Multitarget Information Fusion
of measurements, where Υk+1 (X) and Ck+1 (X) are RFSs of targetgenerated and (possibly state-dependent) clutter measurements, respectively.
3. “True” multisource-multitarget likelihood functions, derived via multitarget
calculus from the measurement models ([179], Chapter 12). Just as the true
single-sensor, single-target likelihood function fk+1 (z|x) can be derived
from the probability mass function
pk+1 (T |x) = Pr(Zk+1 ∈ T |Xk+1|k = x)
(2.49)
of the measurement model Σk+1 via differentiation, so the true multitarget
likelihood function fk (Z|Xk ) can be derived from the belief-mass function
(also known as belief measure)
βk+1 (T |X) = Pr(Σk+1 ⊆ T |Ξk+1|k = X)
(2.50)
of the multisensor-multitarget measurement model via the set derivative.
Here, “true” means that fk+1 (z|x) contains exactly the same information
as the measurement model Zk+1 = ηk+1 (x, Vk+1 ) That is, no modeled
information has been left out or replaced with heuristics; and no information
extraneous to the model has inadvertently been inserted.
4. Formal statistical multitarget motion models ([179], Chapter 13). Just as
single-target motion can be modeled using a motion model
Xk+1|k = φk (x, Wk ),
(2.51)
so the motion of multitarget systems can be modeled using a multitarget
motion model—a randomly varying finite set
Ξk+1|k = Tk (X) ∪ Bk (X)
(2.52)
of predicted targets, where Tk (X) and Bk (X) are RFSs of persisting and
newly appearing targets, respectively.
5. “True” multitarget Markov transition densities, derived via multitarget calculus from the motion models ([179], Chapter 14). Just as the true Markov
transition density fk+1|k (x|x′ ) can be derived from the probability mass
function
pk+1|k (S|x′ ) = Pr(Xk+1|k ∈ S|Xk|k = x′ )
(2.53)
Random Finite Sets
57
of the motion model via differentiation, so the true multitarget Markov transition density fk+1|k (X|X ′ ) can be derived from the belief-mass function
βk+1|k (S|X ′ ) = Pr(Ξk+1|k ⊆ S|Ξk|k = X ′ ),
(2.54)
via set differentiation. Here, “true” means that fk+1|k (x|x′ ) contains exactly
the same information as the motion model Xk+1|k = φk (x, Wk ). No
modeled information has been left out or replaced with heuristics; and no
information extraneous to the model has inadvertently been inserted.
6. Principled statistical approximation ([179], Chapters 16, 17). Just as the
Kalman filter can be derived as a principled approximation of the singlesensor, single-target Bayes filter, so various multitarget filters—for example,
PHD, CPHD, and multi-Bernoulli filters—can be derived as principled approximations of the multisensor-multitarget Bayes filter. Here, “principled”
means that the single-target distributions fk|k (x|Z k ) or the multitarget distributions fk|k (X|Z (k) ) are assumed to have statistically approximate forms
that result in closed-form formulas. Examples of such approximations include linear-Gaussian in the case of fk|k (x|Z k ) and Poisson in the case of
fk|k (X|Z (k) ).
7. Exact derivation of approximate filters using multiobject differential calculus.
Using the multisensor-multitarget measurement model and the multitarget
motion model, the recursive Bayes filter is reformulated in terms of probability generating functionals (p.g.fl.’s). Then functional derivatives are used
to derive the approximate multitarget filters from this p.g.fl. form of the Bayes
filter.
Additional aspects of finite-set statistics will be addressed in Chapter 6
(multiobject metrology) and Chapter 22 (nontraditional measurements):
1. Multiobject miss distances ([179], pp. 510-512). The optimal subpattern
assignment (OSPA) metric (Section 6.2.2) and its generalizations permit the
measurement of distances between finite sets Y = {y1 , ..., yn } of points.
2. Multiobject information-theoretic functionals ([179], pp. 513-513). Multiobject generalizations of the Csiszár family of information-theoretic divergences permit the comparison of one multiobject probability distribution
f1 (Y ) with another multiobject probability distribution f0 (Y ).
3. Formal statistical models for nontraditional measurements ([179], Chapter
4). Just as conventional measurements are modeled as random vectors
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Advances in Statistical Multisource-Multitarget Information Fusion
Zk+1 ∈ Z of a measurement space Z, so nontraditional measurements—
attributes, features, natural-language statements, and inference rules—are
modeled as “generalized measurements”—that is, random closed subsets
Θk+1 ⊆ Z of a measurement space Z.
4. Formal measurement-generation models for nontraditional measurements
([179], Chapters 5 and 6). Just as the generation of single-sensor, singletarget data can be modeled using a measurement model
ηk+1 (x) + Vk+1 = Zk+1 ,
(2.55)
so the generation of nontraditional measurements can be modeled using
generalized measurement models. These have the form
ηk+1 (x) + Vk+1 ∈ Θk+1
(2.56)
if the measurement function ηk+1 (x) is known precisely; and
Θk+1 ∩ Ξx,k+1 ̸= ∅
(2.57)
if otherwise, where a target-related RFS Ξx,k+1 replaces ηk+1 (x).
5. Likelihood functions for nontraditional measurements ([179], Chapters 5 and
6). Just as the measurement models for conventional measurements can
be transformed into likelihood functions fk+1 (z|x), so the measurement
models for nontraditional measurements can be transformed into “generalized
likelihood functions”:
ρk+1 (Θ|x) = Pr(ηk+1 (x) + Vk+1 ∈ Θ)
(2.58)
ρk+1 (Θ|x) = Pr(Θ ∩ Ξx ̸= ∅).
(2.59)
or, alternatively,
Generalized likelihood functions are not conventional likelihood functions.
This is, for example, because ρk+1 (Θ|x) is a probability rather than a probability density. Nevertheless, it has been shown that they are mathematically
rigorous from a strict Bayesian point of view (see [191] and Section 22.3.4).
Chapter 3
Multiobject Calculus
3.1
INTRODUCTION
The multiobject (also known as multitarget) integro-differential calculus, as summarized in this chapter, is the core “mathematical machine” of finite-set statistics. This
calculus and its associated concepts—random finite sets, belief-mass functions,
multiobject probability density functions, and probability generating functionals—
permit:
• The mathematically rigorous derivation of multiobject Markov density functions from mathematically rigorous RFS multitarget motion models.
• The mathematically rigorous derivation of multisensor-multitarget likelihood
functions from mathematically rigorous RFS multisensor-multitarget measurement models.
• The mathematically rigorous derivation of approximate multisensor and multitarget detection, tracking, and identification algorithms such as PHD filters,
CPHD filters, and multi-Bernoulli filters.
The remainder of the chapter is organized as follows:
1. Section 3.2: Basic concepts—set functions, functionals, functional transformations and multiobject density functions.
2. Section 3.3: Set integrals—the fundamental integration concept on hyperspaces of finite sets.
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Advances in Statistical Multisource-Multitarget Information Fusion
3. Section 3.4: Multiobject differential calculus—gradient derivatives, functional derivatives, and set derivatives.
4. Section 3.5: Key formulas of multiobject calculus—the fundamental theorem
of multiobject calculus; change of variables formula for set integrals; set
integrals on joint spaces; and various “turn-the-crank” rules for functional
derivatives (constant rule, sum rule, linear rule, monomial rule, power rule,
product rules, and the general chain rule of Clark and its special cases). (More
complex identities will be described in Chapter 4.)
3.2
BASIC CONCEPTS
The purpose of this section is to introduce several basic concepts: set functions,
functionals, functional transformations, and multiobject density functions.
3.2.1
Set Functions
A set function is a real-valued function ϕ(T ) defined on the measurable subsets T
of Y. Some simple examples:
• Probability-mass function (also known as probability measure):
pY (T ) = Pr(Y ∈ T )
(3.1)
where Y is a random element of Y.
• Possibility measure:
µg (T ) = sup g(y)
(3.2)
y∈T
where g(y) is a fuzzy membership function on Y.
• Belief-mass function or belief measure (see Section 4.2.1):
βΨ (T ) = Pr(Ψ ⊆ T )
(3.3)
where Ψ ⊆ Y is an RFS.
3.2.2
Functionals
A functional on Y is a real-valued (and typically unitless) function F [h] in the
variable h(y), where h(y) is an ordinary real-valued (and typically unitless)
Multiobject Calculus
61
function on y ∈ Y.1 The following are examples of functionals that will be
important in what follows.
• The linear functional induced by a fixed density function f (y) on Y, defined
by
∫
s[h] =
h(y) · f (y)dy.
• The power functional. Let Y ⊆ Y be a finite set. Then
{
if Y = ∅
∏ 1
hY =
.
y∈Y h(y) if Y ̸= ∅
(3.4)
(3.5)
The following identity involving the power functional—a generalization of
the binomial theorem—is sometimes useful:2
∑
(h + h0 )X =
hW · hX−W
(3.6)
0
W ⊆X
or, equivalently,
∑
X
hW = (1 + h) .
(3.7)
W ⊆X
3.2.3
Functional Transformations
A functional transformation is similar to a functional, except that its value (and not
just its argument) is a function. Thus a functional transformation T : h ?→ T [h]
transforms the function h(y) in the variable y ∈ Y into another function T [h](w)
in the variable w ∈ W.
A simple example is the transformation
T [h] = 1 − γ + γ · h
(3.8)
T [h](y) = 1 − γ(y) + γ(y) · h(y),
(3.9)
defined pointwise by
for all y ∈ Y and where 0 ≤ γ(y) ≤ 1 is a unitless function.
1
2
The bracket notation ‘[·]’ has been borrowed from quantum physics [264], which in turn borrowed
it from Volterra [317]. Its purpose is to clearly distinguish functionals F [h] from conventional
functions F (y).
This equation is a consequence of the fact that the superposition of two Poisson RFSs with PHDs
D1 (x), D2 (x) is a Poisson RFS with PHD D1 (x) + D2 (x).
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3.2.4
Multiobject Density Functions
A real-valued function f (Y ) of the finite-set variable Y ⊆ Y is a multiobject
density function if, for each Y , the units of measurement of f (Y ) are u−|Y | ,
where u denotes the units of y.
Thus if Y = R and the units of R are meters (u = m),
• f (∅) is unitless.
• The units of f ({y}) are m−1 .
• The units of f ({y1 , y2 }) with y1 ̸= y2 are m−2 .
• Similarly for larger values of |Y |.
Multiobject density functions can be written in vector notation:3
{ 1
i! · f ({y1 , ..., yi }) if |{y1 , ..., yi }| = i .
fi (y1 , ..., yi ) =
0
if
otherwise
(3.10)
That is, fi (y1 , ..., yi ) vanishes if any two of the y1 , ..., yi are equal.
3.3
SET INTEGRALS
Let f (Y ) be a multiobject density function (as defined in Section 3.2.1). Then its
set integral is defined as:
∫
f (Y )δY
=
∫
∞
∑
1
i!
f ({y1 , ..., yi })dy1 · · · dyi
(3.11)
i=0
=
f (∅) +
∫
∞
∑
1
i!
def.
=
f (∅) +
i=1
∞ ∫
∑
f ({y1 , ..., yi })dy1 · · · dyi
fi (y1 , ..., yi )dy1 · · · dyi
(3.12)
(3.13)
i=1
3
Equation (3.10) involves more than just a change of notation. The finite-set variable Y , as an
instantiation of an RFS Ψ, is defined in terms of the Fell-Matheron topology. However, for each i,
(y1 , ..., yi ) is defined in terms of the product topology on Y × ... × Y (Cartesian product taken
i times). It is necessary to verify that the two notations are topologically consistent—see [94], p.
133, Proposition 2.
Multiobject Calculus
63
where fi (y1 , ..., yi ) was defined in (3.10). Note that by the definition of a
multiobject density function, the products fi (y1 , ..., yi )dy1 · · · dyi are unitless
for every i ≥ 0. Thus the set integral is mathematically well defined.
Let T ⊆ Y be a measurable subset. Then the set integral concentrated in T
is4
∫
∫
f (Y )δY = 1YT · f (Y )δY
(3.14)
T
∫
∞
∑
1
f ({y1 , ..., yi })dy1 · · · dyi
(3.15)
where the power functional hY was defined in (3.5).
Like conventional integrals, the set integral is linear in f :
∫
∫
∫
(a1 f1 (Y ) + a2 f2 (Y )) δY = a1
f1 (Y )δY + a2
f2 (Y )δY.
(3.16)
=
f (∅) +
i=1
i! T × ... × T
? ?? ?
i times
T
T
T
Unlike conventional integrals, it is usually not additive in T . That is, if T1 ∩T2 = ∅
then it is usually the case that
∫
∫
∫
f (Y )δY ̸=
f (Y )δY +
f (Y )δY.
(3.17)
T1 ∪T2
T1
T2
Remark 1 (Caution: Units of measurement) When forming a set integral, care
should be taken with respect to units of measurement in the integrand. For example,
the following multiobject analog of the L2 distance,
√∫
2
∥f1 − f2 ∥2 =
(f1 (Y ) − f2 (Y )) δY ,
(3.18)
is not well defined because of incompatibility of units in the set integral.
4
As is more fully explained in [179], Appendix F.3, pp. 714-715, set integrals are not, strictly
speaking, measure-theoretic integrals. It is possible to define them as measure-theoretic integrals
with respect to a certain extension of the measure on Y to a measure on Y∞ . However, the
definition of this measure requires the introduction of an arbitrary constant c. Conventional practice
in point process theory is to set c = 1 · u, where u denotes the units of measurement in Y.
This measure-theoretic definition of the set integral is potentially problematic for multitarget state
estimation, however. As is explained more fully in [179], pp. 499-500, the size of c should be
approximately equal to the accuracy to which any single-target state y is to be estimated.
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Advances in Statistical Multisource-Multitarget Information Fusion
As an example, suppose that




(1 − q)2
q (1 − q) · (f1 (y) + f2 (y))
f (Y ) =
2
q
·
(f
(y1 ) · f2 (y2 ) + f1 (y2 ) · f2 (y1 ))

1


0
if
if
if
if
Y =∅
Y = {y}
Y = {y1 , y2 }, |Y | = 2
|Y | > 2
(3.19)
where f1 (y), f2 (y) are probability density functions on Y and where 0 < q ≤ 1.
Then
∫
f (Y )δY
(3.20)
∫
∫
1
1
= f (∅) +
f ({y})dy +
f ({y1 , y2 })dy1 dy2
1!
2!
∫
= (1 − q)2 + q (1 − q) (f1 (y) + f2 (y)) dy
(3.21)
∫
q2
+
(f1 (y1 ) · f2 (y2 ) + f1 (y2 ) · f2 (y1 )) dy1 dy2
2
= (1 − q)2 + 2q (1 − q) + q 2
(3.22)
=
3.4
1.
(3.23)
MULTIOBJECT DIFFERENTIAL CALCULUS
The purpose of this section is to introduce the basic concepts of multiobject
differential calculus:
1. Section 3.4.1: Gâteaux directional derivatives—the theoretical basis of
multiobject differential calculus.
2. Section 3.4.2: Volterra functional derivatives—important for the derivation
of approximate multisensor-multitarget filters such as the PHD filter and
CPHD filter.
3. Section 3.4.3: Set derivatives—important for the derivation of multitarget
Markov transition densities and multisensor-multitarget likelihood functions.
Multiobject Calculus
3.4.1
65
Gâteaux Directional Derivatives
Let F [h] be a functional defined on unitless real-valued functions h(y) with
argument y ∈ Y. Let g(y) be another unitless real-valued function with argument
y ∈ Y. Then, if it exists, the Gâteaux directional derivative of F [h] in the
direction of g satisfies the following two properties ([179], Appendix C, pp. 695697):
1. Its value is given by the differential quotient
∂F
F [h + ε · g] − F [h]
[h] = lim
.
ε→0
∂g
ε
(3.24)
2. For each fixed h, the function in g defined by
g ?−→
∂G
[h]
∂g
(3.25)
exists for all g is both linear and continuous.5
Note that if we instead allow g(y) to be a density function, then the units of
ε must be the same as the units of y—so that ε · g is unitless. It follows that, in
this case, (∂F/∂g)[h] has the units of a density function.
The iterated Gâteaux derivative is defined inductively as
∂nF
∂ n+1 F
∂ n+1
[h] =
[h].
∂gn+1 ∂gn · · · ∂g1
∂gn+1 δgn · · · ∂g1
5
(3.26)
If one does not insist on linearity or continuity, (3.24) defines a “Gâteaux differential.” Mathematicians prefer a special case of the Gâteaux derivative called the Frechét derivative (also known as
gradient derivative), which will not concern us in this book.
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Advances in Statistical Multisource-Multitarget Information Fusion
3.4.2
Volterra Functional Derivatives
The Volterra functional derivative of F [h] with respect to a finite subset Y ⊆ Y
is defined to be ([179], Section 11.4.1): 6
δF
[h] = F [h]
δY
(3.27)
if Y = ∅ and, if Y = {y1 , ..., yn } ⊆ Y with |Y | = n > 0,7
δF
∂nF
[h] =
[h]
δY
∂δy1 · · · ∂δyn
(3.28)
where δy (w) denotes the Dirac delta function concentrated at y. In particular,
when Y = {y},8
δF
δF
F [h + εδy ] − F [h]
[h] abbr.
=
[h] = lim
.
ε→0
δy
δ{y}
ε
(3.29)
Note that δy is not an actual function and therefore the expression F [δy ]
is not mathematically well defined. Consequently, (3.28) and (3.29) are intuitive
practitioner heuristics rather than mathematically rigorous definitions. Rigorous
definitions can be found in Appendix C. In this case y ?→(δF/δy)[h] turns
out to be the density function corresponding to the probability measure defined by
6
7
8
The concept of a functional derivative in the sense of (3.28) and (3.29) goes back to Volterra in
1927 ([317], p. 24, Eq. (3)), where the left side of (3.29) is notated as F ′ [h(x); y]. It should be
distinguished from a more casual and careless usage of the term “functional derivative” —that is,
application of the Frechét derivative, Gâteaux derivative, or Gâteaux differential to a functional, as
in (3.24). Both the notation and the definitions in (3.28) and (3.29) are borrowed from quantum
physics (see [264]; [79], p. 406, Eq. (A.15); or [179], Remark 14, p. 376, and Remark 16, p. 382).
As is explained more fully in [179], Appendix F.4, pp. 715-716, general functional derivatives
δF/δY are not quite Radon-Nikodým derivatives of some measure, despite the fact that first-order
functional derivatives can be defined in terms of Radon-Nikodým derivatives.
The Volterra functional derivative (δF/δy)[h] of a functional F [h] is analogous to the partial
derivative (∂f /∂xi )(x) of a function f (x) of a Euclidean vector variable x, with y being a
continuous analog of the finite-valued index i. That is, (3.29) is analogous to
∂f
f (x + εêi ) − f (x)
(x) = lim
ε→0
∂xi
ε
where ê1 , ..., êN is an orthonormal basis. For this reason, the notation (δF/δh(y))[h] is often
employed. The notation (δF/δy)[h] is an abbreviation of this.
Multiobject Calculus
T ?→ (∂F/∂1T )[h]. That is, the defining equation for (δF/δy)[h] is
∫
∂F
δF
[h] =
[h]dy.
∂1T
T δy
67
(3.30)
In other words, the first-order functional derivative δF/δy for all y is defined as
the Radon-Nikodým derivative of the probability measure ∂F/1T .
This definition is problematic from a practical point of view, since it asserts
only that the first-order functional derivatives exist. It does not tell us how to
actually construct them. It turns out that functional derivatives can be constructively
defined in terms of set derivatives—see Appendix C.3 and Remark 2 in Section
3.4.3.
Just as the functional derivative can be regarded as a special kind of Gâteaux
derivative, the Gâteaux derivative can be expressed in terms of functional derivatives:
∫
∂F
δF
[h] = g(y) ·
[h]dy.
(3.31)
∂g
δy
This equation, which generalizes (3.30), is proved in Appendix C.
Several examples of functional derivatives are presented in pp. 377-380 of
[179]. For the sake of illustration, only one is presented here. Let
∫
f [h] = h(y) · f (y)dy
(3.32)
be the linear functional defined in (3.4). Then
δ
f [h]
δy
=
=
=
3.4.3
f [h + εδy ] − f [h]
ε→0
ε
∫
ε · f [δy ]
lim
= δy (w) · f (w)dw
ε→0
ε
f (y).
lim
(3.33)
(3.34)
(3.35)
Set Derivatives
Let F [h] be a functional defined on unitless functions h(y) on y ∈ Y with
0 ≤ h(x) ≤ 1. Define the set function ϕF (T ) for closed T ⊆ Y by
ϕF (T ) = F [1T ]. Then the set derivative of ϕF (T ) with respect to the finite
subset Y ⊆ Y is, if it exists ([179], Section 11.4.2),
δϕF
δF
(T ) =
[1T ].
δY
δY
(3.36)
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Advances in Statistical Multisource-Multitarget Information Fusion
The set derivative can alternatively be defined independently of functionals.
Let ϕ(T ) be a set function defined on closed sets T . Then the set derivative with
respect to y is (see Eq. (11.229) of [179]):
δϕ
ϕ(T ∪ Ey ) − ϕ(T )
(T ) = lim
δy
|Ey |
|Ey |↘0
(3.37)
where Ey is an arbitrarily small closed neighborhood of the point y that is disjoint
from T . (For T that are not disjoint from Ey , a slightly more complex definition
is required—see Appendix C.2 or [94], pp. 145-146, Definition 13.)
Remark 2 (Constructive functional derivatives) As noted earlier, (3.30) defines
the functional derivative only implicitly. However, as is explained in Appendix C.3,
the set derivative can be used to construct functional derivatives:
[
]
δF
δ
[h] =
Fh (T )
δy
δy
T =∅
(3.38)
where the set function Fh (T ) is defined by
Fh (T ) =
∂F
[h].
∂1T
(3.39)
That is: by (3.30), if the functional derivatives (δF/δx)[h] of F [h] exist for all x
and if the function x ?→(δF/δx)[h] is integrable for each fixed h, then Fh (T ) is
a probability measure absolutely continuous with respect to the base measure (see
also Appendix C). Its set derivative must equal the Radon-Nikodým derivative of
the set function Fh (S).
The general set derivative with respect to a finite subset Y = {y1 , ..., yn } ⊆
Y with |Y | = n is defined iteratively:
δϕ
δnϕ
δ
δ n−1 ϕ
(T ) =
(T ) =
(T ).
δY
δyn · · · δy1
δyn δyn−1 · · · δy1
(3.40)
Remark 3 (Set derivative as an inverse Möbius transform) The set derivative is
a continuous-space generalization of the inverse Möbius transform of DempsterShafer theory (see [94], p. 149, Proposition 9, and [179], p. 383, Remark 17).
Multiobject Calculus
69
Remark 4 (Relationship with Moyal’s calculus) Fifty years ago in [207], Moyal
introduced a differential calculus of probability functionals. It is not only different
than the one just described, it is very ill-suited for practical application. This is
for two reasons: (1) it is highly abstract and measure-theoretic; and (2) it provides
only existence proofs for the probability distributions of RFSs, which are defined
only implicitly. It does not provide any explicit means of actually constructing them.
The finite-set statistics calculus is based on set derivatives—which in turn are based
on constructive Radon-Nikodým derivatives. See Appendix J for details.
3.5
KEY FORMULAS OF MULTIOBJECT CALCULUS
The purpose of this section is to summarize the most important mathematical identities associated with multiobject differential and integral calculus. Additional useful
identities can be found in Chapter 4: convolution and deconvolution formulas (Section 4.2.3), Campbell’s theorems (Section 4.2.12), and Radon-Nikodým formulas
(Section 4.2.11).
The section is organized as follows:
1. Section 3.5.1: Fundamental theorem of multiobject calculus—states that set
integrals and functional/set derivatives are inverse operations.
2. Section 3.5.2: Change of variables formula for set integrals—transforms set
integrals into ordinary integrals.
3. Section 3.5.3: Set integrals on disjoint unions of spaces—showing that they
can be transformed into multiple set integrals on the individual spaces.
4. Section 3.5.4: Constant rule—the functional derivatives of a constant functional are zero.
5. Section 3.5.5: Sum rule—showing that the functional derivative is a linear
operator.
6. Section 3.5.6: Linear rule—formula for the functional derivatives of a linear
functional.
7. Section 3.5.7: Monomial rule—formula for the functional derivatives of the
integer powers of a linear functional.
8. Section 3.5.8: Power rule—formula for the first functional derivative of the
power of a functional.
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Advances in Statistical Multisource-Multitarget Information Fusion
9. Section 3.5.9: Product rules—formula for the functional derivatives of the
product of a finite number of functionals.
10. Section 3.5.10: First chain rule—formula for the first functional derivative
of a functional of the form F˜ [h] = f (F1 [h], ..., Fn [h]) where f (x1 , ..., xn )
is an ordinary function and F1 [h], ..., Fn [h] are functionals.
11. Section 3.5.11: Second chain rule—formula for the general functional
derivatives of a functional of the form F˜ [h] = F [T −1 h] where F [h] is
a functional and T is a nonsingular vector transformation.
12. Section 3.5.12: Third chain rule—formula for the first functional derivative
of a functional of the form F˜ [h] = F [fh ] where f (x) is an ordinary
function and fh (y) = f (h(y)).
13. Section 3.5.13: Fourth chain rule—formula for the first functional derivative
of a functional of the form F˜ [h] = F [T [h]] where F [h] is a functional and
T [h] is a functional transformation.
14. Section 3.5.14: Clark’s general chain rule—formula for the general functional derivatives of a functional of the form F˜ [h] = F [T [h]], where F [h]
is a functional and T [h] is a functional transformation.
3.5.1
Fundamental Theorem of Multiobject Calculus
The set integral and functional derivative are inverse operations ([179], Eqs.
(11.246-11.247)):
∫
δF
F [h] =
hY ·
[0]δY
(3.41)
δY
[
]
∫
δ
f (Y ) =
hW · f (W )δW
(3.42)
δY
h=0
where the power functional hY was defined in (3.5).
The corresponding formulas for set derivatives are ([179], Eqs. (11.24411.245)):
∫
δϕ
ϕ(T ) =
(∅)δY
(3.43)
δY
T
[
]
∫
δ
f (Y ) =
f (W )δW
.
(3.44)
δY T
T =∅
Multiobject Calculus
3.5.2
71
Change of Variables Formula for Set Integrals
Let Ψ ⊆ Y be an RFS and fΨ (X) its probability distribution. Let η be a
transformation from finite sets Y ⊆ Y to the elements η(Y ) ∈ W of some other
space W, and let T : W → V be a transformation from the space W to the
elements of some other space V.9 Let PW (w) = PΨ,η (w) be the probability
distribution of the random variable
(3.45)
W = η(Ψ).
Then the change of variables formula for set integrals is ([94], p. 180, Prop. 4):
∫
∫
T (η(Y )) · fΨ (Y )δY = T (w) · PΨ,η (w)dw.
(3.46)
That is, the set integral on the left side can be transformed into the ordinary integral
on the right side, via the change of variables w = η(Y ).
3.5.3
1
Set Integrals on Joint Spaces
s
1
s
Let Y, ..., Y be s spaces, each of which is endowed with an integral. Let Ψ, ..., Ψ
be respective RFSs on these spaces. Then the superposition (set theoretic union) of
these RFSs is an RFS
1
s
˘ = Ψ ⊎ ... ⊎ Ψ
Ψ
(3.47)
on their “joint space,” that is, the disjoint union or “topological sum”
1
s
˘ = Y ⊎ ... ⊎ Y.
Y
(3.48)
˘ have the form
Functions with arguments in Y
j
˘
˘ y)
h(y̆)
= h(
The integral
∫
y̆ = y .
˘ is defined as
·dy̆ on Y
∫
∫
∫
1
1
s
s
˘
˘ y)d
˘ y)d
y.
h(y̆)dy̆
= 1 h(
y + ... + s h(
Y
9
j
if
(3.49)
(3.50)
Y
Note: In this and the following sections, the symbol ‘T ’ will be used to denote a vector or
functional transformation, as well as a subset of the space Y. The proper notational meaning
will be clear from context.
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Advances in Statistical Multisource-Multitarget Information Fusion
˘ is
Thus the set integral on Y
∫
∑ 1 ∫
f˘(Y˘ )δ Y˘ =
f˘({y̆1 , ..., y̆n̆ })dy̆1 · · · dy̆n̆ .
n̆!
(3.51)
n̆≥0
Equation (3.51) can be written in a more mathematically convenient form.
Abbreviate
1
s
1
s
˘˘ (Y ⊎ ... ⊎ Y ) abbr.
˘
f˘Ψ
= f 1 s (Y , ..., Y ).
(3.52)
˘ (Y ) = f Ψ
Ψ,...,Ψ
Then the single set integral on the joint space can be written as a multiple set
integral on the disjoint subspaces:
∫
∫
1
s
1
s
˘
˘
˘
fΨ
f 1 s (Y , ..., Y )δ Y · · · δ Y .
(3.53)
˘ (Y )δ Y =
Ψ,...,Ψ
To demonstrate (3.53), it is sufficient to prove it for s = 2. Abbreviate
1
2
1
2
f (Y , Y ) = f 1 2 (Y , Y ). First note that an induction proof on n̆ (see Section K.1)
Ψ,Ψ
shows that
∫
=
f˘({y̆1 , ..., y̆n̆ })dy̆1 · · · dy̆n̆
∫
∑
1
1
2
2
Cn+n′ ,n f˘({y1 , ..., yn , y1 , ..., yn′ })
(3.54)
n+n′ =n̆
1
1
2
2
·dy1 · · · dyn dy1 · · · dyn′
where Cn+n′ ,n is the binomial coefficient as defined in (2.1). Thus
∫
∑ 1 ∫
f˘(Y˘ )δ Y˘ =
f˘({y̆1 , ..., y̆n̆ })dy̆1 · · · dy̆n̆
n̆!
n̆≥0
∑ ∑ Cn+n′ ,n ∫
1
1
2
2
=
f ({y1 , ..., yn }, {y1 , ..., yn′ })
′ )!
(n
+
n
′
(3.55)
(3.56)
n̆≥0 n+n =n̆
1
=
1
2
2
·dy1 · · · dyn dy1 · · · dyn′
∫
∑
1
1
1
2
2
f ({y1 , ..., yn }, {y1 , ..., yn′ })
′!
n!
·
n
∗
(3.57)
n,n≥0
1
=
1
2
2
·dy1 · · · dyn dy1 · · · dyn′
∫
1
2
1
2
f (Y , Y )δ Y δ Y .
(3.58)
Multiobject Calculus
3.5.4
73
Constant Rule
Let K be a constant functional or constant set function. Then
3.5.5
δ
K = 0.
δY
(3.59)
δ
δF1
δF2
(a1 F1 [h] + a2 F2 [h]) = a1
[h] + a2
[h].
δY
δY
δY
(3.60)
Sum Rule
• Functional form:
• Set-function form:
δ
δϕ1
δϕ2
(a1 ϕ1 (T ) + a2 ϕ2 (T )) = a1
(T ) + a2
(T ).
δY
δY
δY
3.5.6
Linear Rule
• Functional form: Let f [h] =
([179], Eq. (11.261)):
∫
h(y) · f (y)dy be a linear functional. Then

 f [h]
δ
f (y)
f [h] =

δY
0
• Set-function form: Let p(T ) =
∫
T
if
if
if
Y =∅
Y = {y} .
|Y | ≥ 2
(3.62)
f (y)dy. Then ([179], Eq. (11.260)):

 p(T )
δ
f (y)
p(T ) =

δY
0
3.5.7
(3.61)
if
if
if
Y =∅
Y = {y} .
|Y | ≥ 2
(3.63)
Monomial Rule
∫
Let f [h] = h(y) · f (y)dy be a linear functional and let N ≥ 0 be a nonnegative
integer. Then ([179], Eq. (11.11.263)):
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Advances in Statistical Multisource-Multitarget Information Fusion
• Functional form:

f [h]N

δ
N
f [h] =
|Y |! · CN,|Y | · f [h]N −|Y | · f Y

δY
0
if
if
if
Y =∅
0 < |Y | ≤ N .
|Y | > N
(3.64)
was defined in (3.5) and where
where the power-functional notation f Y
CN,n was defined in (2.1).
∫
• Set-function form: Let p(T ) = T f (y)dy and N ≥ 0 a nonnegative
integer. Then ([179], Eq. (11.11.262)):

p(T )N
if
Y =∅

δ
N
p(T ) =
|Y |! · CN,|Y | · p(T )N −|Y | · f Y if 0 < |Y | ≤ N .

δY
0
if
|Y | > N
(3.65)
3.5.8
Power Rule
• Functional form: Let F [h] be a functional and a be a real number. Then
([179], Eq. (11.265)):
δ
δF
F [h]a = a · F [h]a−1 ·
[h].
δy
δy
(3.66)
• Set function form: Let ϕ(T ) be a set function and a be a real number. Then
([179], Eq. (11.264)):
δ
δϕ
ϕ(T )a = a · ϕ(T )a−1 ·
(T ).
δy
δy
3.5.9
(3.67)
Product Rules
• Functional form: Let F [h] = F1 [h] · · · Fn [h] be a product of functionals
and Y ⊆ Y be a finite subset. Then ([179], Eq. (11.274)):
δF
[h] =
δY
∑
δF1
δFn
[h] · · ·
[h]
δW1
δWn
(3.68)
W1 ⊎...⊎Wn =Y
where the summation is taken over all mutually disjoint subsets W1 , ..., Wn
of Y (the empty subset included) such that W1 ∪ ... ∪ Wn = Y .
Multiobject Calculus
75
• Set function form: Let ϕ(T ) = ϕ1 (T ) · · · ϕn (T ) be a product of set functions
and Y ⊆ Y be a finite subset. Then ([179], Eq. (11.273)):
∑
δϕ
(T ) =
δY
δϕ1
δϕn
(T ) · · ·
(T ).
δW1
δWn
(3.69)
W1 ⊎...⊎Wn =Y
When n = 2 the product rule becomes
∑ δF1
δF
δFn
[h] =
[h] ·
[h].
δY
δW
δ(Y − W )
(3.70)
W ⊆Y
If Y = {y} then (3.68) can be rewritten as
n
∑
δF
1
δFi
[h] = F [h]
·
[h].
δy
F [h] δy
i=1 i
(3.71)
δF
δF1
δF2
[h] =
[h] · F2 [h] + F1 [h] ·
[h].
δy
δy
δy
(3.72)
If in addition n = 2,
3.5.10
First Chain Rule
• Functional form: Let F1 [h], ..., Fn [h] be functionals and f (y1 , ..., yn ) a
real-valued function of real arguments y1 , ..., yn . Then ([179], Eq. (11.279)):
n
∑
δ
∂f
δFj
f (F1 [h], ..., Fn [h]) =
(F1 [h], ..., Fn [h]) ·
[h].
δy
∂y
δy
j
j=1
(3.73)
• Set function form: Let ϕ1 (T ), ...., ϕn (T ) be set functions and f (y1 , ..., yn )
a real-valued function of real arguments y1 , ..., yn . Then ([179], Eq.
(11.276)):
n
∑
δ
∂f
δϕj
f (ϕ1 (T ), ..., ϕn (T )) =
(ϕ1 (T ), ..., ϕn (T )) ·
(T ).
δy
∂y
δy
j
j=1
(3.74)
When n = 1 the first chain rule becomes
δ
df
δF
f (F [h]) =
(F [h]) ·
[h].
δy
dy
δy
(3.75)
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3.5.11
Second Chain Rule
• Functional form: Let Y be a vector space, F [h] be a functional, and
T : Y → Y be a nonsingular transformation. For any test function h(y)
define
(T −1 h)(y)
T
=
h(T (y))
(3.76)
Y
=
−1
(3.77)
JTY
=
−1
{T y| y ∈ Y }
∏
JT (y)
(3.78)
y∈Y
where JT (y) is the Jacobian determinant of T . Then ([179], Eq. (11.282)):
δ
1
δF
F [T −1 h] = Y ·
[T −1 h].
−1 Y )
δY
δ(T
JT
(3.79)
• Set function form: Let Y be a vector space, ϕ(S) a set function, and
T : Y → Y a nonsingular transformation. Further define
T −1 S = {T −1 y| y ∈ S}
Then ([179], Eq. (11.281)):
1
δϕ
δ
ϕ(T −1 S) = Y ·
(T −1 S).
δY
JT δ(T −1 Y )
3.5.12
(3.80)
Third Chain Rule
Functional form only: Let f (y) be a real-valued function of the real variable y
and let F [h] be a functional. Define
fh (y) = f (h(y))
(3.81)
for all y. Then ([179], Eq. (11.283)):
δ
δF
df
F [fh ] =
[fh ] ·
(h(y)).
δy
δy
dy
(3.82)
Multiobject Calculus
3.5.13
77
Fourth Chain Rule
The second and third chain rules are actually special cases of this more general
chain rule. Let h ?→ T [h] be a functional transformation (that is, it transforms test
functions h to test functions T [h]). Define the functional derivative of T [h] to
be the pointwise functional derivative. That is, for all y, w ∈ Y, define
δT
δ
T [h + εδy ](w) − T [h](w)
[h](w) =
T [h](w) = lim
.
ε→0
δy
δy
ε
(3.83)
Let F [h] be a functional. Then the following chain rule is true for functional
derivatives ([179], Eq. (11.285)):
∫
δ
δT
δF
F [T [h]] =
[h](w) ·
[T [h]]dw.
(3.84)
δy
δy
δw
Because of the linearity and continuity properties of Gâteaux derivatives, (3.31),
this can be rewritten as a Gâteaux derivative:
δ
∂F
) [T [h]].
F [T [h]] = (
δy
∂ δT [h]
(3.85)
δy
The notation on the right side of this equation means this:
[
]
∂F
∂F
(
) [h] =
[h]
.
∂g
g= δT
[h]
∂ δT
[h]
δy
δy
(3.86)
That is, first compute the gradient derivative (∂F/∂g)[h] of F [h] in the direction
of an arbitrary g(y). Then substitute the functional derivative (δT /δy)[h] of T [h]
at y in place of g.
As a simple example of (3.84), let T [h](x) = 1 − τ (x) + τ (x) · h(x) with
0 ≤ τ (x) ≤ 1 identically and let f (x) = F [1 − τ + τ · x]. Then
f ′ (x)
=
=
=
=
d
F [1 − τ + τ · x]
∫dx
d
δF
(1 − τ (x) + τ (x) · x) ·
[1 − τ + τ · x]dx
dx
δx
∫
δF
τ (x) ·
[1 − τ + τ · x]dx
δx
∂F
[1 − τ + τ · x]
∂τ
(3.87)
(3.88)
(3.89)
(3.90)
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where the final equation follows from (3.31).
3.5.14
Clark’s General Chain Rule
Equation (3.85) is a special case of the most general chain rule for functional
derivatives. This rule, a special case of a still more general chain rule due to D.
Clark [41], is as follows [47]:10
∑
δ
∂ |P| F
( δT
) [T [h]].
F [T [h]] =
δY
∂ W ∈P δW
[h]
(3.91)
P⊟Y
The summation is taken over all partitions P of Y . The notation ‘P ⊟ Y ’ is
shorthand for “P partitions Y into cells”; and |P| denotes the number of cells
in P. Also, the following notational convention is employed:
∂ W ∈{W1 ,...,Wl }
(
δT
[h]
δW
)
abbr.
= ∂
(
)
(
)
δT
δT
[h] · · · ∂
[h] .
δW1
δWl
(3.92)
A summary of the theory of partitions can be found in Appendix D. Equation
(3.91) is proved for functional derivatives in Section K.7 and, for the general case
of topological vector spaces, in [47].
Remark 5 (Faà di Bruno’s formula) Clark’s general chain rule is a generalization of the following general chain rule from vector calculus. Suppose that
f (x), g(x) are real-valued functions on RN . Then the directional derivatives
of the composite function f (g(x)) are
∂n
f (g(x)) =
∂y1 · · · ∂yn
∑
P⊟{1,...,n}
f (|P|) (g(x))
∏
I∈P
∏
∂ |I| g
(x),
i∈I ∂yi
(3.93)
where the
∏ summation is taken over all partitions P of the set {1, ..., n}; and
where
i∈I ∂yi = ∂yi1 · · · ∂yij if I = {i1 , ..., ij } ⊆ {1, ..., n} with |I| = j.
10 Clark’s general chain rule is very general, being true for the “chain derivative” of functions whose
arguments are in an arbitrary topological vector space. (The chain derivative generalizes the
Gâteaux differential.) In particular, the space of “test functions” h(x) on a Hausdorff, locally
compact, and completely separable space X is a topological vector space. It follows that the
general chain rule applies to derivatives of functionals whose arguments are test functions.
Multiobject Calculus
79
As a useful special case, let the functional transformation T [h] be realvalued—that is, it is a functional–and let F [h] = f (x) be an ordinary real-valued
function of a real variable h = x. Then (3.91) reduces to
(
∑
δ
f (T [h]) =
δY
∏ δT
[h]
δW
)
·
d|P| f
(T [h]).
dx|P|
(3.94)
W ∈P
P⊟Y
As a special case of this special case, let f (x) = x−1 . Assuming that
F [h] > 0 for all h, we get the following inverse rule for functionals (also originally
due to Clark):
∑
δ 1
=
δY F [h]
(
∏ δF
[h]
δW
)
(−1)|P| · |P|!
.
F [h]|P|+1
·
(3.95)
W ∈P
P⊟Y
Example 1 (Example of Clark’s general chain rule) As a concrete example of
the use of (3.91), let Y = {y1 , y2 , y3 } with |Y | = 3 and let
(3.96)
T [h] = 1 − ρ + ρ · h
where 0 ≤ ρ(y) ≤ 1 is some unitless function. There are five partitions of Y :
P1
=
{{y1 , y2 , y3 }},
P2 = {{y1 }, {y2 }, {y3 }}
(3.97)
P3
P5
=
=
{{y3 }, {y1 , y2 }},
{{y1 }, {y2 , y3 }}.
P4 = {{y2 }, {y1 , y3 }}
(3.98)
(3.99)
Thus (3.91) becomes
δ
F [T [h]]
δY
∂ |P1 | F
( δT
=
∂ W ∈P1
+
+
δW [h]
|P3 |
) [T [h]] +
∂ |P2 | F
( δT
∂ W ∈P2
δW [h]
|P4 |
) [T [h]] (3.100)
∂
F
∂
F
( δT
) [T [h]] +
( δT
) [T [h]]
∂ W ∈P3 δW
[h]
∂ W ∈P4 δW
[h]
∂ |P5 | F
( δT
∂ W ∈P5
δW [h]
) [T [h]]
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Advances in Statistical Multisource-Multitarget Information Fusion
∂F
=
∂
(
δ3 T
δy1 δy2 δy3 [h]
) [T [h]]
(3.101)
∂3F
) (
) (
) [T [h]]
+ (
δT
δT
δT
∂ δy
[h] ∂ δy
[h] ∂ δy
[h]
1
2
3
∂2F
) (
) [T [h]]
+ (
2T
δT
∂ δy
[h] ∂ δyδ1 δy
[h]
2
2
∂2F
) (
) [T [h]]
+ (
δT
δ2 T
∂ δy
[h]
∂
[h]
δy1 δy3
2
∂2F
) (
) [T [h]].
+ (
δT
δ2 T
∂ δy
[h]
∂
[h]
δy2 δy3
1
However,
δT
[h]
δw
=
T [h + εδw ] − T [h]
ε
(1 − ρ + ρ(h + εδw )) − (1 − ρ + ρh)
lim
ε→0
ε
ερδw
lim
= ρδw = ρ(w) · δw
ε→0
ε
lim
(3.102)
ε→0
=
=
(3.103)
(3.104)
and so
δT
[h] = 0
δW
(3.105)
for all |W | > 1. Thus
δ
F [T [h]]
δY
=
=
=
=
∂3F
) (
) (
) [T [h]]
δT
δT
δT
∂ δy
[h]
∂
[h]
∂
[h]
δy
δy
1
2
3
(
(3.106)
∂3F
[T [h]] (3.107)
∂ (ρ(y1 )δy1 ) ∂ (ρ(y2 )δy2 ) ∂ (ρ(y3 )δy3 )
δ3F
ρ(y1 ) · ρ(y2 ) · ρ(y3 ) ·
[T [h]]
(3.108)
δy1 δy2 δy3
δF
ρY ·
[1 − ρ + ρh].
(3.109)
δY
Chapter 4
Multiobject Statistics
4.1
INTRODUCTION
This chapter summarizes the basic concepts of multiobject statistics: the fundamental and ancillary multiobject statistical descriptors; and the most important multiobject processes. It is organized as follows:
1. Section 4.2: The basic fundamental and ancillary multiobject statistical
descriptors—belief-mass functions, multiobject probability densities, probability generating functionals (p.g.fl.’s), cardinality distributions, probability
generating functions (p.g.f.’s), probability density functions (PHDs), and multiobject factorial moment densities.
2. Section 4.3: Important multiobject processes—Poisson, independently distributed cluster (i.i.d.c.), multi-Bernoulli, and Bernoulli.
3. Section 4.4: Basic ancillary multiobject processes—censored processes and
cluster processes.
4.2
BASIC MULTIOBJECT STATISTICAL DESCRIPTORS
The statistics of an RFS Ψ ⊆ Y are completely specified by three mathematically
equivalent fundamental statistical descriptors, which will be described in this
section.
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1. Section 4.2.1: The belief-mass function βΨ (T )—central to the derivation
of multitarget Markov densities and likelihood functions from multitarget
motion and measurement models (to be described in Sections 5.4 and 5.5).
2. Section 4.2.2: The multiobject probability density function fΨ (Y )—the
fundamental concept for multisensor-multitarget Bayes filtering, and thus for
multisensor-multitarget detection, tracking, and identification.
3. Section 4.2.3: Convolution and deconvolution formulas for multiobject probability density functions.
4. Section 4.2.4: The probability generating functional GΨ [h] or p.g.fl.—
central to the derivation of approximate multisensor-multitarget Bayes filters
such as the PHD filter, CPHD filter, multi-Bernoulli filter, and others (to be
described in Section 5.10).
5. Section 4.2.5: Multivariate p.g.fl.’s GΨ1 ,...,Ψn [h1 , ..., hn ]—important in
the derivation of RFS approximate multitarget filters and in multisensor
problems.
In addition to these fundamental descriptors, various ancillary statistics are
also important:
1. Section 4.2.6: The cardinality distribution pΨ (n)—the probability distribution of the number |Ψ| of elements in Ψ.
2. Section 4.2.7: The probability generating function GΨ (y)—an equivalent
representation of pΨ (n).
3. Section 4.2.8: The probability hypothesis density DΨ (y)—essentially, the
density of objects in Ψ at y.
4. Section 4.2.9: The multiobject factorial moment density DΨ (Y ) (a generalization of the PHD to arbitrary finite sets Y ).
In addition, the following three sections describe important relationships
between the statistical descriptors:
1. Section 4.2.10: The equivalence of the fundamental descriptors (p.g.fl.,
belief-mass function, and multitarget probability density function).
2. Section 4.2.11: Radon-Nikodým-type relationships between the fundamental
descriptors.
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83
3. Section 4.2.12: Linear and quadratic Campbell’s theorems.
4.2.1
Belief-Mass Functions
Let Ψ ⊆ Y be an RFS. Then its belief-mass function is defined as
βΨ (T ) = Pr(Ψ ⊆ T )
(4.1)
for all measurable T .1
Suppose that Ψ1 , ..., Ψn ⊆ Y are independent RFSs and let Ψ = Ψ1 ∪ ... ∪
Ψn . Then
βΨ (T ) = βΨ1 (T ) · · · βΨb (T ).
(4.2)
Belief-mass functions are generalizations of probability-mass functions (Section 2.2.3). Let Ψ = { Y } as in (2.37), where Y ∈ Y is a random variable.
Then
βΨ (T ) = Pr({Y} ⊆ T ) = Pr(Y ∈ T ) = pY (T ).
(4.3)
As another example, consider the “twinkling” random singleton of (2.39), and
assume that Y and ∅q are independent. Then
βΨ (T )
=
=
=
Pr({Y} ∩ ∅q ⊆ T )
Pr(∅q = ∅) + Pr(∅q ̸= ∅) · Pr(Y ∈ T )
1 − q + q · pΨ (T ).
(4.4)
(4.5)
(4.6)
As a final example, consider (2.38): Ψ = {Y1 , ..., Yn } where Y1 , ..., Yn
are independent:
βΨ (T )
1
=
=
Pr({Y1 , ..., Yn } ⊆ T ) = Pr(Y1 ∈ T, ..., Yn ∈ T )
pY1 (T ) · · · pYn (T ).
(4.7)
(4.8)
The belief-mass function is also known as a belief measure. Because of the Choquet-Matheron
theorem, it completely characterizes the statistics of the RFS Ψ ([179], p. 713) under the FellMatheron hit-and-miss topology. For this reason—as well as because of its utility in transforming
multitarget motion and measurement models into multitarget Markov densities and likelihood
functions—it is central to the finite-set statistics approach.
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4.2.2
Multiobject Probability Density Functions
A multiobject probability density function is a multiobject density function f (Y )
(see Section 3.2.4) whose set integral is unity:
∫
f (Y )δY = 1.
(4.9)
Refer to (3.19) for a simple example of a multiobject probability distribution.
If Ψ ⊆ Y is an RFS, its multiobject probability distribution can, if it exists, be
derived from the belief-mass function as follows:2
fΨ (Y ) =
δβΨ
(∅).
δY
(4.10)
Intuitively speaking, fΨ (Y ) is the probability (density) of the event Ψ = Y .3
As an example, consider the belief-mass function of the “twinkling random
singleton,” as in (4.6). The corresponding multitarget probability distribution is
given by
fΨ (∅)
=
fΨ ({y})
=
fΨ (Y )
=
δβΨ
(∅) = βΨ (∅) = 1 − q
δ∅
δβΨ
(∅) = q · fY (y)
δy
0 if |Y | > 1
(4.11)
(4.12)
(4.13)
or, in summary, by
Y
fΨ (Y ) = C1,|Y | · q |Y | (1 − q)|Y | · fY
Y
where the power functional notation fY
coefficient C1,|Y | in (2.1).
2
(4.14)
was defined in (3.5) and the binomial
In point process theory, the density functions
n! · jΨ,n (y1 , ..., yn ) = fΨ ({y1 , ..., yn })
3
are known as the “Janossy densities” [55], [56] of the point process Ψ. They have also been
called “joint multitarget probability densities” or “JMPDs” ([214], p. 27). If they exist (that is, are
integrable finite-valued functions) then Ψ is a “simple” point process ([55], p. 138, Prop. 5.4.V)—
meaning that jΨ,n (y1 , ..., yn ) = 0 whenever yi = yj for some i ̸= j. That is, a simple point
process is essentially the same thing as an RFS. See Section 2.3.1.
If Y is continuously infinite, this is only an intuitive interpretation since, of course, in that case
Ψ = Y is a zero-probability event.
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85
Another example is the multitarget Dirac delta density ([179], p. 366, Eq.
(11.124)). Let Y ′ = {y1′ , ..., yn′ } with |Y ′ | = n and Y = {y1 , ..., yn } with
|Y | = n. Then this density is defined as
∑
′ (y1 ) · · · δy ′ (yn )
δY ′ (Y ) = δ|Y ′ |,|Y |
δyπ1
(4.15)
πn
π
where the summation is taken over all permutations π on the numbers 1, ..., n.
A final example is the multitarget uniform distribution. Let Y = RN and
S ⊆ Y a bounded closed subset with (hyper)volume |S|. Suppose that there can
be no more than n0 objects in S. If Y = {y1 , ..., yn } with |Y | = n then this
distribution is defined as ([94], p. 144; [179], p. 367):
uS,nmax (Y ) =
|Y |! · 1YS · 1|Y |≤n0 +1
|S||Y | · (n0 + 1)
(4.16)
where 1S (y) is the indicator function of S; where 1n≤n0 +1 = 1 if n ≤ n0 + 1
and 1n≤n0 +1 = 0 otherwise; and where the power functional notation hY was
defined in (3.5).
4.2.3
Convolution and Deconvolution
Let Ψ1 , ..., Ψn ⊆ Y
product rule, (3.68),
be statistically independent RFSs. Then by the general
fΨ (Y ) =
∑
fΨ1 (W1 ) · · · fΨn (Wn )
(4.17)
W1 ⊎...⊎Wn =Y
where the summation is taken over all mutually disjoint (and possibly empty)
subsets W1 , ..., Wn of Y , such that W1 ∪ ... ∪ Wn = Y . If n = 2, this
reduces to the convolution-like formula
∑
fΨ (Y ) =
fΨ1 (W ) · fΨ2 (Y − W ).
(4.18)
W ⊆Y
For this reason, (4.17) is called the fundamental convolution theorem for independent RFSs.
As an example, let Y = {y1 , y2 } with y1 ̸= y2 . Then
fΨ (Y )
=
fΨ1 (∅) · fΨ2 ({y1 , y2 }) + fΨ1 ({y1 }) · fΨ2 ({y2 })
(4.19)
+fΨ1 ({y2 }) · fΨ2 ({y1 }) + fΨ1 ({y1 , y2 }) · fΨ2 (∅).
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Next, consider the problem that is inverse to (4.18), also known as the
deconvolution problem:
• fΨ (X) and fΨ1 (X) are known, and we are to determine fΨ2 (X).
Using the quotient rule for functionals, (3.95), D. Clark has answered this
question as follows:
fΨ2 (X) =
∑ ∑ (−1)|P| · |P|!
W ⊆X P⊟W
fΨ1
(∅)|P|+1
· fΨ (X − W ) ·
∏
fΨ1 (V ).
(4.20)
V ∈P
A proof of this fact can be found in Section K.2.
4.2.4
Probability Generating Functionals (p.g.fl.’s)
Let h(y) be a “test function” on Y—that is, h(y) is unitless and 0 ≤ h(y) ≤ 1.4
Let Ψ ⊆ Y be an RFS. Then its probability generating functional (p.g.fl.) is the
functional defined as the expected value5
∫
GΨ [h] = E[hΨ ] = hY · fΨ (Y )δY,
(4.21)
where the power functional hY was defined in (3.5). The p.g.fl. of Ψ has the
following properties:
0
GΨ [1]
GΨ [1T ]
≤
=
=
GΨ [h] ≤ 1
1
βΨ (T ) = Pr(Ψ ⊆ T ).
Also, suppose that Ψ1 , ..., Ψn ⊆ Y are independent RFSs and let
Ψ1 ∪ ... ∪ Ψn . Then the p.g.fl. factors as follows:
GΨ [h] = GΨ1 [h] · · · GΨb [h].
4
5
(4.22)
(4.23)
(4.24)
Ψ =
(4.25)
It is sometimes additionally required that the set {y| h(y) ̸= 0} be closed and bounded.
The formula for the p.g.fl. goes back to Volterra in 1927, who called it a “functional power series”
([317], p. 21, Eq. (1)). Daley and Vere-Jones attribute the introduction of the p.g.fl., in a point
process sense, to the Russian physicist Bogoliubov in 1946 ([56], p.15). Moyal in 1962 attributed
the p.g.fl. and other point process generating functionals to Bartlett and Kendall in the early 1950s
([207], footnote 1, p. 13). The p.g.fl. has since become a textbook-level concept in point process
theory [55], [56].
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87
As an example, the p.g.fl. of the “twinkling random singleton,” whose multiobject probability density function was given in (4.11) through (4.13), is
∫
GΨ [h] =
hY · fΨ (Y )δY
(4.26)
∫
= 1 · f (∅) + h(y) · f ({y})dy + 0
(4.27)
∫
= 1 − q + q h(y) · fY (y)dy
(4.28)
=
1 − q + q · fY [h].
(4.29)
As another example, the p.g.fl. of the multiobject probability density function
of (3.19) is
Gf [h] = (1 − q + q · f1 [h]) · (1 − q + q · f2 [h]) .
(4.30)
For,
Gf [h]
=
=
=
=
=
∫
hY · f (Y )δY
(4.31)
∫
1 · f (∅) + h(y) · f ({y})dy
(4.32)
∫
1
+
h(y1 ) · h(y2 ) · f ({y1 , y2 })dy1 dy2 + 0
2
∫
2
(1 − q) + q (1 − q) h(y) · (f1 (y) + f2 (y)) dy
(4.33)
∫
q2
+
h(y1 ) · h(y2 ) · (f1 (y1 ) · f2 (y2 ) + f1 (y2 ) · f2 (y1 )) dy1 dy2
2
(1 − q)2 + q (1 − q) · (f1 [h] + f2 [h]) + q 2 · f1 [h] · f2 [h]
(4.34)
(1 − q + q · f1 [h]) · (1 − q + q · f2 [h]) .
(4.35)
Remark 6 (Constructing p.g.fl.’s from b.m.f.’s) As a heuristic rule of thumb, a
formula for the p.g.fl. can be constructed from a formula for the belief-mass
function. Rewrite the formula for βΨ (T ) in the variable T as a formula β[1T ]
in the variable 1T . Substituting h for 1T , we get a formula GΨ [h] = β[h] for
the p.g.fl.
Remark 7 (p.g.fl. as a probability) The belief-mass function of an RFS Ψ is
defined directly as a probability: βΨ (T ) = Pr(Ψ ⊆ T ). It turns out that p.g.fl.’s
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Advances in Statistical Multisource-Multitarget Information Fusion
can be regarded as generalized belief-mass functions, as follows ([179], p. 373):
(4.36)
GΨ [h] = Pr(Ψ ⊆ Σα (h)).
Here, α(y) is a uniformly distributed random number in [0, 1] for every y;
def.
Σα (h) = {y| α(y) ≤ h(y)}; and it is assumed that, for any n ≥ 2 and any
distinct y1 , ..., yn , the random variables α(y1 ), ..., α(yn ), Ψ are independent.
The random set Σα (h) is the “asynchronous” random set representation of the
fuzzy membership function h(x)—see (22.38).
Remark 8 (Why emphasize the p.g.fl.?) The p.g.fl. is only one among many fundamental functional statistical descriptors that are employed in point process theory. Others include the characteristic functional, the Laplace functional, and the
factorial moment generating functional [55], [56], [207]. So why is the p.g.fl. emphasized in finite-set statistics in preference to these others? Different functional
descriptors are useful for different purposes, and it turns out that the p.g.fl. is especially useful for multitarget tracking theory and practice. This is partly because
of its close relationship with the belief-mass function, which it generalizes because
of (4.36). But it is useful also partly because so many other statistical descriptors,
such as multiobject probability distributions and the PHD, are easily related to it
via the multiobject calculus.
Remark 9 (Choquet integrals) In (4.21), suppose that we replace the product hY
by its fuzzy logic analog, miny∈Y h(y). Then
)
∫
∫ (
def.
h · dβΨ =
min h(y) · fΨ (Y )δY
(4.37)
y∈Y
is known as the “Choquet integral” of the function h, with respect to the
nonadditive measure (belief-mass function) βΨ (T ) (see [94], p. 179).
4.2.5
Multivariate p.g.fl.’s
1
s
1
s
Let Y1 , ..., Y be spaces, let Ψ, ..., Ψ be respective RFSs on these spaces, and
1
let f 1
s
s
i
(Y , ..., Y ) be their joint multiobject probability distribution, where Y
Ψ1 ,...,Ψ
1
i
s
1
s
denotes a finite subset of Y. Let h(y), ..., h(y) be respective test functions on
1
s
i
i
Y1 , ..., Y, where y ∈ Y. Then the joint multivariate p.g.fl. of the joint process
1
s
1
s
˘ = Ψ ⊎ ... ⊎ Ψ ⊆ Ψ ⊎ ... ⊎ Ψ
Ψ
(4.38)
Multiobject Statistics
89
(where ‘⊎’ denotes disjoint union) is the expected value
1
G1
s
s
(4.39)
[h, ..., h]
Ψ1 ,...,Ψ
1 1
=
=
s s
E[hΨ · · · hΨ ]
∫ 1
1
s s
hY · · · hY · f 1
1
s
Ψ,...,Ψ
s
1
s
(4.40)
(Y , ..., Y )δ Y · · · δ Y .
˘
The joint multivariate p.g.fl. is equivalent to the single-variate p.g.fl. GΨ
˘ [h]
˘
˘
of the RFS Ψ = Ψ1 ⊎ ... ⊎ Ψn . For, let h(y̆) be a test function on the joint space
i i
i
˘
˘ = Y ⊎ ... ⊎ Yn , with h(y̆)
Y
= h(y) when y̆ = y. Then
1
1
˘
GΨ
˘ [h] = G 1
s
s
(4.41)
[h, ..., h].
Ψ,...,Ψ
For, from (3.52),
˘
GΨ
˘ [h]
=
∫
˘ Y˘ · f ˘ (Y˘ )δ Y˘
h
Ψ
=
∫
˘ Y ⊎...⊎Y · f 1
h
1
(4.42)
s
1
s
Ψ,...,Ψ
=
∫
1
s
1
˘Y · · · h
˘Y · f 1
h
s
Ψ,...,Ψ
=
∫
1 1
s s
s
Ψ,...,Ψ
1
=
G1
s
1
s
s
1
s
1
(4.43)
s
(Y , ..., Y )δ Y · · · δ Y
1
hY · · · hY · f 1
s
(Y , ..., Y )δ Y · · · δ Y
(4.44)
s
(Y , ..., Y )δ Y · · · δ Y
(4.45)
s
[h, ..., h].
(4.46)
Ψ,...,Ψ
Two examples of multivariate p.g.fl.’s will be considered: joint multitarget, single-sensor multivariate p.g.fl.’s (Section 4.2.5.1) and joint multitargetmultisensor multivariate p.g.fl.’s (Section 4.2.5.2).
4.2.5.1
Example: Joint Target/Sensor p.g.fl.’s
The following bivariate p.g.fl. plays a central role in the principled approximation of
RFS filters. Let Ξk+1|k ⊆ X be the random predicted multitarget state set at time
tk+1 , and let Σk+1 ⊆ Z be the random multitarget measurement set at time tk+1 .
Assume that current measurements do not depend on the previous measurement
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Advances in Statistical Multisource-Multitarget Information Fusion
history:
fΣk+1 |Ξk+1|k (Z|X) = fΣk+1 |Ξk+1|k ,Σ1 ,...,Σk (Z|X, Z1 , ..., Zk ).
(4.47)
Let
fk+1 (Z|X) abbr.
= fΣk+1 |Ξk+1|k (Z|X)
(4.48)
be the multitarget likelihood function at time tk+1 and let
fk+1|k (X|Z (k) ) abbr.
= fΞk+1|k |Σ1 ,...,Σk (X|Z1 , ..., Zk )
(4.49)
be the distribution of the predicted target process Ξk+1|k , conditioned on the
measurement-stream Z (k) . Then the joint distribution of Σk+1 and Ξk+1|k is
fΣk+1 ,Ξk+1|k (Z, X)
=
fk+1 (Z|X) · fk+1|k (X|Z (k) )
(4.50)
=
(k)
(4.51)
fk+1 (Z, X|Z
).
Thus the joint single-sensor, multitarget p.g.fl. of Ξk+1|k and Σk+1 , conditioned
on Z (k) , is the bivariate p.g.fl.
def.
F [g, h]
=
=
=
=
GΣk+1 ,Ξk+1|k |Σ1 ,...,Σk [g, h|Z (k) ]
∫
g Z · hX · fk+1 (Z, X|Z (k) )δZδX
∫
g Z · hX · fk+1 (Z|X) · fk|k (X|Z (k) )δZδX
∫
hX · Gk+1 [g|X] · fk|k (X|Z (k) )δZδX
where
Gk+1 [g|X] =
∫
g Z · fk+1 (Z|X)δZ
(4.52)
(4.53)
(4.54)
(4.55)
(4.56)
is the p.g.fl. of fk+1 (Z|X). Note that F [g, h] is normal—that is, F [1, 1] = 1—
and is, therefore, a p.g.fl.
4.2.5.2
Example: Joint Target/Multisensor p.g.fl.’s
Generalize the previous example by assuming that, instead of a single sensor, we
1
have s sensors with respective measurement spaces
s
Z, ..., Z. At time tk+1 ,
Multiobject Statistics
91
1
s
suppose that we have respective random measurement sets Σk+1 , ..., Σk+1 . If
the sensors are independent (measurements are conditionally independent of target
states), then
1
1
s
s
1
s
(4.57)
fk+1 (Z, ..., Z|X) = f k+1 (Z|X) · · · f k+1 (Z|X).
1
s
Thus the joint multitarget-multisensor p.g.fl. of Ξk+1|k , Σk+1 , ..., Σk+1 is
1
=
∫
=
·fk+1|k (X|Z (k) )δ Z · · · δ ZδX
∫
1
s
1
s
hX · Gk+1 [g|X] · · · Gk+1 [g|X]
s
F [g, ..., g, h]
1
s
1
s
1
s
g Z · · · g Z · hX · f k+1 (Z|X) · · · f k+1 (Z|X) (4.58)
1
s
(4.59)
·fk+1|k (X|Z (k) )δX
where
j
j
Gk+1 [g|X] =
∫
j
j
j
j
j
g Z · f k+1 (Z|X)δ Z.
(4.60)
Remark 10 (An erroneous factorization) The claim has been made, in Eq. (20) of
1
s
[282], that, if the sensors are independent, then F [g, ..., g, h] factors as follows:
1
s
1
s
(4.61)
F [g, ..., g, h] = F [g, h] · · · F [g, h].
This equation is true only if s = 1—that is, only when there is a single sensor.6
Furthermore, it is untrue even under the assumptions made in [282]—namely, that
the multiobject state process has a special “traffic process” form. Specifically, a
“traffic process” Ξ ⊆ X with PHD DΞ (x) is defined as follows: (1) it has the
1
s
1
j
s
form Ξ = Ξ∪...∪ Ξ; (2) the Ξ, ..., Ξ are independent; (3) each Ξ is Poisson with
j
ℓ
PHD D j (x) = β(x)·DΞ (x) ([282], Eq. (17)); and (4) the β(x) are nonnegative
Ξ
∑s ℓ
functions such that
ℓ=1 β(x) = 1 identically for all x. The erroneousness of
(4.61) can be demonstrated via a counterexample—see Section K.3.
6
As a consequence, the claimed main result of [282]—that is, the intensity-function measurementupdate in Eq. (26)—is untrue. In particular, the claimed computational linearity of Eq. (26) in
the number s of sensors is spurious, because Eq. (26) is not, as claimed, a theoretically valid
measurement-update equation.
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4.2.6
Advances in Statistical Multisource-Multitarget Information Fusion
Cardinality Distributions
The cardinality distribution of an RFS Ψ ⊆ Y is
pΨ (n)
=
=
=
Pr(|Ψ| = n)
∫
fΨ (Y )δY
|Y |=n
∫
1
fΨ ({x1 , ..., dxn })dx1 · · · dxn .
n!
(4.62)
(4.63)
(4.64)
The number pΨ (n) is the probability that Ψ contains n elements.
As an example, consider the multiobject probability density function of
(4.14). Then its cardinality distribution is, for n = 0, 1,
pΨ (n) = C1,n · q n (1 − q)1−n
(4.65)
where Cn′ ,n is the binomial coefficient as defined in (2.1).
4.2.7
Probability Generating Functions (p.g.f.’s)
Setting h = y where 0 ≤ y ≤ 1 is a scalar, the p.g.fl. reduces to the probability
generating function (p.g.f.) of Ψ:
∫
∑
GΨ (y) = [GΨ [h]]h=y = y |Y | · fΨ (Y )δY =
pΨ (n) · y n .
(4.66)
n≥0
The p.g.f. is so-named because, for all n ≥ 0, the cardinality distribution can be
generated from it,
1 (n)
pΨ (n) = GΨ (0),
(4.67)
n!
(n)
where GΨ (y) denotes the nth derivative of GΨ (y).
The expected value µΨ , the second factorial moment µΨ,2 , and the variance
2
σΨ
of GΨ (y) (equivalently, of pΨ (n)) are given by:
∑
(1)
µΨ = GΨ (1) =
n · pΨ (n)
(4.68)
n≥1
µΨ,2
=
(2)
GΨ (1) =
∑
(4.69)
n(n − 1) · pΨ (n)
n≥2
2
σΨ
=
µΨ,2 − µ2Ψ + µΨ = −µ2Ψ +
∑
n≥1
n2 · pΨ (n).
(4.70)
Multiobject Statistics
93
Suppose that Ψ1 , ..., Ψn ⊆ Y are independent RFSs and let Ψ = Ψ1 ∪ ... ∪
Ψn . Then the p.g.f. factors as
GΨ (y) = GΨ1 (y) · · · GΨn (y)
(4.71)
and the mean and variance of Ψ are given by
µΨ
2
σΨ
4.2.8
=
=
µΨ1 + ... + µΨn
2
2
σΨ
+ ... + σΨ
.
1
n
(4.72)
(4.73)
Probability Hypothesis Densities (PHDs)
The probability hypothesis density (PHD) of an RFS Ψ is an ordinary density
function DΨ (y) on single objects y ∈ Y (see [179], Section 16.2). Intuitively
speaking:
• The number DΨ (y) is the density of the objects at y.
• DΨ (y)dy is the number of objects contained in the infinitesimal region dy
centered at y.
• DΨ (y) is the probability (density) of the zero-probability event y ∈ Ψ,
in the same sense that an ordinary probability density function fY (y) of a
random variable Y ∈ Y is the probability (density) of the zero-probability
event Y = y. (If Y is a discrete space then DΨ (y) = Pr(y ∈ Ψ).) See
[179], pp. 576-580.
Formally, the PHD is defined as a set integral ([179], Eq. (16.26)):
)
∫
∫ (∑
DΨ (y) = fΨ ({y} ∪ W )δW =
δw (y) · fΨ (Y )δY.
(4.74)
w∈Y
The PHD can also be expressed in terms of functional derivatives and set derivatives
([179], Eqs. (16.35,16.36)):
DΨ (y) =
δGΨ
δ log GΨ
δβΨ
δ log βΨ
[1] =
[1] =
(Y) =
(Y).
δy
δy
δy
δy
(4.75)
The two logarithmic formulas are often to be preferred, since they often result in
algebraically simpler formulas.
The PHD has the following important properties:
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Advances in Statistical Multisource-Multitarget Information Fusion
• For any closed T ⊆ Y, the expected number E[|Ψ∩T |] of objects contained
in T is given in terms of the PHD by
∫
∫
|Ψ ∩ T | · fΨ (Y )δY =
DΨ (y)dy.
(4.76)
T
• In particular, if T = Y then the total expected number of objects E[|Ψ|] in
the scene is the integral of the PHD:
∫
NΨ = DΨ (y)dy.
(4.77)
• Suppose that Ψ1 , ..., Ψn are independent RFSs and let Ψ = Ψ1 ∪ ... ∪ Ψn .
Then
DΨ (y) = DΨ1 (y) + ... + DΨn (y).
(4.78)
As a simple example of a PHD, consider the multitarget probability density
f (Y ) of (3.19). Its PHD is given by
∫
1
1
D(y) =
f ({y}) +
f ({y, w})dw + 0
(4.79)
0!
1!
= q (1 − q) · (f1 (y) + f2 (y))
(4.80)
∫
+q 2 (f1 (y) · f2 (w) + f1 (w) · f2 (y)) dw
=
=
q (1 − q) · (f1 (y) + f2 (y)) + q 2 · (f1 (y) + f2 (y))
q · f1 (y) + q · f2 (y).
(4.81)
(4.82)
Remark 11 (PHDs: Terminology and interpretations) In point process theory,
the PHD is more commonly known as a “first-moment density,” “intensity density,”
or “intensity function” [55], [56]. To avoid possible confusion, the lattermost
usage has been avoided because of the very large number of alternative meanings
of “intensity” in engineering and physics. In classical thermodynamics, the PHD
is known as a “phase-space density” (see, for example, [235], p. 31). As such, the
PHD turns out to play a central role in the traffic-flow theory (TFT) approach to
multitarget tracking in urban environments [187]. The name “probability hypothesis density” is unique to multitarget tracking for historical reasons [165]. The
concept and name were first proposed, at an intuitive level, by M. Stein and C.
Winter [274], [275]. It was subsequently shown by Mahler to be the same thing as
a first-moment density in the point process sense ([94], pp. 168-169).
Multiobject Statistics
4.2.9
95
Factorial Moment Density
Let Ψ ⊆ Y be an RFS. Then the multitarget factorial moment density of the
multiobject distribution fΞ (Y ) is ([168],[165, p.1162, Eq. (60)], [179], Section
16.2.5):
∫
δGΨ
δβΨ
DΨ (Y ) = fΨ (Y ∪ W )δW =
[1] =
(Y).
(4.83)
δY
δY
It generalizes the concept of a PHD, since
DΨ (x) = DΨ ({x}).
(4.84)
The multiobject distribution fΨ (Y ) can be recovered from its factorial
moment density DΨ (Y ) via the following inversion formula (see [165, p.1162,
Eq. (60)], Eq. (68); and [179], Eq. (16.88)):
fΨ (Y ) =
∫
(−1)|W | · DΨ (Y ∪ W )δW.
(4.85)
The factorial moment density also obeys the identity ([168], p. 142, [55], p. 222):
GΨ [1 + h] =
4.2.10
∫
hY · DΨ (Y )δY.
(4.86)
Equivalence of the Fundamental Descriptors
Let Ψ ⊆ Y be an RFS. Then the following are all equivalent representations of
the probability law of Ψ: the probability density fΨ (Y ), the belief-mass function
βΨ (T ), and the p.g.fl. GΨ [h]. This is because they can all be recovered from each
other:
∫
βΨ (T ) =
fΨ (Y )δY = GΨ [1T ]
(4.87)
T
fΨ (Y )
=
GΨ [h]
=
δβΨ
δGΨ
(∅) =
[0]
δY
δY
∫
∫
δβΨ
hY · fΨ (Y )δY = hY ·
[0]δY.
δY
(4.88)
(4.89)
Thus fΨ (Y ), βΨ (T ), and GΨ [h] all completely characterize the statistics of Ψ
and can be used interchangeably without losing any information about Ψ.
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Advances in Statistical Multisource-Multitarget Information Fusion
4.2.11
Radon-Nikodým Formulas
Let Ψ ⊆ Y be an RFS and let fΨ (Y ), βΨ (T ), and GΨ [h] be, respectively,
its probability distribution, belief-mass function, and p.g.fl. The general RadonNikodým theorem for p.g.fl.’s is ([179], Eq. (11.251)):
∫
δGΨ
[h] = hY · fΨ (W ∪ Y )δY
(4.90)
δW
where the power functional hY was defined in (3.5).
A special case for belief-mass functions is obtained by setting h = 1T :
∫
δβΨ
(T ) =
fΨ (W ∪ Y )δY.
(4.91)
δW
T
When W = ∅ this reduces to ([179], Eq. (11.248)):
∫
βΨ (T ) =
fΨ (Y )δY.
(4.92)
T
This is the multiobject analog of the single-target equation
∫
pY (T ) =
fY (y)dy.
(4.93)
T
4.2.12
Campbell’s Theorems
These theorems show us how to simplify formulas involving sums of test functions.
Let us be given a function h(y). Then:
• Linear form of Campbell’s theorem: The expected value E[A] of the random
number
A = hΨ
(4.94)
where
hY abbr.
=
∑
h(y)
(4.95)
h(y) · DΨ (y)dy
(4.96)
y∈Y
is:
∫
hY · fΨ (Y )δY =
∫
where DΨ (y) is the PHD of Ψ as defined in (4.75). Equation (4.96) is
proved in Section K.6.
Multiobject Statistics
97
As an example, (4.76) is a direct consequence of Campbell’s theorem. For,
set h(y) = 1T (y) and let Y ⊆ Y be finite. Then
|Y ∩ T | =
∑
(4.97)
1T (y)
y∈Y
and so by Campbell’s theorem,
E[|Ψ ∩ T |]

∫
=
=
∫
=
∫

∑
y∈Y

1T (y) · fΨ (Y )δY
(4.98)
1T (y) · DΨ (y)dy
(4.99)
DΨ (y)dy.
(4.100)
T
• Quadratic form of Campbell’s theorem (scalar and vector versions): Given a
second test function h′ (y), the expected value E[A] of the random number

A = hΨ · h′Ψ = 
is
∫
hΨ · h′Ψ · fΨ (Y )δY
=
∑
y∈Ψ

h(y) 
∑
y∈Ψ

h′ (y)
(4.101)
∫
h(y) · h′ (y) · DΨ (y)dy
(4.102)
∫
+ h(y1 ) · h′ (y2 ) · DΨ ({y1 , y2 })dy1 dy2
where DΨ (Y ) is the factorial moment density of Ψ as defined in (4.83).
Equation (4.102) is proved in Section K.6. Equation (4.102) immediately
generalizes to the case when h(x) is vector-valued and thus A is matrixvalued:



∑
∑
A = hΨ · (h′Ψ )T = 
h(y) 
h′ (y)T 
(4.103)
y∈Ψ
y∈Ψ
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Advances in Statistical Multisource-Multitarget Information Fusion
4.3
IMPORTANT MULTIOBJECT PROCESSES
The purpose of this section is to introduce four families of RFSs that will have
prominent roles throughout the book:
1. Section 4.3.1: Poisson RFSs—central to the theory of PHD filters.
2. Section 4.3.2: i.i.d.c. RFSs—central to the theory of CPHD filters.
3. Section 4.3.3: Bernoulli RFSs—central to the theory of Bernoulli filters.
4. Section 4.3.4: Multi-Bernoulli RFSs—central to the theory of multi-Bernoulli
filters.
4.3.1
Poisson RFSs
Intuitively speaking, an RFS Ψ is Poisson if its instantiations are constructed as
follows. We are given a PHD DΨ (y) and therefore also the expected number of
targets
∫
NΨ =
DΨ (y)dy.
(4.104)
NΨn
n!
(4.105)
From the Poisson distribution
pΨ (n) = e−NΨ ·
draw an integer ν ∼ pΨ (·). Then from the spatial distribution of objects,
sΨ (y) =
DΨ (y)
,
NΨ
(4.106)
draw ν elements y1 , ..., yν ∼ sΨ (·). Given this, Ψ = {y1 , ..., yν } is a particular
instantiation of Ψ.
Stated somewhat differently, a Poisson RFS is one in which the objects of
Ψ are spatially distributed according to the distribution sΨ (y), and in which the
number |Ψ| of objects in Ψ is Poisson-distributed.
More formally, an RFS Ψ ⊆ Y is Poisson if its p.g.fl. has the form
GΨ [h] = eDΨ [h−1]
where the linear-functional notation DΨ [h − 1] was defined in (3.4).
Other statistics associated with a Poisson RFS Ψ are:
(4.107)
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99
• Multiobject probability distribution:
Y
fΨ (Y ) = e−NΨ · DΨ
(4.108)
Y
where the power-functional notation DΨ
was defined in (3.5).
• p.g.f.:
GΨ (y) = eNΨ ·(y−1) .
(4.109)
2
σΨ
= NΨ .
(4.110)
• Variance:
Remark 12 (“Independent increments” property) Let T1 , T2 ⊆ Y be closed
subsets such that T1 ∩ T2 = ∅. Let Ψ be Poisson with GΨ [h] = eDΨ [h−1] and
define two new RFSs
Ψ 1 = Ψ ∩ T1 ,
Ψ 2 = Ψ ∩ T2 .
(4.111)
Ordinarily, it should not be possible for Ψ1 and Ψ2 to be independent, since
they are both defined in terms of Ψ and thus are correlated. Because Ψ is
a Poisson RFS, however, Ψ1 and Ψ2 are actually independent. This follows
from the fact that Ψ1 , Ψ2 are Poisson with GΨ1 [h] = e(1T1 DΨ )[h−1] and
GΨ2 [h] = e(1T2 DΨ )[h−1] —see Section K.5.
4.3.2
Identical, Independently Distributed Cluster (i.i.d.c.) RFSs
Intuitively speaking, an RFS Ψ is i.i.d.c. if its instantiations are constructed as
follows. We are given a probability distribution pΨ (n) on the number of targets,
and a probability density sΨ (y) on the targets themselves. From pΨ (n), draw
an integer ν ∼ pΨ (·). Then from sΨ (y), draw ν elements y1 , ..., yν ∼ sΨ (·).
Then Ψ = {y1 , ..., yν } is a particular instantiation of Ψ.
Stated somewhat differently, an i.i.d.c. RFS is one in which the objects of
Ψ are spatially distributed according to the distribution sΨ (y), and in which the
probability distribution of the number of objects in Ψ is pΨ (n).
An i.i.d.c. RFS is thus a direct generalization of the concept of a Poisson RFS.
If the cardinality distribution pΨ (n) is Poisson then an i.i.d.c. RFS Ψ is a Poisson
RFS.
Formally, an RFS Ψ ⊆ Y is i.i.d.c. if its p.g.fl. has the form
GΨ [h] = GΨ (sΨ [h])
(4.112)
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where
GΨ (y) =
∑
pΨ (n) · y n
(4.113)
n≥0
is the p.g.f. of Ψ and where the linear-functional notation sΨ [h] was defined in
(3.4).
Other statistics associated with an i.i.d.c RFS Ψ are:
• Multiobject probability distribution:
fΨ (Y ) = |Y |! · pΨ (|Y |) · sYΨ
(4.114)
where the power functional notation sYΨ was defined in (3.5).
• Expected number of objects:
(1)
NΨ = GΨ (1) =
∑
n · pΨ (n).
(4.115)
n≥0
• PHD:
DΨ (y) = NΨ · sΨ (y).
• Variance:
(2)
2
σΨ
= GΨ (1) − NΨ2 + NΨ .
(4.116)
(4.117)
Note that the only interaction between the p.g.f. GΨ (y) and the PHD
DΨ (y) is given by (4.115). That is, the integral of the PHD and the expected
value of the cardinality distribution must be equal.
4.3.3
Bernoulli RFSs
An RFS Ψ is a Bernoulli RFS if |Ψ| ≤ 1, in which case we define
Pr(|Ψ| = 1) = qΨ .
(4.118)
It is characterized by two items: a spatial distribution (a probability density) sΨ (y)
and a probability of existence qΨ . The statistical descriptors of a Bernoulli RFS are:
• p.g.fl.:
GΨ [h] = 1 − qΨ + qΨ · sΨ [h].
(4.119)
Multiobject Statistics
101
• Multiobject probability distribution:
fΞ (Y ) =


1 − qΨ
qΨ · sΨ (y)

0
if
if
if
Y =∅
Y = {y} .
|Y | ≥ 2
(4.120)
• p.g.f.:
GΨ (y) = 1 − qΨ + qΨ · y.
(4.121)
DΨ (y) = qΨ · sΨ (y).
(4.122)
n
pΨ (n) = C1,n · qΨ
· (1 − qΨ )1−n
(4.123)
• PHD:
• Cardinality distribution:
where the binomial coefficient Cn,i was defined in (2.1).
• Expected number of objects:
NΨ = qΨ .
(4.124)
2
σΨ
= qΨ · (1 − qΨ ).
(4.125)
• Variance:
4.3.4
Multi-Bernoulli RFSs
An RFS is multi-Bernoulli if it is the union (superposition) of a finite number nΨ
of independent Bernoulli RFSs. Its p.g.fl. therefore has the form
(
)
nΨ
nΨ
1
1
GΨ [h] = 1 − qΨ
+ qΨ
· s1Ψ [h] · · · (1 − qΨ
+ qΨ
· snΨΨ [h]) .
(4.126)
A multi-Bernoulli process can be more intuitively understood as follows:
nΨ
1
• There are nΨ independent random objects YΨ
, ..., YΨ
∈ Y with respecnΨ
1
tive probability distributions sΨ (y), ..., sΨ (y) and respective probabilities
nΨ
1
of existence qΨ
, ..., qΨ
.
Let Y = {y1 , ..., yn } with |Y | = n. Then the other statistics associated
with a multi-Bernoulli RFS Ψ are:
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• Multiobject probability distribution:
(n
)
Ψ
∏
i
fΨ (Y ) =
(1 − qΨ )
(4.127)
i=1
·
∑
1≤i1 ̸=...̸=in ≤nΨ
i1
qΨ
· siΨ1 (y1 )
q in · sin (yn )
· · · Ψ Ψ in .
i1
1 − qΨ
1 − qΨ
• p.g.f.:
(
)
nΨ
nΨ
1
1
GΨ (y) = 1 − qΨ
+ qΨ
· y · · · (1 − qΨ
+ qΨ
· y) .
(4.128)
• PHD:
nΨ
1
DΨ (y) = qΨ
· s1Ψ (y) + ... + qΨ
· snΨΨ (y).
(4.129)
• Cardinality distribution: If n > nΨ then pΨ (n) = 0 and, if otherwise,
(n
)
( 1
)
Ψ
nΨ
∏
qΨ
qΨ
i
pΨ (n) =
(1 − qΨ ) · σnΨ ,n
(4.130)
1 , ..., 1 − q nΨ
1 − qΨ
Ψ
i=1
where σn,i (x1 , ..., xn ) is the elementary homogeneous symmetric function
of degree i in n variables.
• Expected number of objects:
nΨ
1
NΨ = qΨ
+ ... + qΨ
.
(4.131)
nΨ
nΨ
2
1
1
(1 − qΨ
).
σΨ
= qΨ
(1 − qΨ
) + ... + qΨ
(4.132)
• Variance:
Equation (4.127) follows from [179], Eq. (11.133).7
Note that the variance of a multi-Bernoulli RFS cannot exceed the expected
number of objects:
2
σΨ
≤ NΨ .
(4.133)
One consequence of this is that multi-Bernoulli filters cannot provide high-accuracy
approximations of the multitarget Bayes filter, if target number is being poorly
estimated.
Equation (3.19) provides a specific example of (4.127), assuming that nΨ = 2
1
2
and qΨ
= qΨ
= q.
7
Erratum: Eq. (11.134) in [179] does not follow from Eq. (11.133) and is not correct.
Multiobject Statistics
4.4
103
BASIC DERIVED RFSs
Certain important RFSs can be derived from those already considered. Two that
will play important roles later in this book are:
1. Section 4.4.1: Censored RFSs—important for the theory of target prioritization as described in Section 25.14.
2. Section 4.4.2: Cluster RFSs—important for the theory of extended targets
and group targets, as considered in Chapter 21.
4.4.1
Censored RFSs
Let Ψ ⊆ Y be an RFS and let T0 ⊆ Y be a closed subset. Then Ψ ∩ T0 is
also an RFS, one in which Ψ has been “censored” by excluding from it all of the
elements of T0c .
The statistical descriptors of Ψ ∩ T0 can be shown to be ([94], pp. 164-165;
[179], Eq. 14.302):
• Multiobject probability distribution:
fΨ∩T0 (Y ) = 1YT0 ·
δβΨ c
(T0 )
δY
(4.134)
where the power functional notation 1YT0 was defined in (3.5).
• p.g.fl.:
GΨ∩T0 [h] = GΨ [1 − 1T0 + h · 1T0 ].
(4.135)
• Belief-mass function:
βΨ∩T0 (T ) = βΨ∩T0 (T ∪ T0c ).
(4.136)
GΨ∩T0 (y) = GΨ [1 − 1T0 + y · 1T0 ].
(4.137)
• p.g.f.:
• Cardinality distribution:
∫
δ n βΨ
1
n
pΨ∩T0 (n) =
(T c )dy1 · · · dyn .
??
? δy1 · · · δyn 0
n! ?
T0 × ... × T0
(4.138)
• PHD:
DΨ∩T0 (x) = 1T0 (x) · DΨ (x).
(4.139)
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4.4.2
Cluster RFSs
Let us be given two spaces Y (a “mother space”) and D (a “daughter space”). A
cluster RFS is an RFS of D that consists of two parts:
• A mother RFS (also known as “germ process”) Ψ ⊆ Y.
• A family ∆y ⊆ D of daughter RFSs (also known as “grain process”),
parametrized by y ∈ Y.
It is usually assumed that, for any n ≥ 2 and any distinct y1 , ..., yn , the
∆y1 , ..., ∆yn are independent (“independent daughters”).
The cluster process determined by the mother and daughter processes is
defined to be the RFS ∆ of D defined by
∆=
∪
∆y .
(4.140)
y∈Ψ
That is, an instantiation of ∆ is constructed by
1. Drawing a sample Y = {y1 , ..., yn } ∼ fΨ (·) from the mother process;
2. Drawing samples Dyi = {dyi ,1 , ..., dyi ,n(yi ) } ∼ f∆yi (·) from the daughter
processes; and
3. Constructing the sample D = Dy1 ∪ ... ∪ Dyn of the cluster process.
Because of the independence assumption, it can be shown that the joint
probability distribution of Ψ and ∆ is
GΨ,∆ [g, h] = GΨ [h · G∆∗ [g]]
(4.141)
where GΨ [h] is the p.g.fl. of the mother RFS Ψ; where
abbr.
T [g](y) = G∆y [g] =
∫
g D · f∆y (D)δD
(4.142)
are the p.g.fl.’s of the daughter RFSs; and where GΨ [h · G∆∗ [g]] is shorthand for
GΨ [h · T [g]].
In turn, the p.g.fl. of the cluster process ∆ itself is
G∆ [g] = GΨ,∆ [g, 1] = GΨ [G∆∗ [g]].
(4.143)
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105
Equation (4.141) is true because
∫
GΨ,∆ [g, h] =
g D · hY · fΨ,∆ (Y, D)δDδY
∫
=
g D · hY · f∆|Ψ (D|Y ) · fΨ (Y )δDδY
∫
=
hY · G∆|Ψ [g|Y ] · fΨ (Y )δY
(4.144)
(4.145)
(4.146)
If Ψ = Y = {y1 , ..., yn } with |Y | = n, then G∆|Ψ [g|Y ] is the p.g.fl. of
∆ = ∆y1 ∪ ... ∪ ∆yn . Since the daughters are independent, G∆|Ψ [g|Y ] factors
into the product of the p.g.fl.’s of the daughters:
G∆|Ψ [g|Y ] =
n
∏
G∆yi [g] =
i=1
∏
T [g](y).
(4.147)
y∈Y
Thus
GΨ,∆ [g, h]
=
∫

hY · 
∏
y∈Y
=
∫
=
GΨ [h · T [g]].

T [g](y) · fΨ (Y )δY
(hT [g])Y · fΨ (Y )δDδY
(4.148)
(4.149)
(4.150)
Example 2 (Measurement RFSs) The random measurement set Σ generated by
a random target process Ξ is one example of a cluster RFS, with the state space X
being the mother space and the measurement space Z being the daughter space.
Let Σx be the set of measurements generated by a target with state x. Then Ξ
is the mother RFS, the Σx are the daughter RFSs, and the total cluster RFS is (see
(7.7)):
∪
Σ=
Σx .
(4.151)
x∈Ξ
See Remark 17 in Section 8.2 for a more concrete example.
Example 3 (Spawning models) The random target set Ξk+1|k of those targets
at time tk+1 that are spawned by a random target set Ξk|k at time tk is another
example of a cluster RFS. In this case the state space X is both the mother space
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Advances in Statistical Multisource-Multitarget Information Fusion
and the daughter space. Let Ξx be the set of targets that were spawned by a target
with state x. Then Ξk|k is the mother RFS, the Ξx are the daughter RFSs, and
the cluster RFS is
∪
Ξk+1|k =
Ξx .
(4.152)
x∈Ξk|k
Example 4 (Group targets) A group target (see Section 5.5) is a set of targets that
belong to a single coordinated tactical group. Let X be the state space for the
conventional targets, and let ˚
X be the state space for the group targets. Let
Ξx̊ ⊆ X be the RFS of targets associated with the group target with state x̊,
and let ˚
Ξ ⊆ X be the RFS of group targets. Then the RFS of all conventional
targets is
∪
Ξ=
Ξx̊ .
(4.153)
x̊∈˚
Ξ
Chapter 5
Multiobject Modeling and Filtering
5.1
INTRODUCTION
The finite-set statistics approach provides an explicit methodology for deriving the
optimal solutions for multisource-multitarget problems. It consists of three main
steps:
1. Step 1: Construction of multisensor-multitarget motion and measurement
models in terms of RFSs.
2. Step 2: Construction, using multiobject calculus, of “true” multitarget
Markov densities and “true” multisensor-multitarget likelihood functions
from these RFS models.
3. Step 3: Utilization of the true multitarget Markov density and the true
multitarget likelihood function in a multitarget Bayes filter.
The purpose of this chapter is to describe these three steps in greater detail. It
is organized as follows:
1. Section 5.2: The multisensor-multitarget Bayes filter.
2. Section 5.3: Multitarget Bayes optimality and Bayes-optimal multitarget
state estimators.
3. Section 5.4: RFS multitarget motion models.
4. Section 5.5: RFS multitarget measurement models.
5. Section 5.6: Multitarget Markov density functions.
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6. Section 5.7: Multisensor-multitarget likelihood functions.
7. Section 5.8: The multisensor-multitarget Bayes filter in p.g.fl. form.
8. Section 5.9: The Bayes filter for “mixed-state” systems—that is, systems
with a state variable of the form (x̊, X) where x̊ ∈ ˚
X is the state of a single
object and X ⊆ X is a finite set of conventional target states.
9. Section 5.10: A summary of principled approximate multisource-multitarget
filters—PHD, CPHD, multi-Bernoulli, and Bernoulli.
5.2
THE MULTISENSOR-MULTITARGET BAYES FILTER
The multisensor-multitarget recursive Bayes filter is the theoretical foundation for
multisensor-multitarget detection, tracking and identification ([179], Chapter 14).
Let Z (k) : Z1 , ..., Zk be a time sequence (sample path) of measurement sets
collected by all of the sensors. Then this filter propagates a multitarget posterior
distribution fk|k (X|Z (k) ) through time:1
fk+1|k+1 (X|Z (k+1) )
→ ...
It is defined by the time-update and measurement-update equations
∫
fk+1|k (X|Z (k) ) =
fk+1|k (X|X ′ ) · fk|k (X ′ |Z (k) )δX ′
(5.1)
... →
fk|k (X|Z (k) )
fk+1|k (X|Z (k) )
→
fk+1|k+1 (X|Z (k+1) )
=
→
fk+1 (Zk+1 |X) · fk+1|k (X|Z (k) )
fk+1 (Zk+1 |Z (k) )
(5.2)
fk+1 (Zk+1 |X) · fk+1|k (X|Z (k) )δX
(5.3)
where
fk+1 (Zk+1 |Z
1
(k)
)=
∫
Note: In two other commonly used systems of notation, the multitarget probability density
fk+1|k (X|Z (k) ) can be written as
fk+1|k (X|Z (k) ) = f (Xk+1 |Z1:k )
or as
fk+1|k (X|Z (k) ) = fΞk+1|k |Σ1 ,....,Σk (X|Z1 , ..., Zk )
where Ξk+1|k is the predicted multitarget RFS at time tk+1 and Σj is the measurement RFS
at time tj .
Multiobject Modeling and Filtering
109
is the Bayes normalization factor and where the integrals are set integrals, as defined
in (3.11).
The multitarget Bayes filter requires two a priori distributions:
1. The multitarget Markov transition density
MX (X ′ ) = fk+1|k (X|X ′ ),
(5.4)
which is the probability (density) that targets with state set X will be present
at time tk+1 if the targets at time tk had state set X ′ .
2. The multitarget likelihood function
LZ (X) = fk+1 (Z|X),
(5.5)
which is the probability (density) that the measurement set Z will be
collected at time tk+1 if the targets at time tk+1 have state set X.
The question then becomes: How do we construct “true” formulas for these
two items (in a sense to be defined shortly)? In analogy with Section 2.2, we first
construct:
1. A RFS multitarget motion model Ξk+1|k = Ξk+1|k (X ′ ), which is the RFS
of targets at time tk+1 , given that the targets at time tk have state set X ′ .
This process is described in Section 5.4.
2. An RFS multitarget measurement model Σk+1 = Σk+1 (X), which is the
RFS of measurements at time tk+1 , given that the targets at time tk+1 have
state set X. This process is described in Section 5.5.
Then we:
• Convert the RFS motion model into a “true” multitarget Markov density.
• Convert the RFS measurement model into a “true” multitarget likelihood
function.
By “true” is meant the following:
• fk+1|k (X|X ′ ) contains exactly the same information as the original RFS
multitarget motion model—no more and no less. That is,
– No information in the model has been lost. If information has been
lost, then this means that fk+1|k (X|X ′ ) does not faithfully preserve
the information contained in the model.
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Advances in Statistical Multisource-Multitarget Information Fusion
– No information extraneous to the motion model has been inadvertently
added. For if information has been added, then an unrecognized statistical bias has been inserted into fk+1|k (X|X ′ ).
• fk+1 (Z|X) contains exactly the same information as the original RFS
multisensor-multitarget measurement model. That is,
– No information has been lost. If information has been lost, then
fk+1 (Z|X) does not faithfully preserve the model.
– No extraneous information has been introduced. If information has been
introduced, fk+1 (Z|X) will introduce an unrecognized statistical bias
into the Bayesian analysis.
The construction of true multitarget Markov densities and true multitarget
likelihood functions requires the multitarget integral and differential calculus, which
was introduced in Chapter 3. The two procedures are described in Sections 5.6 and
5.7, respectively.
5.3
MULTITARGET BAYES OPTIMALITY
Information of interest—the number of targets, their positions, velocities, types, and
so on—can be extracted from the posterior distribution fk+1|k+1 (X|Z (k+1) ) using
a Bayes-optimal multitarget state estimator.
ˆ
A multitarget state estimator is a function X(Z),
the values of which are
state sets and the argument of which is a measurement set Z. Let C(X, Y ) ≥ 0
be a multitarget cost function defined on state sets X, Y . This means that
C(X, Y ) = C(Y, X) and that C(X, Y ) = 0 implies X = Y . The posterior
ˆ given the measurement set Z, is the cost averaged with respect to the
cost of X,
posterior distribution:
∫
¯ X|Z)
ˆ
ˆ
C(
= C(X, X(Z))
· fk+1|k+1 (X|Z (k) , Z)δX.
(5.6)
The Bayes risk is the average posterior cost (with respect to all possible measurement sets):
∫
ˆ
¯ X|Z)]
ˆ
¯ X|Z)
ˆ
R(X)
= EZ [C(
= C(
· fk+1 (Z|Z (k) )δZ
(5.7)
∫
ˆ
=
C(X, X(Z))
· fk+1 (Z|X) · fk+1|k (X|Z (k) )δXδZ. (5.8)
Multiobject Modeling and Filtering
111
ˆ is Bayes-optimal (with respect to the cost
The multitarget state estimator X
function C(X, Y )) if it minimizes the Bayes risk ([94], pp. 189-190, and [179], p.
63). This is the only theoretically rigorous meaning of the term “Bayes-optimal”
when applied to multitarget filtering.
An example of a Bayes-optimal multitarget state estimator is the joint multitarget (JoM) estimator ([179], Section 14.5.3, pp. 498-505):
|X|
ˆ k+1|k+1 = arg sup c
X
· fk+1|k+1 (X|Z (k+1) )
X∈X∞ |X|!
(5.9)
where c > 0 is a constant with the same units of measurement as x. Another
example is the marginal multitarget (MaM) estimator ([179], Section 14.5.2, pp.
497-498)
ˆ k+1|k+1 =
X
arg sup
fk+1|k+1 ({x1 , ..., xn̂k+1|k+1 }|Z (k+1) )
(5.10)
x1 ,...,xn̂k+1|k+1 ∈X
where
n̂k+1|k+1 = arg sup pk+1|k+1 (n|Z (k+1) )
(5.11)
n≥0
and where pk+1|k+1 (n|Z (k+1) ) is, as defined in (4.62), the cardinality distribution
of fk+1|k+1 (X|Z (k+1) ).
Remark 13 (Transient events) In a few applications, measurements are available
only at a single instant of time—at tk , say. This occurs, for example, in static
data-clustering (see Section 21.6). The targets are considered to be motionless and
the goal is to localize rather than track them. If the multitarget prior distribution
f0 (X) is unknown, two approaches can be employed. The first is to assume that
f0 (X) is a multitarget uniform distribution—that is, a distribution that is uniform
in regard to both target number and target state (see [179], pp. 366-367). Then the
JoM or MaM estimators can be applied to the corresponding posterior distribution.
Alternatively, one can abandon the Bayesian approach in favor of a maximumlikelihood approach. If Z0 is the measurement set that has been collected, the
multitarget state can be estimated using the multitarget version of the maximum
likelihood estimator (MMLE):
ˆ 0 = arg sup f0 (Z0 |X)
X
(5.12)
X
where f0 (Z|X) is the multitarget likelihood function for the sensor (see [179], pp.
500-501).
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Advances in Statistical Multisource-Multitarget Information Fusion
RFS MULTITARGET MOTION MODELS
Just as single-target motion can be modeled using a motion model
Xk+1|k = φk (x, Wk ),
(5.13)
so the motion of multitarget systems can be modeled using a multitarget motion
model of the general form
previous targets
new targets
? ?? ? ? ?? ?
Ξk+1|k = Tk+1|k (X ′ ) ∪ Bk+1|k .
(5.14)
Let X ′ = {x′1 , ..., x′n } with |X ′ | = n. Then it is typically assumed that
Tk+1|k (X ′ ) = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n )
(5.15)
where Tk+1|k (x′ ) is the RFS of targets at time tk+1 that originated in some
fashion from a target at time tk with state x′ . In turn, Tk+1|k (x′ ) can have the
form
per
sp
Tk+1|k (x′ ) = Tk+1|k
(x′ ) ∪ Tk+1|k
(x′ )
(5.16)
sp
where Tk+1|k
(x′ ) is the RFS of new targets spawned by x′ ; and where
per
per
′
Tk+1|k (x ) models the persistence of x′ itself—that is, either Tk+1|k
(x′ ) = ∅
per
(the target x′ disappeared from the scene), or |Tk+1|k (x′ )| = 1 (the target x′
persisted).
There are two cases to consider:
1. Uncoordinated multitarget motion: The motions of all targets are statistically
independent—that is, unrelated to each other moment-to-moment. A simple
example will be presented shortly in Section 5.6.
2. Coordinated multitarget motion ([179], pp. 478-482): The motions of the
targets are related in some manner, and thus are not statistically independent.
The following special case of uncoordinated motion is of particular interest:
• Standard Multitarget Motion Model without Spawning:
sp
per
– Tk+1|k
(x′ ) = ∅ for all x′ , in which case Tk+1|k (x′ ) = Tk+1|k
(x′ ).
– Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k are statistically independent.
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– Bk+1|k is Poisson.
• Standard Multitarget Motion Model with Spawning:
sp
– Tk+1|k
(x′ ) ̸= ∅ for all x′ .
per
per
sp
sp
– Tk+1|k
(x′1 ), ..., Tk+1|k
(x′n ), Tk+1|k
(x′1 ), ..., Tk+1|k
(x′n ), Bk+1|k
statistically independent.
are
– Bk+1|k is Poisson.
It should also be pointed out that the target appearance RFS Bk+1|k can
be used to implement optimal search, the purpose of which is to detect currently
undetected targets. In conjunction with sensor management and platform management, Bk+1|k can be chosen so as to implement whatever search strategy has been
chosen.
5.5
RFS MULTITARGET MEASUREMENT MODELS
Just as single-sensor, single-target data can be modeled using a measurement model
of the form
Zk+1 = ηk+1 (x, Vk ),
(5.17)
so multitarget multisensor data can be modeled using a multisensor-multitarget
measurement model. Assume that the measurements from s sensors are collected
at approximately the same time, tk+1 .
First, the random RFS Σk+1 of measurements has the form
1
s
(5.18)
Σk+1 = Σk+1 ⊎ ... ⊎ Σk+1
j
j
where Σk+1 ⊆ Z is the random measurement set generated by the jth sensor,
1
s
and where ‘⊎’ indicates disjoint union. Typically, Σk+1 , ..., Σk+1 are assumed
to be conditionally independent of the multitarget state.
j
Second, each of the Σk+1 has the form
target-generated measurements
j
Σk+1 =
?
j
??
?
Υk+1 (X)
clutter
?
j
??
?
∪ C k+1 (X)
(5.19)
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where Υk+1 (X) is the random set of target-generated measurements and
Ck+1 (X) is the random set of clutter-generated measurements. The latter functionally depends on X because, in some applications, clutter statistics can depend
on the states of the targets.
The basic issue, then, is the form of Υk+1 (X)—where hereafter the sensor
index j will be suppressed for the sake of notational clarity.
Most familiar sensors rely on wave phenomena such as acoustic waves and
electromagnetic waves. When these waves impinge on targets, sensor signatures
are generated and collected by the sensor. Such signatures can be modeled using a
superpositional measurement model that generalizes the single-target (2.22):
∑
Zk+1 =
ηk+1 (x) + Vk+1 .
(5.20)
x∈X
Here, X = {x1 , ..., xn }, n ≥ 0, is a set of target states; and Zk+1 is a random
signature (for example, an image) or a random real- or complex-valued vector. That
is, the generated measurement is a superposition of signals generated by all of the
targets present (if any). Superpositional models will be discussed in more detail in
Chapter 19.
Equation (5.20) is typically so computationally involved that it must first be
simplified using some sort of preprocessing methodology. Typical examples are:
• In radar, computing the real part Re(Zk+1 ) of Zk+1 and then applying
some sort of peak detector (threshold) to it.
• Applying a “blob detector” to a camera image, and selecting the centroids of
the blobs.
• Applying a wavelet-coefficient detector to a high range-resolution radar
(HRRR) signature.
In each of these cases, each signature will typically generate a finite set of
“detection measurements” or “detections.” A detection measurement model results
if we assume that any measurement is generated by at most a single target. In this
case, the measurement model will have the general form
measurements
meas’s (target x1 )
meas’s (target xn )
? ?? ?
? ?? ? ?clutter
? ?? ?
?? ?
Σk+1 = Υk+1 (x1 ) ∪... ∪ Υk+1 (x1 ) ∪ Ck+1
(5.21)
where Υk+1 (x) is the RFS of measurements generated by a target with state x;
and where Ck+1 is the RFS of remaining measurements—that is, false detections
and/or clutter.
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Given (5.21), there are four possibilities (see [179], p. 433, Figure 12.2):
1. Point (also known as “small”) targets: Targets are far enough away from the
sensor that each generates at most a single measurement—that is, |Υk+1 (x)|
is no larger than 1. At the same time, they are separated enough (relative to
sensor resolution) that they are resolvable as separate targets. Point targets
are the focus of Part II.
2. Extended targets ([179], pp. 427–432): Each measurement originates with
a single physical target, but this target can generate multiple measurements.
That is, the number |Υk+1 (x)| of measurements generated by a single target
can be arbitrarily large or small. This is because, typically, the target is close
enough to the sensor that multiple measurements are generated by discernible
scatterers distributed on the target’s surface.
(a) The state variables of an extended target can include centroid, centroidal
velocity, target type, and target-shape parameters.
(b) Extended targets are addressed in Chapter 21.
3. Group targets. Once again |Υk+1 (x)| can be arbitrary. But in this case, measurements are generated by autonomous or semi-autonomous point targets
that, collectively, constitute a tactically integrated “meta-target.” Examples
include platoons or regiments, aircraft sorties, and aircraft carrier groups.
(a) The state variables of a group target can include centroid, centroidal
velocity, formation type, number of targets in the group, and groupshape parameters.
(b) Group targets differ from extended targets primarily in that the former
can interpenetrate whereas the latter, having physical extent, cannot
overlap.
(c) Group targets are addressed in Chapter 21.
4. Unresolved targets ([179], pp. 432-444): Targets are so far away that they
appear to be colocated at a single point. Mathematically speaking, they can
be modeled as pairs x̊ = (n, x) where n is the number of targets colocated
at the same state x. Thus the value of |Υ(n, x)| can be arbitrary. Unresolved
targets are one of the subjects of Chapter 21.
Two special cases of these general models are of particular interest:
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• Standard Multitarget Measurement Model (described in more detail in Section 7.2):
– Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent.
– |Υk+1 (x)| ≤ 1
for all
x (the small-target case).
– Ck+1 is Poisson.
• Generalized Standard Multitarget Measurement Model:
– Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent.
– |Υk+1 (x)| can be arbitrary (the extended-target or group target case),
for all x.
– Ck+1 can be arbitrary.
It is necessary to discuss another situation that can give rise to the generalized
standard model. In general, it is possible for clutter to be dependent on the state of
the target. For example, clutter may be more dense in the vicinity of a target than
elsewhere. Or, multiple returns may be created because of multipath conditions. In
this case, the clutter-generated targets can have the form
Ck+1 (X)
state-dependent clutter
independent clutter
? ?? ?
1
Ck+1
(X)
? ?? ?
0
Ck+1
=
=
(
∪
1
Ck+1
(x)
∪
)
0
∪ Ck+1
(5.22)
(5.23)
x∈X
1
where Ck+1
(x) is the random set of clutter measurements associated with
0
the target with state x. Also, Ck+1
is the random set of clutter measurements that have no dependence on target states. Typically, if |X| = n then
1
1
0
Ck+1
(x1 ), ..., Ck+1
(xn ), Ck+1
are assumed to be independent. But most typically,
0
it is assumed that Ck+1 (X) = Ck+1
—that is, clutter has no dependence on the
target states.
A simple example will be presented shortly in Section 5.7. State-dependent
clutter is considered in greater detail in Section 8.7.
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5.6
117
MULTITARGET MARKOV DENSITIES
Just as the Markov transition density fk+1|k (x|x′ ) can be derived from the
probability mass function
pk+1|k (S|x′ ) = Pr(Xk+1|k ∈ S|Xk|k = x′ )
(5.24)
of the single-target motion model, so the true multitarget Markov transition density
fk+1|k (X|X ′ ) can be derived from the belief-mass function
βk+1|k (S|X ′ ) = Pr(Ξk+1|k ⊆ S|Ξk|k = X ′ )
(5.25)
of the RFS multitarget motion model. This is accomplished using the set derivative:
fk+1|k (X|X ′ ) =
δβk+1|k
(∅|X ′ ).
δX
(5.26)
As an example, assume that targets do not spawn other targets. That is, a
per
target either persists or it disappears, in which case |Tk+1|k (x′ )| = |Tk+1|k
(x′ )| ≤
1. Define the probability of target survival to be
pS (x′ ) abbr.
= pS,k+1|k (x′ ) def.
= Pr(Tk+1|k (x′ ) ̸= ∅).
(5.27)
Then the belief-mass function of Tk+1|k (x′ ) is
=
βk+1|k (S|x′ )
Pr(Tk+1|k (x′ ) ⊆ S)
=
=
Pr(Tk+1|k (x′ ) = ∅) + Pr(Tk+1|k (x′ ) ̸= ∅, Tk+1|k (x′ ) ⊆ S) (5.29)
1 − pS (x′ ) + pS (x′ ) · Pr(Tk+1|k (x′ ) ⊆ S|Tk+1|k (x′ ) ̸= ∅). (5.30)
(5.28)
Since Tk+1|k (x′ ) is a singleton set if it is nonempty, the final factor is a probabilitymass function pk+1|k (S|x′ ) with density function fk+1|k (x|x′ ), and so
βk+1|k (S|x′ ) = 1 − pS (x′ ) + pS (x′ ) · pk+1|k (S|x′ ).
(5.31)
There are different formulas for fk+1|k (X|X ′ ), depending on the number of
elements in X ′ . Only the cases X ′ = ∅ and X ′ = {x′ } will be considered
here—for the general case, see (7.66).
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• X ′ = ∅:
fk+1|k (X|∅) =
{
1
0
if
if
X′ = ∅
X ′ ̸= ∅
(5.32)
with p.g.fl.
(5.33)
Gk+1|k [h|∅] = 1.
′
′
• X = {x }:


1 − pS (x′ )
pS (x ) · fk+1|k (x|x′ )
fk+1|k (X|{x }) =

0
′
′
if
if
if
X=∅
X = {x}}
|X| ≥ 2
(5.34)
with p.g.fl.
Gk+1|k [h|x′ ] = 1 − pS (x′ ) + pS (x′ ) · Mh (x′ )
where
′
def.
Mh (x ) =
5.7
∫
h(x) · fk+1|k (x|x′ )dx.
(5.35)
(5.36)
MULTISENSOR-MULTITARGET LIKELIHOOD FUNCTIONS
Just as the single-sensor, single-target likelihood function fk+1 (z|x)
derived from the probability mass function
pk+1 (T |x) = Pr(Zk+1 ∈ T |Xk+1|k = x)
can be
(5.37)
of the measurement model, so the single-sensor, multitarget likelihood function
fk (Z|Xk ) can be derived from the belief-mass function
βk+1 (T |X) = Pr(Σk+1 ⊆ T |Ξk+1|k = X)
(5.38)
of the multisensor-multitarget measurement model. This is accomplished using the
set derivative:
δβk+1
LZ (X) abbr.
= fk+1 (Z|X) =
(∅|X).
(5.39)
δZ
Multisensor case: In the multisensor case, a measurement set will have the
form
1
s
Zk+1 = Z k+1 ⊎ ... ⊎ Z k+1
(5.40)
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119
j
where Z k+1 is the measurement set collected by the jth sensor. Assuming that
measurements are conditionally independent of state, the corresponding multisensor
likelihood function will be
1
s
1
s
fk+1 (Z|X) = f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X)
j
(5.41)
j
where f k+1 (Z k+1 |X) is the likelihood function for the jth sensor.
As an example, assume that a target generates either a single measurement or
no measurement at all. Define the probability of target detection to be
pD (x) abbr.
= pD,k+ (x) def.
= Pr(Υk+1 (x) ̸= ∅).
(5.42)
Then the belief-mass function of Υk+1 (x) is
βk+1 (T |x)
=
=
=
Pr(Υk+1 (x) ⊆ T )
Pr(Υk+1 (x) = ∅) + Pr(Υk+1 (x) ̸= ∅, Υk+1 (x) ⊆ T )
1 − pD (x) + pD (x) · Pr(Υk+1 (x) ⊆ T |Υk+1 (x) ̸= ∅)
(5.43)
(5.44)
(5.45)
=
1 − pD (x) + pD (x) · pk+1 (T |x)
(5.46)
where pk+1 (T |x) is a probability-mass function with density function Lz (x) =
fk+1 (z|x).
There are different formulas for fk+1 (Z|X), depending on the number of
elements in X. Only the cases X = ∅ and X = {x} will be considered here—
for the general case, see (7.21).
• X = ∅:
fk+1 (Z|∅) =
{
1
0
if
if
Z=∅
Z ̸= ∅
(5.47)
with p.g.fl.
(5.48)
Gk+1 [g|∅] = 1.
• X = {x}:
fk+1 (Z|{x}) =


1 − pD (x)
pD (x) · fk+1 (z|x)

0
if
if
if
Z=∅
Z = {z}}
|Z| ≥ 2
(5.49)
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with p.g.fl.
Gk+1 [g|x] = 1 − pD (x) + pD (x) · Lg (x)
where
Lg (x) =
5.8
∫
g(z) · fk+1 (z|x)dz.
(5.50)
(5.51)
THE MULTITARGET BAYES FILTER IN p.g.fl. FORM
The finite-set statistics approach is based on the derivation of approximate multisensormultitarget filters from the p.g.fl. form of the multisensor-multitarget Bayes filter.
The first step is to replace the multitarget Bayes filter
... →
fk|k (X|Z (k) )
→
fk+1|k (X|Z (k) )
→
fk+1|k+1 (X|Z (k+1) )
→ ...
with a filter on the p.g.fl.’s of its multitarget distributions
... →
Gk|k [h|Z (k) ]
→
Gk+1|k [h|Z (k) ]
→
Gk+1|k+1 [h|Z (k+1) ]
→ ...
The p.g.fl. form of the filter neither loses information nor inadvertently introduces
extraneous information, as compared with the multitarget Bayes filter. This is
because of the relationship (see (4.88)) that relates p.g.fl.’s with their corresponding
multitarget probability distributions:
fk|k (X|Z
(k)
[
]
δGk|k
(k) def. δGk|k
(k)
)=
[0|Z ] =
[h|Z ]
.
δX
δX
h=0
(5.52)
The second step is to express Gk+1|k [h|Z (k) ] in terms of Gk|k [h|Z (k) ] and
Gk+1|k+1 [h|Z (k+1) ] in terms of Gk+1|k [h|Z (k) ], as explained in the next two
subsections.
5.8.1
The p.g.fl. Time Update Equation
This is
Gk+1|k [h|Z
(k)
]=
∫
Gk+1|k [h|X ′ ] · fk|k (X ′ |Z (k) )δX ′
where
′
Gk+1|k [h|X ] =
∫
hX · fk+1|k (X|X ′ )δX
(5.53)
(5.54)
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121
is the p.g.fl. of the multitarget Markov transition density fk+1|k (X|X ′ ).
Equation (5.53) immediately follows from set-integration of both sides of
(5.1):
∫
Gk+1|k [h|Z (k) ] =
hX · fk+1|k (X|Z (k) )δX
(5.55)
)
∫ (∫
=
fk+1|k (X|X ′ )δX · fk|k (X ′ |Z (k) )δX ′ (5.56)
∫
=
Gk+1|k [h|X ′ ] · fk|k (X ′ |Z (k) )δX ′ .
(5.57)
5.8.2
The p.g.fl. Measurement Update Equation
The p.g.fl. form of Bayes’ rule is
[
Gk+1|k+1 [h|Z (k+1) ] =
]
δFk+1
δFk+1
δZk+1 [g, h]
δZk+1 [0, h] def.
] g=0 ,
= [
δFk+1
δFk+1
[0,
1]
δZk+1
δZk+1 [g, h]
g=0,h=1
(5.58)
where the bivariate p.g.fl. Fk+1 [g, h] of the joint target-measurement RFS Σk+1 ⊎
Ξk+1|k ⊆ Z ⊎ X is defined as2
Fk+1 [g, h] =
∫
hX · Gk+1 [g|X] · fk+1|k (X|Z (k) )δX
where
Gk+1 [g|X] =
∫
g Z · fk+1 (Z|X)δZ
is the p.g.fl. of the multisensor-multitarget likelihood function fk+1 (Z|X).
2
See Section 4.2.5 for a discussion of multivariate p.g.fl.’s).
(5.59)
(5.60)
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Equation (5.2) follows from
[
]
δFk+1
[g, h]
δZk+1
g=0
=
[∫
hX ·
]
δGk+1
[g|X] · fk+1|k (X|Z (k) )δX (5.61)
δZk+1
g=0
=
∫
hX · fk+1 (Zk+1 |X) · fk+1|k (X|Z (k) )δX
(5.62)
=
fk+1 (Zk+1 |Z (k) )
∫
· hX · fk+1|k+1 (X|Z (k+1) )δX
(5.63)
=
fk+1 (Zk+1 |Z (k) ) · Gk+1|k+1 [h|Z (k+1) ].
(5.64)
For, we then get
δFk+1|k+1
[0, h]
δZ
δFk+1|k+1
[0, 1]
δZ
5.9
=
fk+1 (Zk+1 |Z (k) ) · Gk+1|k+1 [h|Z (k+1) ]
fk+1 (Zk+1 |Z (k) ) · Gk+1|k+1 [1|Z (k+1) ]
(5.65)
=
Gk+1|k+1 [h|Z (k+1) ].
(5.66)
THE FACTORED MULTITARGET BAYES FILTER
In some applications, the state of the system will have the mixed form (x̊, X) where
x̊ ∈ ˚
X is the state of a single object and X is a finite subset of states x ∈ X.
Examples of such applications include:
• Simultaneous localization and mapping (SLAM) [210], [208], [1] in which
case:
– X is the set of reference landmarks or reference features (usually
assumed to be static).
– x̊ is the state-vector of the robot.
• Detection and tracking of single group targets (Section 21.9.3), in which case:
– x̊ is the state of the group target.
– X is the set of (conventional) targets of which it is comprised.
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123
• Joint multitarget tracking and sensor registration (Chapter 12), in which
case:
– X is the usual multitarget state set.
– x̊ is the vector of sensor biases for all sensors.
• Sensor management (Part V), in which case
– X is the multitarget state.
– x̊ is the joint state of all of the sensors.
The optimal Bayes filter for this problem has the form
... → fk|k (x̊, X|Z (k) ) → fk+1|k (x̊, X|Z (k) ) → fk+1|k+1 (x̊, X|Z (k+1) ) → ...
where
fk+1|k (x̊, X|Z
(k)
)
=
∫
fk+1|k (x̊, X|x̊′ , X ′ )
(5.67)
·fk|k (x̊′ , X ′ |Z (k) )dx̊′ δX ′
fk+1|k+1 (x̊, X|Z (k+1) )
=
fk+1 (Zk+1 |Z (k) )
=
fk+1 (Zk+1 |x̊, X) · fk+1|k (x̊, X|Z (k) )
(5.68)
fk+1 (Zk+1 |Z (k) )
∫
fk+1 (Zk+1 |x̊, X)
(5.69)
·fk+1|k (x̊, X|Z (k) )dx̊δX
and where fk+1|k (x̊, X|x̊′ , X ′ ) and fk+1 (Z|x̊, X) are, respectively, the Markov
transition density and likelihood function for the hybrid-state system.
This mixed-state Bayes filter can, under certain assumptions, be restated in an
often more useful “factored” form.3 By Bayes’ rule,
fk|k (x̊, X|Z (k) ) = fk|k (x̊|Z (k) ) · fk|k (X|Z (k) ,x̊)
(5.70)
where fk|k (x̊|Z (k) ) is a probability distribution on x̊ and fk|k (X|x̊, Z (k) ) is
a multitarget probability distribution on X. The Markov density can similarly be
3
The concept of dimensional reduction using density factorization appears to be due to Murphy and
Russell in 1996, who employed it for Rao-Blackwellization of particle filters [213]. In robotics, it is
the theoretical basis for both the FastSLAM algorithm [204] and the RFS approach to SLAM [210],
[208].
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factored as
fk+1|k (x̊, X|X ′ ,x̊′ ) = fk+1|k (x̊|x̊′ , X ′ ) · fk+1|k (X|x̊,x̊′ , X ′ ).
(5.71)
Assume that the future single-object state does not depend on the original
multiobject state, and that the future multiobject state does not depend on the
original single-object state:
fk+1|k (x̊|x̊′ , X ′ )
fk+1|k (X|x̊,x̊′ , X ′ )
=
=
fk+1|k (x̊|x̊′ )
fk+1|k (X|X ′ ,x̊′ ).
(5.72)
(5.73)
Then it is shown in Section K.4 that the hybrid-state Bayes filter can be written in
factored form as two coupled filters:
... → fk|k (x̊|Z (k) )
... → fk|k (X|Z (k) ,x̊)
→ fk+1|k (x̊|Z (k) ) →
↑↓
→ fk+1|k (X|Z (k) ,x̊) →
fk+1|k+1 (x̊|Z (k+1) ) → ...
↑↓
fk+1|k+1 (X|Z (k+1) ,x̊) → ...
These filters are defined by the following equations:
• Mixed-state time-update:
fk+1|k (x̊|Z (k) )
fk+1|k (X|Z
(k)
,x̊)
=
∫
fk+1|k (x̊|x̊′ ) · fk|k (x̊′ |Z (k) )dx̊′ (5.74)
=
∫
f˜k+1|k (X|Z (k) ,x̊′ )
(5.75)
·fk|k+1 (x̊′ |x̊, Z (k) )dx̊′
where the top equation is a conventional single-target time-update; where, for
fixed x̊′ ,
∫
f˜k+1|k (X|Z (k) ,x̊′ ) = fk+1|k (X|X ′ ,x̊′ ) · fk|k (X ′ |Z (k) ,x̊′ )δX ′ (5.76)
is a conventional multitarget time-update; and where
fk|k+1 (x̊′ |x̊, Z (k) ) =
fk+1|k (x̊|x̊′ ) · fk|k (x̊′ |Z (k) )
fk+1|k (x̊|Z (k) )
is a reverse Markov density (“retrodictive density”).
(5.77)
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125
• Mixed-state measurement-update:
(
fk+1|k+1 (x̊|Z (k+1) )
=
fk+1|k+1 (X|Z (k+1) ,x̊)
=
)
fk+1 (Zk+1 |x̊, Z (k) )
·fk+1|k (x̊|Z (k) )
fk+1 (Zk+1 |Z (k) )
(
)
fk+1 (Zk+1 |x̊, X)
·fk+1|k (X|Z (k) ,x̊)
fk+1 (Zk+1 |Z (k) ,x̊)
(5.78)
(5.79)
where the top equation is a conventional single-target measurement-update;
where the bottom equation is, for fixed x̊, a conventional multitarget
measurement-update; and where
fk+1 (Zk+1 |Z
(k)
,x̊)
fk+1 (Zk+1 |Z (k) )
=
∫
fk+1 (Zk+1 |x̊, X)
(5.80)
=
·fk+1|k (X|Z (k) ,x̊)δX
∫
fk+1 (Zk+1 |Z (k) ,x̊)
(5.81)
·fk+1|k (x̊|Z (k) )dx̊.
5.10
APPROXIMATE MULTITARGET FILTERS
The purpose of this section is to summarize the finite-set statistics strategy for
deriving approximate multitarget filters. It is organized as follows:
1. Section 5.10.1: The p.g.fl. time-update equation, assuming conditionally
independent Time evolution of targets.
2. Section 5.10.2: The p.g.fl. measurement-update equation, assuming conditionally independent generation of measurements.
3. Section 5.10.3: The finite-set statistics methodology for devising approximate multitarget filters.
4. Section 5.10.4: PHD filters in the general sense of the term.
5. Section 5.10.5: CPHD filters in the general sense of the term.
6. Section 5.10.6: Multi-Bernoulli filters in the general sense of the term.
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7. Section 5.10.7: Bernoulli filters in the general sense of the term.
5.10.1
The p.g.fl. Time Update for Independent Targets
The p.g.fl. form of the multitarget Bayes filter was described in Section 5.8. The
purpose of this section is to derive, under fairly general assumptions, a concrete
formula for the predicted p.g.fl. Gk+1|k [h|Z (k) ] in terms of the previous p.g.fl.
Gk|k [h|Z (k) ]—see (5.83).
RFS multitarget motion models were discussed in Section 5.4. Let X ′ =
′
{x1 , ..., x′n } with |X| = n and assume that the RFS motion model has the form
Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n ) ∪ Bk+1|k
(5.82)
where:
• Tk+1|k (x′ ) is the RFS of targets at time tk+1 , originating with a target with
state x′ at time tk ;
• Bk+1|k is the random set of newly appearing targets.
• Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k are statistically independent.
Given this, the predicted p.g.fl. Gk+1|k [h] is related to the original p.g.fl.
Gk|k [h] by
Gk+1|k [h] = GB
(5.83)
k+1|k [h] · Gk|k [Qk+1|k [h]],
where GB
k+1|k [h] is the p.g.fl. of Bk+1|k ; where h ?→ Qk+1|k [h] is the functional
transformation defined by
Qk+1|k [h](x′ ) = Gk+1|k [h|x′ ];
and where Gk+1|k [h|x′ ] is the p.g.fl. of Tk+1|k (x′ ).
To see why (5.83) is true, note that from (5.53),
Gk+1|k [h] =
∫
Gk+1|k [h|X ′ ] · fk|k (X ′ |Z (k) )δX ′ .
(5.84)
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127
Because of independence,
Gk+1|k [h|X ′ ]
GB
k+1|k [h]
=
∏
Gk+1|k [h|x′ ]
(5.85)
Qk+1|k [h](x′ )
(5.86)
x′ ∈X ′
GB
k+1|k [h]
=
∏
x′ ∈X ′
X
GB
k+1|k [h] · Qk+1|k [h]
=
′
(5.87)
where the power-functional notation hX was defined in (3.5). Thus
Gk+1|k [h]
∫
′
Qk+1|k [h]X · fk|k (X ′ |Z (k) )δX ′
=
GB
k+1|k [h]
=
GB
k+1|k [h] · Gk|k [Qk+1|k [h]].
(5.88)
(5.89)
As a simple example, assume that there are no target appearances of any kind.
In this case Bk+1|k = ∅ and |Tk+1|k (x′ )| ≤ 1. (This is what was assumed for the
example presented at the end of Section 5.6.) From (5.35) we know that the p.g.fl.
of Tk+1|k (x′ ) is
Qk+1|k [h](x′ ) = Gk+1|k [h|x′ ] = 1 − pS (x′ ) + pS (x′ ) · Mh (x′ )
(5.90)
or, alternatively,
(5.91)
Qk+1|k [h] = 1 − pS + pS · Mh
where
Mh (x′ ) =
∫
h(x) · fk+1|k (x|x′ )dx.
(5.92)
Thus
Gk+1|k [h|X ′ ] = (1 − pS + pS Mh )
X′
(5.93)
and so the p.g.fl. of the predicted-target RFS Ξk+1|k is:
Gk+1|k [h]
=
GB
k+1|k [h] · Gk|k [1 − pS + pS Mh ]
(5.94)
=
Gk|k [1 − pS + pS Mh ].
(5.95)
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Advances in Statistical Multisource-Multitarget Information Fusion
5.10.2
The p.g.fl. Measurement Update for Independent Measurements
The p.g.fl. form of the multitarget Bayes filter was described in Section 5.8. The
purpose of this section is to derive, under fairly general assumptions, a concrete
formula for the measurement-updated p.g.fl. Gk+1|k+1 [h|Z (k+1) ] in terms of the
predicted p.g.fl. Gk+1|k [h|Z (k) ]—see (5.99).
RFS multitarget measurement models were discussed in Section 5.5. Let
X = {x1 , ..., xn } with |X| = n and assume that the single-sensor, multitarget
measurement model has the form
Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1
(5.96)
where:
• Υk+1 (x) is the RFS of measurements (including both target-generated
measurements and state-dependent clutter measurements) associated with a
target that has state x at time tk+1 .
• Ck+1 is the random set of clutter measurements generated by the background
at time tk+1 .
• Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent.
Given this, the measurement-updated p.g.fl. Gk+1|k+1 [h] can be expressed in
terms of the predicted p.g.fl. Gk+1|k [h]. Specifically, the bivariate p.g.fl. of (5.59)
has the form
Fk+1 [g, h] = Gκk+1 [g] · Gk+1|k [h · Tk+1 [g]]
(5.97)
where Gκk+1 [g] denotes the p.g.fl. of Ck+1 , and where the functional transformation g ?→ Rk+1 [g] is defined as
(5.98)
Rk+1 [g](x) = Gk+1 [g|x],
where Gk+1 [g|x] is the p.g.fl. of Υk+1 (x). Consequently, from (5.58) the
measurement-updated p.g.fl. has the form
[
( κ
)]
δ
G
[g]
·
G
[h
·
R
[g]]
k+1
k+1|k
k+1
δZk+1
g=0
.
(5.99)
Gk+1|k+1 [h] = [
( κ
)]
δ
G
[g]
·
G
[h
·
R
[g]]
k+1
k+1|k
k+1
δZk+1
g=0,h=1
As a simple example, suppose that there is no clutter of any kind, so that
Ck+1 = ∅ and |Υk+1 (x)| ≤ 1. (This is what was assumed for the example at the
Multiobject Modeling and Filtering
129
end of Section 5.7.) From (5.50) we know that the p.g.fl. of Υk+1 (x) is
(5.100)
Rk+1 [g](x) = Gk+1|k [g|x] = 1 − pD (x) + pD (x) · Lg (x)
or, alternatively,
(5.101)
Rk+1 [g](x) = 1 − pD + pD Lg
where
Lg (x) =
∫
g(z) · fk+1 (z|x)dz.
(5.102)
X
(5.103)
Thus the p.g.fl. of Σk+1 is
Gk+1 [g|X] = (1 − pD + pD Lg )
and so
Fk+1 [g, h]
5.10.3
=
=
Gκk+1 [g] · Gk+1|k [h(1 − pD + pD Lg )]
Gk+1|k [h(1 − pD + pD Lg )].
(5.104)
(5.105)
A Principled Approximation Methodology
Given suitable independence assumptions, we know from (5.83) and (5.99) that the
time-update and measurement-update equations for the p.g.fl. Bayes filter are
=
Gk+1|k [h] = GB
k+1|k [h] · Gk|k [Qk+1|k [h]]
Gk+1|k+1 [h]
[
( κ
)]
δ
G
[g]
·
G
[h
·
R
[g]]
k+1
k+1|k
k+1
δZk+1
g=0
[
( κ
)]
δ
G
[g]
·
G
[h
·
R
[g]]
k+1
k+1|k
k+1
δZk+1
(5.106)
(5.107)
.
g=0,h=1
Suppose that we are given a multitarget motion model and a single-sensor,
multitarget measurement model. Then we can derive various approximate multitarget filters by:
1. Choosing carefully specified simplifying approximate forms for Gk+1|k [h]
and Gk+1|k+1 [h].
2. Applying the product rule for functional derivatives, (3.70), followed by
3. Application of Clark’s general chain rule, (3.91).
The most common simplifying assumptions employ the Poisson, i.i.d.c., and
multi-Bernoulli processes as discussed in Sections 4.3.1, 4.3.2, and 4.3.4.
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Advances in Statistical Multisource-Multitarget Information Fusion
5.10.4
Poisson Approximation: PHD Filters
This section introduces the following concepts:
• PHD filters in the general sense.
• The generalized classical PHD filter.
• The classical PHD filter.
• Nonclassical PHD filters.
5.10.4.1
PHD Filters in the General Sense
The following six-step procedure is used to derive PHD filters in the general sense
of the term:
1. Assume that the evolving multitarget RFS is approximately Poisson, as in
(4.107). That is, for every k ≥ 0,
Gk|k [h]
=
eDk|k [h−1]
(5.108)
Gk+1|k [h]
=
eDk+1|k [h−1]
(5.109)
fk|k (X|Z (k) )
=
X
e−Nk|k · Dk|k
(5.110)
fk+1|k (X|Z (k) )
=
X
e−Nk+1|k · Dk+1|k
.
(5.111)
or, equivalently, that
2. Given that Gk|k [h] is Poisson, use the formulas for the multitarget motion
model to determine the formula for Gk+1|k [h].
3. Use (4.75) to determine the PHD of Gk+1|k [h]:
Dk+1|k (x) =
δGk+1|k
[1].
δx
(5.112)
4. Given that Gk+1|k [h] is Poisson with the PHD as determined in (5.112), use
the multitarget measurement model to determine the formula for Gk+1|k+1 [h].
5. Use (4.75) to determine the PHD of Gk+1|k+1 [h]:
Dk+1|k+1 (x) =
δGk+1|k+1
[1].
δx
(5.113)
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131
6. Given the previous steps, the Time evolution
... →
Dk|k (x)
→
Dk+1|k (x)
→
Dk+1|k+1 (x)
→ ...
defines a PHD filter in the general sense, corresponding to the given target
and sensor models.
In general, closed-form formulas for PHD filters can be derived only if simplifying assumptions are made in regard to the multitarget motion and measurement
models.
5.10.4.2
The Generalized Classical PHD Filter
Specifically, suppose that:
• The motion model is the “standard” one with spawning, as described at the
end of Section 5.4.
• The measurement model is the generalized standard one described at the end
of Section 5.5.
• Gk|k [h] is not assumed to be Poisson.
Then it is possible to derive closed-form formulas for a generalized classical PHD filter for these models. This filter is described in Section 8.2. The
measurement-update equation for the general PHD filter involves a combinatorial
sum and is not computationally tractable in general.
5.10.4.3
The Classical PHD Filter
Assume in addition that:
• The measurement model is the “standard” model, as described at the end of
Section 5.5.
Then the resulting PHD filter is the “classical” PHD filter, which will be
discussed further in Section 8.4. It should be emphasized that, as with the general
PHD filter,
• The time-update formula for the classical PHD filter is exact—that is, it is not
necessary to assume that Gk|k [h] is Poisson.
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Advances in Statistical Multisource-Multitarget Information Fusion
5.10.4.4
Nonclassical PHD Filters
The following variants of the PHD filter—ones that address nonstandard multitarget
motion or measurement models—will be addressed later in the book:
• Multisensor versions of the general PHD filter (Chapter 10).
• Generalizations of the classical PHD filter for addressing rapidly maneuvering targets (Section 11.4).
• Generalizations of the classical PHD filter for addressing nontraditional (for
example, human-mediated) measurements (Section 22.10).
• Variants of the PHD filter for addressing extended, cluster, group, or unresolved targets (Chapter 21).
• Variants of the PHD filter for addressing superpositional sensors (Chapter
19).
5.10.5
i.i.d.c. Approximation: CPHD Filters
This section introduces the following concepts:
• CPHD filters in the general sense.
• The classical CPHD filter.
• Nonclassical CPHD filters.
5.10.5.1
CPHD Filters in the General Sense
The following six-step procedure is used to derive CPHD filters in the general sense
of the term:
1. Assume that the evolving multitarget RFS is approximately i.i.d.c., as in
(4.112). That is, for every k ≥ 0,
Gk|k [h]
=
Gk|k (sk|k [h])
(5.114)
Gk+1|k [h]
=
Gk+1|k (sk+1|k [h])
(5.115)
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133
or, equivalently,
fk|k (X|Z (k) )
fk+1|k (X|Z
(k)
)
=
|X|! · pk|k (|X|) · sX
k|k
(5.116)
=
|X|! · pk+1|k (|X|) · sX
k+1|k .
(5.117)
2. Given that Gk|k [h] is i.i.d.c., use the formulas for the multitarget motion
model and (5.106) to determine the formula for Gk+1|k [h].
3. Use (4.62) and (4.75) to determine the cardinality distribution and PHD of
Gk+1|k [h]:
pk+1|k (n)
=
Dk+1|k (x)
=
[
]
1 dn
Gk+1|k [x]
n! dxn
x=0
δGk+1|k
[1].
δx
(5.118)
(5.119)
4. Given that Gk+1|k [h] is i.i.d.c. with the cardinality and spatial distributions
as in (5.118) and (5.119), use the formulas for the multitarget measurement
model and (5.108) to determine the formula for Gk+1|k+1 [h].
5. Use (4.62) and (4.75) to determine the cardinality distribution and PHD of
Gk+1|k+1 [h]:
pk+1|k+1 (n)
=
Dk+1|k+1 (x)
=
[
]
1 dn
G
[x]
k+1|k+1
n! dxn
x=0
δGk+1|k+1
[1].
δx
(5.120)
(5.121)
6. Given this, the Time evolution
... →
Dk|k (x)
... →
pk|k (n)
→
↓
→
Dk+1|k (x)
pk+1|k (n)
→
↑↓
→
Dk+1|k+1 (x)
→ ...
pk+1|k+1 (n)
→ ...
is a CPHD filter in the general sense, corresponding to the given target and
sensor models.
Once again, closed-form formulas for CPHD filters can be derived only if
simplifying assumptions are made for the multitarget motion and measurement
models.
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Advances in Statistical Multisource-Multitarget Information Fusion
5.10.5.2
The Classical CPHD Filter
Assume in addition that:
• The motion model is the “standard” one without spawning, as described at
the end of Section 5.4.
• The measurement model is the “standard” model described at the end of
Section 5.5.
Then the resulting CPHD filter is the “classical” CPHD filter, which is
discussed further in Section 8.5.
5.10.5.3
Nonclassical CPHD Filters
In addition, the following variants of the CPHD filter—addressing nonstandard
multitarget motion or measurement models—will be addressed in this book:
• Multisensor versions of the general and classical CPHD filters (Chapter 10).
• Generalizations of the classical CPHD filter for addressing rapidly maneuvering targets (Section 11.5).
• Generalizations of the classical CPHD filter for addressing nontraditional
measurements (Section 22.10).
• Variants of the CPHD filter for unknown clutter and unknown detection
profiles (Chapters 18 and 17, respectively).
• Variants of the CPHD filter for superpositional sensors (Chapter 19).
5.10.6
Multi-Bernoulli Approximation: Multi-Bernoulli Filters
This section introduces the following concepts:
• Multi-Bernoulli filters in the general sense.
• The MeMBer filter.
• The CBMeMBer filter.
Multiobject Modeling and Filtering
5.10.6.1
135
Multi-Bernoulli Filters in the General Sense
The following six-step procedure is used to derive multi-Bernoulli filters in the
general sense of the term:
1. Assume that the evolving multitarget RFS is approximately multi-Bernoulli,
as in (4.126). That is, for every k ≥ 0,
Gk|k [h]
Gk+1|k [h]
=
νk|k (
=
i=1
νk+1|k (
∏
∏
i
i
1 − qk|k
+ qk|k
· sik|k [h]
)
(5.122)
)
i
i
1 − qk+1|k
+ qk+1|k
· sik+1|k [h] . (5.123)
i=1
2. Given that Gk|k [h] is multi-Bernoulli, use the formulas for the multitarget
motion model and (5.106) to determine the formula for Gk+1|k [h].
3. Use some procedure to, at least approximately, determine from Gk+1|k [h]
i
the multi-Bernoulli parameters νk+1|k and qk+1|k
, sik+1|k (x) for
i = 1, ..., νk+1|k .
4. Given that Gk+1|k [h] is multi-Bernoulli with these parameters, use the
formulas for the multitarget measurement model and (5.108) to determine
the formula for Gk+1|k+1 [h].
5. Use some procedure to, at least approximately, determine from Gk+1|k+1 [h]
i
the multi-Bernoulli parameters νk+1|k+1 and qk+1|k+1
, sik+1|k+1 (x) for
i = 1, ..., νk+1|k+1 .
6. Given this, the Time evolution
... → νk|k
v
k|k
i
... → {qk|k
}i=1
v
k|k
... → {sik|k (x)}i=1
→ νk+1|k →
↑↓
vk+1|k
i
→ {qk+1|k
}i=1
→
↑↓
vk+1|k
→ {sik+1|k (x)}i=1
→
νk+1|k+1 → ...
↑↓
vk+1|k+1
i
{qk+1|k+1
}i=1
→ ...
↑↓
vk+1|k+1
{sik+1|k+1 (x)}i=1
→ ...
is a multi-Bernoulli filter in the general sense, corresponding to the given
target and sensor models.
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5.10.6.2
The Multitarget Multi-Bernoulli (MeMBer) Filter
This filter was proposed by Mahler in Chapter 17 of [179]. The derivation there
of the MeMBer filter measurement-update equations included an ill-considered
approximation—namely the first-order Taylor’s linearization in Eq. (17.176) on
p. 681 of [179]. This approximation resulted in a significant upward bias in the
estimate of target number.
5.10.6.3
The Cardinality Balanced MeMBer (CBMeMBer) Filter
This bias was noticed by Vo, Vo, and Cantoni [310]. They subsequently devised
a corrected MeMBer filter, the cardinality-balanced multi-Bernoulli (CBMeMBer)
filter [310]. This filter is described in Section 13.4.
5.10.6.4
Other Multi-Bernoulli Filters
The following multi-Bernoulli filter variant will be addressed later in the book:
• A multi-Bernoulli filter for “raw” image data (Chapter 20).
5.10.7
Bernoulli Approximation: Bernoulli Filters
Assume that the evolving target RFS is approximately Bernoulli, as in (4.119)—that
is, for every k ≥ 0,
Gk|k [h]
Gk+1|k [h]
=
=
(5.124)
(5.125)
1 − qk|k + qk|k · sk|k [h]
1 − qk+1|k + qk+1|k · sk+1|k [h].
Then the Time evolution
... →
qk|k
... →
sk|k (x)
→
↑↓
→
qk+1|k
sk+1|k (x)
→
↑↓
→
qk+1|k+1
→ ...
sk+1|k+1 (x)
→ ...
is the Bernoulli filter in the general sense corresponding to the given target and
sensor models.
For the standard multitarget motion and measurement models, the “classical”
Bernoulli filter was independently proposed by B.-T. Vo [298] and by Mahler
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137
([179], pp. 514-528). 4 The time-update and measurement-update steps for this filter
can be found in Section 13.2.
A tutorial introduction to the Bernoulli filter can be found in Ristic et al. [262].
See also Ristic’s book, Particle Filters for Random Set Models [250].
4
In [179], the Bernoulli filter was called the “joint target-detection and tracking (JoTT)” filter. This
book adopts Vo’s more technically descriptive and correct terminology, “Bernoulli filter.”
Chapter 6
Multiobject Metrology
6.1
INTRODUCTION
Metrology refers to the process of determining the degree of similarity or dissimilarity of entities of interest. It is central to information fusion, whether we are to
compare competing algorithms with each other or, within an algorithm, to determine
the influence of internal parameters on the algorithm’s performance.
In the single-sensor, single-target realm, two general metrological paradigms
dominate:
1. Measurement of the distance between points: Suppose that a single-target
tracking algorithm generates a time sequence of state estimates that are to be
compared to ground truth:
Tracker:
Ground truth:
x1|1 , ..., xk|k
g1 , ..., gk .
(6.1)
(6.2)
The tracker’s instant-by-instant performance can be compared using a distance metric such as Euclidean distance: d(xk|k , gk ) = ∥xk|k − gk ∥. Its
trackwise performance can be measured using a suitable generalization of
this, such as the root-mean-square (RMS) miss distance:
?
? k
?1 ∑
k
k
d({xi|i }i=1 , {gk }i=1 ) = ?
d(xi|i , gi )2 .
k i=1
139
(6.3)
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Advances in Statistical Multisource-Multitarget Information Fusion
2. Measurement of the distance between probability distributions: Suppose that
the tracker generates track distributions, such as:
fk|k (x) = NPk|k (x − xk|k ).
(6.4)
Suppose also that ground truth can be represented as a distribution
gk (x) = NCk (x − gk )
(6.5)
where Ck is determined using (for example) the Cramer-Rao bound. Then
instant-by-instant performance can be compared using, as two possible illustrations, the Hellinger distance
∫ (√
)2
√
d(fk|k , gk ) =
fk|k (x) − gk (x) dx
(6.6)
or the Kullback-Leibler cross-entropy
(
)
∫
fk|k (x)
KL(fk|k ; gk ) = fk|k (x) · log
dx.
gk (x)
(6.7)
This chapter outlines analogous approaches for multisensor-multitarget problems:
• Comparison of the distance between multitarget state sets X = {x1 , ..., xn }.
• Comparison of the distance between multitarget distributions fk|k (X).
The chapter is organized as follows:
1. Section 6.2: Multiobject miss distance—Hausdorff distance, Wasserstein distance, optimal subpattern assignment (OSPA) distance, and generalizations of
OSPA.
2. Section 6.3: Multiobject information-theoretic functionals: the Csiszár family of information-theoretic functionals and the Cauchy-Schwartz functional;
and specific formulas for them for use with PHD and CPHD filters.
6.2
MULTIOBJECT MISS DISTANCE
In single-sensor, single-target statistics, target states x and sensor measurements z
are points. If we want to determine the performance of a single-sensor, single-target
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filter, the most common approach is to first choose a distance metric d(x, x′ ) on
states. Most commonly, x, x′ are Euclidean vectors and d(x, x′ ) is a Mahalanobis
distance
√
d(x, x′ ) = (x − x′ )T C −1 (x − x′ ).
(6.8)
The instantaneous performance of the filter can then be determined by calculating
the distance d(xk|k , gk ) between the filter’s current state estimate xk|k and the
current ground truth gk .
In multitarget problems, however, multitarget states X and multisensormultitarget measurements Z are finite sets of points. This section addresses the
following question:
• How can the concept of distance be usefully extended to multitarget states
X, X ′ —or, more generally, to the finite subsets of any space that is equipped
with an underlying distance metric?
Three possible answers to this question are described in what follows: Hausdorff distance, Wasserstein distance, and the optimal subpattern assignment (OSPA)
metric.
The section is organized as follows:
1. Section 6.2.1: A short history of the development of the concept of “multiobject miss distance.”
2. Section 6.2.2: An introduction to the optimal subpattern assignment (OSPA)
multiobject miss distance.
3. Section 6.2.3: The generalization of OSPA to algorithms that produce estimates of state uncertainty as well as states.
4. Section 6.2.4: The generalization of OSPA to labeled tracks.
5. Section 6.2.5: The generalization of OSPA to temporally-connected tracks.
6.2.1
Multiobject Miss Distance: A History
Finite-set statistics is based on hyperspaces Y∞ , whose elements are the finite
subsets of some underlying space Y. For metrological purposes, it is desirable
to have a way of computing the distance d(Y, Y ′ ) between two finite subsets
Y, Y ′ ⊆ Y. Here the term “distance” has a specific meaning—a metric in the
mathematical sense. That is, it must have the following properties:
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• Nonnegativity: d(Y, Y ′ ) ≥ 0.
• Symmetry: d(Y, Y ; ) = d(Y ′ , Y ).
• Definiteness: d(Y, Y ′ ) = 0 if and only if Y = Y ′ .
• Triangle inequality: d(Y, Y ′′ ) ≤ d(Y, Y ′ ) + d(Y ′ , Y ′′ ) for any Y, Y ′ , Y ′′ .
Remark 14 (The triangle inequality and practice) Because of the triangle inequality’s seeming abstractness, some practitioners have been inclined to dismiss
it as a fussy mathematical nicety. For this reason, it is necessary to clarify its
practical meaning and importance.1 Suppose that two multitarget trackers A and
B are being fed the same data. Let their respective outputs be XA and XB .
These are to be compared using a distance-type Measure of Performance (MoP),
denoted d(X, Y ). Suppose that A’s estimate is “close” to ground truth G—that
is, d(XA , G) is small. Suppose further that B’s estimate is “close” to A’s—
that is, d(XB , XA ) is small. If d(·, ·) is to be meaningful from a practical
point of view, then B’s estimate must also be “close” to ground truth—that is,
d(XB , G) must also be small. Any distance-type MoP d(·, ·) that does not satisfy
this property would be “metrically incoherent”—it would not measure “closeness”
in an intuitively reasonable manner. The triangle inequality ensures that d(·, ·) is
metrically coherent, because it forces d(XB , G) to be small:
d(XB , G) ≤ d(XB , XA ) + d(XA , G).
6.2.1.1
(6.9)
Hausdorff Distance
The Hausdorff distance is the most familiar distance metric for subsets, finite or
otherwise. It is defined by dH (Y, Y ′ ) = ∞ if either Y = ∅ or Y ′ = ∅ and, if
otherwise,
{
}
H
′
′
′
d (Y, Y ) = max max min
d (y, y ) , max
min d (y, y )
(6.10)
′
′
′
′
y∈Y y ∈Y
y ∈Y y∈Y
where d(y, y′ ) is some underlying metric on Y.
The Hausdorff distance is statistically consistent with finite-set statistics,
since its metric topology is the Fell-Matheron topology ([201], pp. 3,12).2 However,
it is not entirely adequate as a metric for performance estimation in multitarget
1
2
This discussion is adapted from Ristic, Vo, and Clark [261], Section 1.
For a general discussion of metrics that generate the Fell-Matheron topology, see [225].
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tracking. This is because it tends to be insensitive to cardinality and to the effect of
statistical outliers ([267], Section II-A). In particular, it is possible for d(Y, Y ′ ) to
be small even if |Y | and |Y ′ | are very different.
6.2.1.2
Wasserstein Distance
Let Y = {y1 , ..., yn } and Y ′ = {y1′ , ..., yn′ } with |Y | = n, |Y ′ | = n′ , and
n = n′ . Then the following definition of miss distance, originally proposed by
Drummond [62], is intuitively natural:
?
? n
?1 ∑
′
′ ∥2
d(Y, Y ) = min ?
∥yi − yπi
π
n i=1
(6.11)
where the minimum is taken over all permutations π on the numbers 1, ..., n. That
is, d(Y, Y ′ ) is the smallest root-mean-square (RMS) error between the elements of
Y and the elements of Y ′ .
How might this definition be extended to the situation when n ̸= n′ —but in
such a manner that one ends up with a true metric on Y? In 2002, Mahler proposed
the family of Wasserstein distances as an answer.
Let d(y, y′ ) be a metric on y, y′ ∈ Y. Then the Wasserstein distance of
power p is defined by (see [109], [110], [179], Section 14.6.3, pp. 510-512):3
?
? n n′
?∑ ∑
p
W
′ def.
d (Y, Y ) = inf ?
C
p
C
′ p
i,i′ · d(yi , yi′ )
(6.12)
i=1 i′ =1
where the infimum is taken over all n × n′ “transportation matrices” C. A matrix
C is a transportation matrix if, for all i = 1, ..., n and i′ = 1, ..., n′ , Ci,i′ ≥ 0
and
n
n′
∑
∑
1
1
′
Ci,i = ′ ,
Ci,i′ = .
(6.13)
n
n
′
i=1
i =1
When n = n′ , p = 2, and d(y, y′ ) = ∥y − y′ ∥, (6.12) reduces to
Drummond’s (6.11).
3
The metric topology of the Wasserstein metric is not the restriction of the Matheron topology to
finite subsets.
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6.2.2
The Optimal Sub-Pattern Assignment (OSPA) Metric
Schuhmacher, Vo, and Vo discovered that, from the point of view of practical
performance evaluation, Wasserstein distance4 suffers from a number of subtle
non-intuitive behaviors (see [267], Section II-B). In its place, they proposed a
new Wasserstein-like metric that not only avoids these difficulties, but is also more
mathematically intuitive and more easily computed [267].
Let us be given the following:
• A baseline metric d(x, x′ ) defined on single-target states x, x′ .
• A real number c > 0—the “association cutoff radius”—which has the same
units of measurement as x.
• A unitless real number p ≥ 1.
Let
dc (x, x′ ) = min{c, d(x, x′ )}
(6.14)
′
be the “cutoff metric” associated with d(x, x ). Let X = {x1 , ..., xn } be the
estimated track set and G = {g1 , ..., gm } the ground-truth track set, with |X| = n
and |G| = m. First assume that 0 < n ≤ m. Then the OSPA distance is defined
as:5
OSPA
dp,c
(X, G) =
(
n
1 ∑
cp
dc (xi , gπi )p +
· (m − n)
min
π m
m
i=1
)1/p
(6.15)
where the minimum is taken over all permutations π on 1, ..., m. If 0 = n ≤ m,
OSPA
then by convention set dp,c
(∅, G) = c. If n > m, then define dp,c (X, G) =
dp,c (G, X). Also, define dOSPA
p,c (X, G) = 0 if n = m = 0.
It follows that
OSPA
0 ≤ dp,c
(X, G) ≤ c.
(6.16)
The number c determines the importance assigned to target-number accuracy,
as compared to the importance assigned to localization accuracy. The number p
determines the sensitivity of the metric to statistical “outliers.” The larger the value
4
5
In [267], Schuhmacher et al. refer to the Wasserstein metric as the “optimal mass transfer” (OMAT)
metric.
As with the Wasserstein metric, the metric topology of the OSPA metric is not the restriction of the
Fell-Matheron topology to finite subsets. Rather, it is the “vague topology”—the topology most
commonly used in the counting-measure formulation of point process theory. See [267], p. 3451.
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of p, the more that a multitarget tracking algorithm is “punished” for arriving
at poor target state estimates. Equation (6.15) is easily computed using standard
optimal-assignment algorithms such as Munkres, JVC, and so on [267].
The definition of the OSPA metric has two parts. The first part
n
min
π
1 ∑
dc (xi , gπi )p
m i=1
(6.17)
measures truth-to-track localization accuracy, in essentially the same manner as
Drummond’s formula (6.11). However, dc (x, g) does not distinguish between
state-vectors x, g that are too far apart to be candidates for association. The second
part
cp
· (m − n)
(6.18)
m
measures the degree of accuracy in estimating target number. If c is small then
localization accuracy is more strongly emphasized than cardinality accuracy.
6.2.2.1
Constructive Interpretation of OSPA
OSPA
Intuitively speaking, if 0 < n ≤ m then dp,c
(X, G) is the distance between the
elements of G that are most tightly associated with the elements of X—but with
the proviso that an element of G must be qualified to associate with an element of
X—that is, it must be within distance c of it.
This interpretation can be understood more completely using the following
three-step procedure for constructing dOSPA
p,c (X, G). Suppose that 0 < n ≤ m.
Then:
1. Find that subset
GX = {gπ̂1 , ..., gπ̂n } ⊆ G
(6.19)
of G with n elements which is closest to X, using the following
generalization
n
1 ∑
dp (X, G) = min
d(xi , gπi )p
(6.20)
π m
i=1
of Drummond’s truth-to-track association formula (6.11); and where π̂ is the
corresponding optimal assignment xi ↔ gπ̂i for i = 1, ..., n.
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2. For each g ∈ G, let
δg =
{
c
min{c, d(xi , gπ̂i )}
if
if
g∈
/ GX
g = gπ̂i
(6.21)
be the “association cutoff distance” between g and its optimal-association
match in X. That is, if g has no match then δg = c. If g has a match
but is insufficiently near it then δg = c. If g has a match and is sufficiently
near it, then δg is the distance to its match.
3. Compute the pth-order average of the cutoff distances δg :
√
p
1 ∑ p
δg
|G|
(6.22)
g∈G
=
?


?
?
∑ p
∑ p
1
?
p

?
δg +
δg 
|G|
g∈GX
6.2.2.2
g∈G
/ X
=
? (
)
?
n
∑
?1
p
?
min
dc (xi , gπi )p + cp · (m − n)
π
m
i=1
(6.23)
=
dOSPA
p,c (X, G).
(6.24)
The “Components” of OSPA
The OSPA metric can be decomposed into two “components.” These are not metrics, since they do not satisfy the triangle inequality. Nevertheless, they provide
valuable additional information about the degree to which localization versus cardinality contribute to an OSPA score. The first, the localization-error component
loc
ep,c
(X, G), measures the contribution of localization accuracy alone. When n ≤ m
it is defined by
loc
ep,c
(X, G) =
(
n
1 ∑
min
dc (xi , gπi )p
π m
i=1
)1/p
(6.25)
loc
and, if n > m, by ep,c
(G, X) = eloc
p,c (X, G). The second, the cardinality-error
crd
component ep,c (X, G), measures the contribution of cardinality accuracy alone.
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When n ≤ m it is defined by
crd
ep,c (X, G) = c ·
(
m−n
m
)1/p
(6.26)
crd
crd
and, if n > m, by ep,c
(G, X) = ep,c
(X, G).
6.2.3
Extension of OSPA to Covariance (COSPA)
Most modern multitarget tracking algorithms produce track outputs of the form
X = {(x1 , P1 ), ..., (xn , Pn )}, where Pi is the error-covariance matrix corresponding to the state estimate xi . To accommodate such algorithms, it is necessary to
generalize the OSPA metric in a suitable manner. The purpose of this section is to
describe techniques for doing so.
The basic approach is to extend the base metric d(x, x′ ) to a metric
d((x, P ), (x′ , P ′ )) defined on pairs (x, P ), where x is a state-vector and P is
a covariance matrix. This extension must have the following consistency property:
d((x, P ), (x′ , P ′ )) → d(x, x′ )
(6.27)
as P → 0 and P ′ → 0.
The simplest such extension is
d((x, P ), (x′ , P ′ )) = d(x, x′ ) + d(P, P ′ )
(6.28)
where d(P, P ′ ) is any metric defined on positive-definite matrices P, P ′ . The
most obvious choice for d(P, P ′ ) is the Frobenius metric
dF (P, P ′ ) =
√
tr(P − P ′ )2
(6.29)
arising from the Frobenius matrix scalar product ⟨P, P ′ ⟩F = tr(P T P ′ ).
There is a potential difficulty with (6.28), however. One would expect that
the closeness of x and x′ should be influenced by the closeness of P and P ′ ,
and vice versa. For example, in practice (x, P ) arises from a track distribution
fk|k (x|Z k ) which inherently imposes a statistical coupling between x and P :
fk|k (x|Z k ) = NPk|k (x − xk|k ).
In (6.28), however, x and P are completely independent of each other.
(6.30)
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A second metric on covariance matrices has found application in image
processing (see [38], Eq. (11)). It is the metric on the Riemannian manifold of
covariance matrices defined by [87], [203]:
?
? d
?∑
−1
d(C1 , C2 ) = ∥ log(C1 C2 )∥F = ? (log λi )2
(6.31)
i=1
√
where log C denotes the matrix logarithm, ∥C∥F = ⟨C, C⟩F is the Frobenius
norm, and λi are the eigenvalues of C1−1 C2 . Besides being a distance metric,
d(C1 , C2 ) is invariant under affine transformation and under matrix inversion. That
is,
d(BC1 B T , BC2 B T ) = d(C1 , C2 )
(6.32)
for any regular matrix B, and
d(C1−1 , C2−1 ) = d(C1 , C2 ).
(6.33)
A third approach is to compare the track distribution fk|k (x|Z k ) with another
distribution f (x|Gk ) that represents the current ground truth Gk = {g1 , ..., gγ }.
This is the approach adopted by Nagappa, Clark, and Mahler [221]. They define
d((x, P ), (x′ , P ′ )) to be the Hellinger distance (see (6.67)) between two track
distributions. For linear-Gaussian distributions NP (y − x) and NP ′ (y − x′ ),
the Hellinger distance can be computed in closed form:
′
′
d((x, P ), (x , P ))
=
=
1−
∫ √
√
NP (y − x) · NP ′ (y − x′ )dy
(6.34)
√
det P P ′
(6.35)
det 12 (P + P ′ )
(
)
1
· exp − (x − x′ )T (P + P ′ )−1 (x − x′ ) .
4
1−
To compare an estimated track (x, P ) with a ground truth track (g, C), one must
first choose C. Nagappa et al. proposed that C should be specified in terms of
the Cramer-Rao lower bound (CRLB), as determined using an estimation process.
See [221] for more details.
There is a potential problem with (6.34) and (6.35): they do not necessarily
satisfy the consistency property of (6.27). This failure occurs, for example, when
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both covariances P, P ′ are very small. Thus, more generally, let E be a fixed
covariance matrix and define
∫ √
′
′
dE ((x, P ), (x , P )) = 1 −
NP +E (y − x) · NP ′ +E (y − x′ )dy (6.36)
√√
det(P + E)(P ′ + E)
= 1−
(6.37)
det 12 (P + P ′ + 2E)
(
)
1
′ T
′
−1
′
· exp − (x − x ) (P + P + 2E) (x − x ) .
4
Then
dE (x, x′ )
=
lim dE ((x, P ), (x′ , P ′ ))
(
)
1
′ T −1
′
1 − exp − (x − x ) E (x − x )
8
(6.38)
P,P ′ →0
=
(6.39)
is essentially just a Mahalanobis distance on x, x′ .
6.2.4
OSPA for Labeled Tracks (LOSPA)
The OSPA metric measures only the instantaneous distance, at a particular time tk ,
between the current track set Xk|k and the current ground-truth target set Gk . In
general, we must determine how well a multitarget tracker estimates ground truth
over an extended period of time. This is the problem addressed by the temporal
generalization of OSPA due to Ristic, Vo, and Clark [261], to be described shortly
in Section 6.2.5. For convenience it will be referred to here as the “TOSPA metric.”
Before we can understand the TOSPA metric, however, we must first consider an
intermediary metric: OSPA for labeled tracks. This is the purpose of this section.
The track of a single target is not just a time sequence x1|1 , ..., xK|K or,
alternatively, (x1|1 , P 1|1 ), ..., (xk|k , P K|K ). (For the sake of notational clarity,
in what follows we will consider only the former situation.) It is, rather, a time
sequence of the form ([179], Section 14.5.6, pp. 505-508)
(x1|1 , ℓ), ..., (xK|K , ℓ)
(6.40)
where the integer label ℓ ∈ {1, ...., L} uniquely identifies each (xk|k , ℓ) as
belonging to a single, temporally connected track-trajectory. Thus the output of a
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multitarget tracker will be a time sequence
X (K) : X1|1 , ..., XK|K
(6.41)
of track sets, where now each Xi|i has the form
X = {(x1 , ℓ1 ), ..., (xn , ℓn )}
(6.42)
with labels ℓi ∈ {1, ..., L}. It follows that, for each k, the instantaneous track set
Xk|k at time tk can be partitioned as
1
L
Xk|k = Xk|k
⊎ ... ⊎ Xk|k
(6.43)
ℓ
ℓ
where Xk|k
is the set of all (x, ℓ) such that (x, ℓ) ∈ Xk|k . Either Xk|k
contains
ℓ
a single element (the unique track with label ℓ at time tk ) or Xk|k = ∅ (that is,
the track with label ℓ has been dropped). Stated differently, the sequence
ℓ
ℓ
X1|1
, ..., XK|K
(6.44)
is the ℓth track sequence as determined by the multitarget tracker.
Now assume that ground truth has been provided. Then we have a time
sequence
G(K) : G1|1 , ..., GK|K
(6.45)
of ground truth track sets, each of which consists of elements of the form (g, γ)
where g is a ground truth state and γ ∈ {1, ..., L} is a ground-truth label. Each
ground truth track set can be partitioned as
Gk|k = G1k|k ⊎ ... ⊎ GL
k|k ,
(6.46)
Gγ1|1 , ..., GγK|K
(6.47)
and
is the track sequence for the ground-truth track with label γ.
Next, suppose for the moment that the tracker correctly and exactly estimated
all ground truth track sequences. Even given this fact, the tracker’s track labeling
convention will not be the same as the ground truth labeling convention. That is,
ℓ
there will be a permutation σ on 1, ..., L such that Xk|k
= Gσℓ
for all
k|k
k = 1, ..., K and all ℓ = 1, ..., L.
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More generally, suppose that the tracker’s estimates of ground truth are not
ℓ
exact. Then there will be a σ such that Xk|k
and Gσℓ
k|k are as “close to each
other as possible” (in an OSPA sense) for all k, ℓ.
The question then becomes: When it comes to labeled track sequences, what
does “close to each other” mean in a quantifiable sense? Ristic et al. begin by
replacing the original base metric d(x, x′ ) on target states with a new base metric
on labeled states (x, ℓ) as follows:
√
d˜p,α ((x, ℓ), (x′ , ℓ′ )) = p d(x, x′ )p + αp · (1 − δℓ,ℓ′ )
(6.48)
where δℓ,ℓ′ is the Kronecker delta. The number α > 0 controls the relative
weighting of the labeling error 1 − δℓ,ℓ′ .
Now suppose that
X
G
=
=
{(x1 , ℓ1 ), ..., (xn , ℓn )}
{(g1 , γ1 ), ..., (gm , γm )}
(6.49)
(6.50)
with |X| = n and |G| = m. Here, ℓj ∈ {1, ...., N } are the estimated labels
of the estimated tracks; and γj ∈ {1, ...., N } are the known labels of the ground
truth tracks. Denote the result of stripping labels from the tracks as
X↓
G↓
=
=
{x1 , ..., xn }
{g1 , ..., gm }.
(6.51)
(6.52)
Substitute d˜p,α (·, ·) in (6.48) into (6.15) whenever d(·, ·) occurs. What results is
an explicit formula for the OSPA metric with labels, denoted as “LOSPA”:
(
)1/p
n
p ∑
α
↓
↓ p
dLOSPA
dOSPA
(1 − δℓi ,γπ̂i )
(6.53)
p,c,α (X, G) =
p,c (X , G ) +
m i=1
where the permutation π̂ on 1, ..., n is defined by
π̂ = arg min
π
n
∑
dc (xi , gπi )p .
(6.54)
i=1
Thus, one first finds the optimal association π̂ of tracks in X with ground truth
tracks∑
in G, irrespective of labels. If all labels were estimated correctly, then the
LOSPA
OSPA
sum
i (1 − δℓi ,γπ̂i ) will be zero and dp,c,α (X, G) = dp,c,α (X, G). If none are
LOSPA
correct, the sum will be n and the value of dp,c,α (X, G) will be increased by an
amount determined by the size of α.
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6.2.5
Temporal OSPA (TOSPA)
The LOSPA metric measures the distance between labeled track sets at a particular
instant of time. To address the general track-fusion problem, we must be able to
measure the distance between time sequences of labeled track sets. In particular,
we must address the fact that the assignment of labels to evolving track sequences
is essentially arbitrary.
This is the purpose of the temporal OSPA (TOSPA) metric, introduced by
Ristic, Vo, and Clark [261]. Given the labeled ground truth track set G =
{(g1 , γ1 ), ..., (gm , γm )} and a permutation σ on 1, ..., L, define
σ
(6.55)
G = {(g1 , σγ1 ), ..., (gm , σγm )}.
Then given the ground truth multitrack sequence G(K) and the multitrack sequence
X (K) produced by a multitarget tracker, the TOSPA metric is defined by
dTOSPA
p,c,α (X
(K)
,G
(K)
)=
(
min
σ
K
∑
σ
dLOSPA
p,c,α (Xk|k , Gk|k )
p
)1/p
.
(6.56)
k=1
Intuitively speaking, the TOSPA metric searches for the best match, in a
temporally global sense, between the ground truth tracks and the tracks created by
the multitarget tracker. It measures the following aspects of tracker performance:
• Localization accuracy.
• Accuracy in determining target number (which encompasses accuracy in
regard to dropped tracks or false tracks).
• Accuracy in track labeling throughout the timespan of an entire scenario.
Suppose that the algorithm being measured provides error-covariance matrices as well as states. Then the approach described in Section 6.2.3 immediately
allows the TOSPA metric to be suitably generalized.
The minimization operation in (6.56) is an optimal track-to-track association
procedure. It requires minimization of the total distances between all ground truth
and all estimated tracks, in both space and time. It is therefore computationally
intractable in general, and some approximation must be devised. The approach of
Ristic et al. [261] is to reassign labels to the estimated track sequences, with the
purpose of correctly aligning them with the labels already assigned to the ground
truth track sequences.
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Gγ1|1 , ..., GγK|K
(6.57)
Let
be a ground truth track sequence with label γ—which means that, for any k, either
Gγk|k = ∅ or Gγk|k = {(gk|k , γ)}. Likewise, let
ℓ
ℓ
X1|1
, ..., XK|K
(6.58)
ℓ
be an estimated track sequence with label ℓ—so that, for any k, either Xk|k
=∅
ℓ
or Xk|k = {(xk|k , ℓ)}. Let
eℓk =
{
1
0
if
if
ℓ
Xk|k
̸= ∅
,
otherwise
ẽγk =
{
1
0
if
if
Gγk|k ̸= ∅
.
otherwise
(6.59)
Then define the pairwise cost of assigning a ground truth track γ with an estimated
track ℓ to be:
{ ∑K ℓ γ
∑K ℓ γ
k=1 ek ẽk ·∥xk −gk ∥
∑K
if
ℓ ẽγ
k=1 ek ẽk > 0
exp
e
(
)
c(ℓ, γ) =
.
(6.60)
k=1 k k
∞
if
otherwise
The c(l, γ) are used to form the assignment matrix in the two-dimensional
assignment algorithm. Once truth-to-track assignments have been made using this
heuristic, an estimated track sequence has either (1) been given the label of the
ground truth track sequence that was associated with it; or (2) assigned a completely
new label if it could not be associated with any ground truth track sequence.
Once this has been accomplished, labeled track sets Xk|k for the estimated
tracks are created for each time tk —but now equipped with the newly-assigned
labels. The LOSPA or labeled-COSPA metrics are then applied to these labeled
track sets; and their value(s) are displayed, instant-by-instant, for the entire scenario.
Equation (6.60) causes longer duration estimated tracks to be assigned to ground
truth tracks.
6.3
MULTIOBJECT INFORMATION FUNCTIONALS
Single-target distance metrics address the problem of measuring the similarity or
dissimilarity of two states x, x′ . A more general concept is that of measuring the
similarity or dissimilarity of two probability distributions f1 (x), f0 (x) defined on
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states. Perhaps the most familiar approach is information-theoretic—for example,
the Kullback-Leibler discrimination functional:
(
)
∫
f1 (x)
KL(f1 ; f0 ) = f1 (x) · log
dx.
(6.61)
f0 (x)
This section addresses the following question:
• How can information-theoretic functionals be extended to multitarget probability distributions f1 (X), f0 (X) defined on multitarget states X—or,
more generally, to probability distributions defined on the finite subsets of
any space?
The approach described in this section is the infinite family of multiobject
Csiszár information-theoretic functionals, introduced by Zajic and Mahler in 1999
([331], pp. 96-97). They are useful not only for performance evaluation, but also
for the approach to sensor and platform management to be described later in Part V.
The section is organized as follows:
1. Section 6.3.1: The family of Csiszár information functionals.
2. Section 6.3.2: Csiszár functionals for Poisson processes.
3. Section 6.3.3: Csiszár functionals for i.i.d.c. processes.
6.3.1
Csiszár Information Functionals
Let c(x) be a convex kernel—that is, a unitless, nonnegative convex function of a
nonnegative variable x ≥ 0, such that c(1) = 0 and such that c(x) is strictly
convex at x = 1 (that is, c(2) (1) > 0).
Let f1 (Y ) and f0 (Y ) be two multiobject probability distributions on
some space Y. Then the multiobject Csiszár information-discrimination functional
associated with c(x) is [331]:
Ic (f1 ; f0 ) =
∫
c
(
f1 (Y )
f0 (Y )
)
· f0 (Y )δY.
(6.62)
It has the property that Ic (f1 ; f0 ) ≥ 0, with equality occurring only if f1 (Y ) =
f0 (Y ) almost everywhere [52], [53].6
6
Caution: If f1 or f0 is not a multiobject probability density function, then neither of these
properties remains valid. See [52], [53].
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155
The correspondence c ?→ Ic (f1 ; f0 ) is not one-to-one, since
(6.63)
Ic2 (f1 ; f0 ) = Ic1 (f1 ; f0 ).
whenever
c2 (x) = c1 (x) + K · (x − 1)
7
for any constant K.
Examples of multiobject Csiszár information functionals are as follows:
• Kullback-Leibler—c(x) = 1 − x + x log x:
Ic (f1 ; f0 ) =
∫
f1 (Y ) · log
(
f1 (Y )
f0 (Y )
)
(6.64)
δY.
• Chi-squared—c(x) = (x − 1)2 :
∫
f1 (Y )2
δY.
f0 (Y )
(6.65)
|f1 (Y ) − f0 (Y )|δY.
(6.66)
∫ √
(6.67)
Ic (f1 ; f0 ) = −1 +
• L1 metric—c(x) = |x − 1| :
Ic (f1 ; f0 ) =
∫
√
2
• Hellinger—c(x) = ( x − 1) :
Ic (f1 ; f0 ) = 2 − 2
f1 (Y ) · f0 (Y )δY.
• Information deviation—c(x) = αx+1−α−x
α(1−α)
(3,4)):
Ic (f1 ; f0 ) =
7
(1)
1
α (1 − α)
(
1−
∫
α
([323], Eq. (6), and [117], Eqs.
f1 (Y )α · f0 (Y )1−α δY
)
.
(6.68)
If one sets K = −c1 (1), then c2 (x) will have a unique minimum at x = 1. In this case,
c2 (x) = 0 implies x = 1.
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If the Kullback-Leibler functional is expanded in f1 around f0 in a
Taylor’s series expansion, then the chi-square discrimination is seen to be the
second-order Taylor’s approximation of the Kullback-Leibler discrimination. Also,
chi-square discrimination and Hellinger discrimination provide bounds for the
Kullback-Leibler discrimination ([92], pp. 12,13):
(
)
I(√x−1)2 (f1 ; f0 )2 ≤ I1−x+x log x (f1 ; f0 ) ≤ log 1 + I(x−1)2 (f1 ; f0 ) . (6.69)
The information deviation functional is defined for 0 ≤ α ≤ 1. It converges
to Kullback-Leibler discrimination when α → 0 or α → 1. It is closely related
to Chernoff information, which in multiobject form is defined as
(
)
∫
ω
1−ω
C(f1 ; f0 ) = sup − log f1 (Y ) · f0 (Y )
δY .
(6.70)
0≤ω≤1
It is also closely related to the Rényi α-divergence, which in multiobject form is
defined as
∫
1
Rα (f1 ; f0 ) =
log f1 (Y )α · f0 (Y )1−α δY
(6.71)
α−1
for α > 0. Specifically, if c(x) is the convex function corresponding to
information deviation then
Rα (f1 ; f0 ) =
1
· log [1 − α (1 − α) · Ic (f1 ; f0 )] .
α−1
(6.72)
A final information functional does not seem to be a Csiszár divergence, but
should be discussed because of increasing interest in it for information fusion applications [64]. This is the Cauchy-Schwartz divergence functional. Its popularity
is due to the fact that (1) it behaves much like Kullback-Leibler discrimination; and
(2) can be evaluated essentially in closed form if f1 (y) and f0 (y) are Gaussian
mixtures [132]. The multiobject version of the Cauchy-Schwartz divergence is:
∫ |Y |
c · f1 (Y ) · f0 (Y )δY
√∫
CS(f1 ; f0 ) = − log √∫
(6.73)
c|Y | · f1 (Y )2 δY ·
c|Y | · f0 (Y )2 δY
where c is a positive real number that has the same units of measurement as the
elements y of Y.8 Equation (6.73) has the following geometric meaning:
CS(f1 ; f0 ) = − log cos θf1 ,f0
8
(6.74)
This constant is not required in the single-object version of the Cauchy-Schwartz discrimination. It
is necessary for the multiobject version because, without it, the indicated set integrals would not be
mathematically well defined.
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157
where θf1 ,f0 is the angle between f1 and f0 , when they are regarded as vectors in
the inner product space L2 (Y∞ ) of square-integrable multiobject density functions
f (Y ) on Y ∈ Y∞ , with respect to the inner product
⟨f1 , f2 ⟩ =
6.3.2
∫
c|Y | · f1 (Y ) · f2 (Y )δY.
(6.75)
Csiszár Functionals for Poisson Processes
The purpose of this section is to provide explicit formulas for the information
functionals described in the previous section, given that f1 , f0 are both Poisson
(as defined in Section 4.3.1). That is,
f1 (Y ) = e−N1 · D1Y ,
f0 (Y ) = e−N0 · D0Y
(6.76)
where D1 (y), D0 (y) are the respective PHDs of f1 (Y ), f0 (Y ); N1 , N0 are the
respective integrals of D1 (y), D0 (y); and the power-functional notation D Y was
defined in (3.5). Also, define
s1 (y) = N1−1 · D1 (y),
s0 (y) = N0−1 · D0 (y).
(6.77)
The mathematical derivations of the following formulas can be found in Section
K.9.
• Kullback-Leibler divergence ( c(x) = 1 − x + x log x ):
Ic (f1 ; f0 )
=
=
N0 − N1 + Ic (D1 ; D0 )
( ( )
)
N1
N1
N0 · c
+
· Ic (s1 ; s0 ) .
N0
N0
(6.78)
(6.79)
• Chi-squared divergence ( c(x) = (x − 1)2 ):
log (1 + Ic (f1 ; f0 ))
=
=
∫
D1 (x)2
N0 − 2N1 +
dx
D0 (x)
( ( )
)
N12
N0
· c
+ Ic (s1 ; s0 ) .
N0
N1
(6.80)
(6.81)
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• Information deviation ( c(x) = α−1 (1 − α)
−1
· (αx + 1 − α − xα ) ):
log (1 − α (1 − α) · Ic (f1 ; f0 ))
−αN1 − (1 − α)N0
∫
+ D1 (y)α · D0 (y)1−α dy
( )
(
)
1
N0 · c N
N0
−α(1 − α) ·
.
+N1α N01−α · Ic (s1 ; s0 )
=
=
(6.82)
(6.83)
• Rényi α-divergence:9
Rα (f1 ; f0 )
=
=
α
· N1 + N0
α−1∫
1
+
D1 (y)α · D0 (y)1−α dy
α−1
( )
N1
αN0 · c
+ αN1α N01−α · Ic (s1 ; s0 )
N0
−
(6.84)
(6.85)
where c(x) is the convex kernel corresponding to information deviation.
For completeness, the following result is included:
• Cauchy-Schwartz divergence:
c
CS(f1 ; f0 ) =
2
6.3.3
∫
(D1 (y) − D0 (y))2 dy.
(6.86)
Csiszár Functionals for i.i.d.c. Processes
The purpose of this section is provide explicit formulas for various information
functionals, assuming that f1 , f0 are both i.i.d.c. (as defined in Section 4.3.2):
f1 (Y ) = |Y |! · p1 (|Y |) · sY1 ,
f0 (Y ) = |Y |! · p0 (|Y |) · sY0 .
(6.87)
The derivations of the following formulas are given in Section K.10.
• Kullback-Leibler divergence—c(x) = 1 − x + x log x:
Ic (f1 ; f0 ) = Ic (p1 ; p0 ) + N1 · Ic (s1 ; s0 )
9
Equation (6.84) is originally due to Ristic, Vo, and Clark—see [260], Eq. (18).
(6.88)
Multiobject Metrology
where
Ic (p1 ; p0 ) =
∑
p1 (n) · log
159
(
p1 (n)
p0 (n)
)
.
(6.89)
n≥0
• Chi-squared divergence—c(x) = (x − 1)2 :
Ic (f1 ; f0 ) = −1 + I˜c (p1 ; p0 ) · Gp̃ (I˜c (s1 ; s0 ))
(6.90)
I˜c (f1 ; f0 ) = I˜c (p1 ; p0 ) · Gp̃ (I˜c (s1 ; s0 ))
(6.91)
or equivalently
where Gp̃ (y) is the p.g.f. of the probability distribution p̃(n) defined by
x(n) =
1
p1 (n)2
·
I˜c (p1 ; p0 ) p0 (n)
(6.92)
and where
I˜c (f1 ; f0 )
=
I˜c (s1 ; s0 )
=
I˜c (p1 ; p0 )
=
∫
f1 (Y )2
δY
f0 (Y )
∫
s1 (y)2
dy
s0 (y)
∑ p1 (n)2
.
p0 (n)
(6.93)
(6.94)
(6.95)
n≥0
• Information deviation—c(x) = α−1 (1 − α)
Ic (f1 ; f0 ) =
−1
· (αx + 1 − α − xα ):
[
]
1
· 1 − I˜c (p1 ; p0 ) · Gp̃ (I˜c (s1 ; s0 )
α (1 − α)
(6.96)
I˜c (f1 ; f0 ) = I˜c (p1 ; p0 ) · Gp̃ (I˜c (s1 ; s0 )
(6.97)
or, equivalently,
where Gp̃ (y) is the p.g.f. of the probability distribution p̃(n) defined by
p̃(n) =
p1 (n)α · p0 (n)1−α
I˜c (p1 ; p0 )
(6.98)
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Advances in Statistical Multisource-Multitarget Information Fusion
and where
I˜c (f1 ; f0 )
=
I˜c (p1 ; p0 )
=
∫
f1 (Y )α · f0 (Y )1−α δY
∑
p1 (n)α · p0 (n)1−α
(6.99)
(6.100)
n≥0
I˜c (s1 ; s0 )
=
∫
s1 (y)α · s0 (y)1−α dy.
(6.101)
• Rényi α-divergence:10
Rα (f1 ; f0 ) =
1
1
· log I˜c (p1 ; p0 ) +
· log Gp̃ (I˜c (s1 ; s0 )) (6.102)
α−1
α−1
where c(x), I˜c (p1 ; p0 ), I˜c (s1 ; s0 ), and Gp̃ (y) are defined as in (6.96).
10 Equation (6.102) is originally due to Ristic, Vo, and Clark—see [260], Eq. (14).
Part II
RFS Filters: Standard
Measurement Model
161
Chapter 7
Introduction to Part II
The chapters in Part II describe multitarget algorithms that presume the “standard”
multitarget motion and measurement models (introduced, respectively, in Sections
5.4 and 5.5. These algorithms are:
• The “classical” PHD and CPHD filters and their properties and behavior—for
example, “spooky action at a distance.”
• Multisensor classical PHD and CPHD filters.
• The cardinality-balanced multi-Bernoulli (CBMeMBer) filter.
• Jump-Markov PHD and CPHD filters for rapidly maneuvering targets.
• PHD smoothers.
• Extension of the PHD filter to joint multitarget tracking and estimation of
unknown spatial biases in the sensors.
• The Vo-Vo exact closed-form solution of the general multitarget Bayes filter.
The purpose of this introductory chapter is to set the stage for Part II by
describing the standard measurement model in greater detail, and by elucidating
the relationship between it and the conventional measurement-to-track association
(MTA) approach to multitarget tracking (which was introduced in Section 1.1.3).
The remainder of the chapter is organized as follows:
1. Section 7.1: A summary of the major lessons learned in this chapter.
2. Section 7.2: The standard multitarget measurement model.
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3. Section 7.3: An approximation of the multitarget likelihood function for the
standard multitarget measurement model.
4. Section 7.4: The standard multitarget motion model.
5. Section 7.5: The standard multitarget motion model with target spawning.
6. Section 7.6: The organization of Part II.
7.1
SUMMARY OF MAJOR LESSONS LEARNED
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• The formula for the p.g.fl. of the standard multitarget measurement model
(see (7.19)):
Gk+1 [g|X] = eκk+1 [g−1] · (1 − pD + pD Lg )X .
(7.1)
• The formula for the corresponding multitarget likelihood function (see
(7.21)):
fk+1 (Z|X)
=
κk+1 (Z) · (1 − pD )X
∑ ∏
pD (xi ) · fk+1 (zθ(i) |xi )
·
.
(1 − pD (xi )) · κk+1 (zθ(i) )
θ
(7.2)
i:θ(i)>0
• An approximate formula for this likelihood function, assuming that targets
are not too close together (see (7.50)):

fk+1 (Z|X) ∼
= κk+1 (Z) · 1 − pD +
∑
z∈Zk+1
X
pD L z 
.
κk+1 (z)
(7.3)
• A formula expressing the relationship between measurement-to-track association (MTA) and the multitarget likelihood function for the standard multitarget measurement model (see (7.48)):
quasi-uniform prior
likelihood of the MTA θ
∫ ? RFS likelihood
??
?
? ?? ?
??
?
∑ ?
fk+1 (Zk+1 |X) ·
f0 (X)
δX =
ℓZk+1 |Xk+1|k (θ) .
θ
(7.4)
Introduction to Part II
165
• The formula for the p.g.fl. of the standard multitarget motion model, without
spawning (see (7.64)):
′
Gk+1 [h|X ′ ] = ebk+1 [h−1] · (1 − pS + pS Mh )X .
(7.5)
• The formula for the multitarget Markov density function for the standard
multitarget motion model, without spawning (see (7.66)):
fk+1|k (X|X ′ )
′
=
bk+1|k (X) · (1 − pS )X
(7.6)
′
∑ ∏
pS (xi ) · fk+1|k (xθ(i) |xi )
·
.
(1 − pS (x′i )) · bk+1|k (xθ(i) )
θ
7.2
i:θ(i)>0
STANDARD MULTITARGET MEASUREMENT MODEL
The standard multitarget measurement model ([179], pp. 408-422) was introduced
in Section 5.5. At time tk+1 , the measurement RFS is (see (5.21))
Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 ,
(7.7)
where:
• x1 , ..., xn are the distinct states of n targets present at time tk+1 .
• Ck+1 is the Poisson clutter RFS.
• The Bernoulli RFS Υk+1 (xi ) is the set of measurements generated by the
ith target, with |Υk+1 (xi )| ≤ 1.
• Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent.
It follows that Σk+1 is the union of a multi-Bernoulli RFS (the targetgenerated measurements) and a Poisson RFS (clutter). Since |Υk+1 (xi )| ≤ 1,
each target either generates a measurement (a “target detection” or just “detection”)
or it does not (it is “not detected” or it is a “missed detection”).
This section describes the standard measurement model in more detail than in
Section 5.5. It consists of the following subsections:
1. Section 7.2.1: The component submodels of the standard multitarget measurement model.
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2. Section 7.2.2: The p.g.fl. and multitarget likelihood function of the standard
multitarget measurement model.
3. Section 7.2.3: Some special cases of the standard multitarget measurement
model.
4. Section 7.2.4: A review of the theory of measurement-to-track association
(MTA).
5. Section 7.2.5: The relationship between the standard RFS measurement
model and MTA.
7.2.1
Standard Multitarget Measurement Submodels
From the RFS measurement model we get the following model functions:
• Probability of detection. The probability of collecting a (single) measurement
from a target with state x:
pD (x) abbr.
= pD,k+1 (x) = Pr(Υk+1 (x) ̸= ∅).
(7.8)
• Single-target likelihood function. This is the probability density function
Lz (x) abbr.
= fk+1 (z|x) =
δpk+1
(∅|x)
δz
(7.9)
of the probability measure
pk+1 (T |x) = Pr(Υk+1 (x) ⊆ T |Υk+1 (x) ̸= ∅).
(7.10)
It is the probability (density) that a target with state x at time tk+1 will
generate measurement z.
• Clutter intensity function. The PHD (first-moment density) of the clutter
RFS:
κk+1 (z) =
δβCk+1
(∅)
δz
where
βCk+1 (T ) = Pr(Ck+1 ⊆ T ). (7.11)
• Clutter rate. The expected number of clutter measurements:
∫
λk+1 = κk+1 (z)dz.
(7.12)
Introduction to Part II
167
• Clutter spatial distribution. The spatial distribution of the clutter measurements:
κk+1 (z)
ck+1 (z) =
.
(7.13)
λk+1
• Clutter p.g.f. and cardinality distribution. These are
Gκk+1 (z)
=
pκk+1 (m)
=
Gκk+1 [z] = eλk+1 ·(z−1) ,
m
1 dm Gκk+1
−λk+1 λk+1
(0)
=
e
·
m! dz m
m!
(7.14)
(7.15)
where
Gκk+1 [g] = eκk+1 [g−1]
(7.16)
is the p.g.fl. of the Poisson clutter RFS, where
κk+1 [g − 1] =
∫
(g(z) − 1) · κk+1 (z)dz
and where it must be the case that
]
[ κ
dGk+1
λk+1 =
(z)
.
dz
z=1
7.2.2
(7.17)
(7.18)
Standard Multitarget Measurement Model: p.g.fl. and Likelihood
Given this, the fundamental statistical descriptors for the standard model are as
follows. Let X = {x1 , ..., xn } with |X| = n and Zk+1 = {z1 , ..., zm } with
|Z| = m. Then:
• p.g.fl. of the standard measurement model ([179], Eq. (12.151)):
Gk+1 [g|X] = eκk+1 [g−1] · (1 − pD + pD Lg )X
(7.19)
where the power functional notation hX was defined in (3.5); and where
Lg (x) =
∫
g(z) · fk+1 (z|x)dz.
(7.20)
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• Multitarget likelihood function for the standard measurement model ([179],
Eq. (12.139)):
fk+1 (Z|X)
=
κk+1 (Z) · (1 − pD )X
∑ ∏
pD (xi ) · fk+1 (zθ(i) |xi )
·
(1 − pD (xi )) · κk+1 (zθ(i) )
θ
(7.21)
i:θ(i)>0
where
−λk+1
κk+1 (Z) = e−λk+1 · κZ
k+1 = e
∏
κk+1 (z);
(7.22)
z∈Z
and where the summation is taken over all measurement-to-track associations
(MTAs) or association hypotheses θ.
MTAs are functions
θ : {1, ..., n} → {0, 1, ..., m}
such that θ(i) = θ(i′ ) > 0 implies i = i′ . For a given θ,
• θ(i) = 0 indicates that the target xi is hypothesized to have been undetected.
• By convention, the product in (7.21) equals 1 for the unique association such
that θ(i) = 0 identically (that is, when none of the targets are detected).
• θ(i) > 0 indicates that the target xi is hypothesized to have generated
measurement zθ(i) .
7.2.3
Standard Multitarget Measurement Model: Special Cases
Three special cases of the multitarget likelihood function are of interest.
• Multitarget likelihood function—no clutter, λk+1 = 0: Then fk+1 (Z|X) =
0 if m > n and, otherwise ([179], Eq. (12.136)):
fk+1 (Z|X)
=
(1 − pD )X
∑
·
(7.23)
m
∏
pD (xij ) · fk+1 (zj |xij )
1≤i1 ̸=...̸=im ≤n j=1
=
1 − pD (xij )
(1 − pD )X
∑ ∏ pD (τ (z)) · fk+1 (z|τ (z))
·
1 − pD (τ (z))
τ :Z→X z∈Z
(7.24)
Introduction to Part II
169
where the second summation is taken over all injections (that is, one-to-one
functions) τ : Z → X.
• Multitarget likelihood function—no missed detections, pD (x) = 1:
fk+1 (Z|X) = κk+1 (Z)
∑
∏
θ
i:θ(i)>0
fk+1 (zθ(i) |xi )
.
κk+1 (zθ(i) )
(7.25)
• Multitarget likelihood function—no clutter and no missed detections, λk+1 =
0, pD = 1. In this case ([179], Eq. (12.1108)):
fk+1 (Z|X) = δn,m
∑
fk+1 (z|xπ1 ) · · · fk+1 (z|xπn )
(7.26)
π
where the summation is taken over all permutations π on the numbers
1, ..., n.
7.2.4
Measurement-to-Track Association (MTA)
The multitarget likelihood function for the standard multitarget measurement model
has close connections with the theory underlying conventional multitarget tracking
algorithms, such as the multiple-hypothesis tracker (MHT). The purpose of this and
the following section is to describe this relationship in greater detail. As in Section
1.1.3, the following discussion is conceptual. It is not intended to be a description of
the internal logic of any particular conventional multitarget tracking approach. For
further information about such approaches, see the book by Blackman and Popoli
[24].
It is first necessary to establish some notation. If θ is a MTA, then:
• Zθ def.
= {zi | θ(i) > 0} is the set of target detections (target-generated
measurements), with mθ def.
= |Zθ |.
• Z − Zθ is the set of false detections and/or clutter measurements, with
|Z − Zθ | = m − mθ .1
• Xθ def.
= {xi | θ(i) > 0} is the set of detected tracks, with |Xθ | = mθ .
• X − Xθ = {xi | θ(i) = 0} is the set of undetected tracks, with |X − Xθ | =
n − mθ .
1
Note that Z − Zθ could also contain measurements from previously unknown targets. This
possibility is ignored for the sake of conceptual clarity.
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Suppose that Xk+1|k = {x1 , ..., xn } with |Xk+1|k | = n are the predicted
tracks at time tk+1 , with respective track distributions
fk+1|k (x|1), ..., fk+1|k (x|n).
Also suppose that Zk+1 = {z1 , ..., zm } with |Z| = m is the set of new
measurements collected at time tk+1 . In conventional multitarget tracking theory,
the goal is to determine which measurements originated with which predicted tracks
or, alternatively, which measurements are clutter-generated.
In what follows, let z ∈ Zk+1 . Then the total likelihood that z is associated
with the ith predicted track, given that the track was detected, is
∫
ℓk+1 (z|i) = pD (x) · fk+1 (z|x) · f (x|i)dx.
(7.27)
This measures the degree to which the measurement distribution matches the track
distribution, given the degree to which the track can be detected. Also,
∫
ℓk+1 (∅|i) = (1 − pD (x)) · f (x|i)dx
(7.28)
is the total likelihood that the ith track will not be detected at all.
The following discussion summarizes the basic elements of the theory of
MTA (also known as data association).
• No clutter, no missed detections: In this case m = n and a MTA θ is just
a permutation on the numbers 1, ..., n, and pD (x) = 1. The quantity
ℓZk+1 |Xk+1|k (θ)
=
=
ℓk+1 (zθ(1) |1) · · · ℓk+1 (zθ(n) |n)
∏
ℓk+1 (zθ(i) |i)
(7.29)
(7.30)
i:θ(i)>0
is the likelihood—the global association likelihood—that the following association is true: zθ(1) associates with x1 , zθ(2) associates with x2 , and
so on. The larger the number of good associations, the larger the value of the
global association likelihood.
• With clutter but no missed detections: In this case, m ≥ n and
clutter
detections
??
?
??
? ?∏
?
Zk+1 −Zθ
·
ℓk+1 (zθ(i) |i)
ℓZk+1 |Xk+1|k (θ) = e−λk+1 κk+1
i:θ(i)>0
(7.31)
Introduction to Part II
171
is the associated global association likelihood, where the power functional
notation κZ was defined in (3.5). It is the likelihood that the association
zθ(i) ⇔ xi is true for all i, and thus that the remaining measurements in
Zk+1 are clutter-generated.
• With clutter and missed detections: In this case the global association
likelihood as three contributing factors:
missed detections
clutter
detections
? ?∏
??
?
?
??
? ? ∏ ??
−λk+1 Zk+1 −Zθ
ℓZk+1 |Xk+1|k (θ) = e
κk+1
·
ℓk+1 (∅|i) ·
ℓk+1 (zθ(i) |i) .
i:θ(i)=0
i:θ(i)>0
(7.32)
• With clutter and constant probability of detection: Suppose that pD (x) = pD
is constant. Then (7.27) and (7.28) become
ℓk+1 (z|i)
ℓk+1 (∅|i)
where
ℓ˜k+1 (z|i) =
∫
pD · ℓ˜k+1 (z|i)
1 − pD
(7.33)
(7.34)
fk+1 (z|x) · f (x|i)dx.
(7.35)
=
=
Thus (7.32) reduces to
∏
n−mθ
θ
ℓZk+1 |Xk+1|k (θ) = κk+1 (θ)·pm
D (1−pD )
ℓ˜k+1 (zθ(i) |i) (7.36)
i:θ(i)>0
where
Z
k+1
κk+1 (θ) = e−λk+1 · κk+1
−Zθ
.
(7.37)
• Global association probabilities: There is no a priori reason to prefer one
association over another. Thus the prior distribution p0 (θ) on associations
can be assumed to be uniform. The global association probability—the
posterior probability that θ is the correct association—is the posterior
distribution
ℓZk+1 |Xk+1|k (θ)
pZk+1 |Xk+1|k (θ) = ∑
.
(7.38)
′
θ ′ ℓZk+1 |Xk+1|k (θ )
• Linear-Gaussian case with constant probability of detection. In this case,
assume that
fk+1 (z|x) = NR (z − Hx),
fk+1|k (x|i) = NPi (x − xi ).
(7.39)
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Then (7.27) and (7.28) reduce to
ℓk+1 (z|i)
=
pD · NR+HPi H T (z − Hxi )
(7.40)
ℓk+1 (∅|i)
=
1 − pD .
(7.41)
In this case the global association probability becomes
pZk+1 |Xk+1|k (θ)
∝
n−mθ
θ
κk+1 (θ) · pm
D (1 − pD )
·QZk+1 |Xk+1|k (θ) · e
(7.42)
− 12 dZk+1 |Xk+1|k (θ)2
where κk+1 (θ) was defined in (7.37) and where
=
dZk+1 |Xk+1|k (θ)2
(7.43)
∑
T
T −1
(zθ(i) − Hxi ) (R + HPi H ) (zθ(i) − Hxi )
i:θ(i)>0
and where
1
QZk+1 |Xk+1|k (θ) = ∏
i:θ(i)>0
√
.
(7.44)
det 2π(R + HPi H T )
The quantity dZk+1 |Xk+1|k (θ) is the global association distance. The
association θ that minimizes dZk+1 |Xk+1|k (θ) is the best association
in a global nearest-neighbor sense.
To see why (7.42) is true, note that under linear-Gaussian assumptions, (7.36)
becomes
ℓZk+1 |Xk+1|k (θ)
=
(7.45)
n−mθ
θ
κk+1 (θ) · pm
D (1 − pD )
·
∏
NR+HPi H T (zθ(i) − Hxi )
i:θ(i)>0
=
n−mθ
θ
κk+1 (θ) · pm
(7.46)
D (1 − pD )


∏
1

√
·
det 2π(R + HPi H T )
i:θ(i)>0


∑
1
· exp −
(zθ(i) − Hxi )T (R + HPi H T )−1 (zθ(i) − Hxi )
2
i:θ(i)>0
Introduction to Part II
173
from which (7.42) follows.
7.2.5
Relationship Between the MTA and RFS Approaches
Assume a priori that the n predicted tracks are equally likely to be actual targets.
Then the multitarget prior distribution that describes the predicted tracks is the
“quasi-uniform” multitarget prior distribution. It is defined as f0 (X) = 0 if
|X| ̸= n and, if otherwise with X = {x1 , ..., xn } and |X| = n,
∑
f0 (X) =
fk+1|k (x1 |π1) · · · fk+1|k (xn |πn)
(7.47)
π
where the summation is taken over all permutations π on the numbers 1, ..., n.
If fk+1 (Z|X) is defined as in (7.21), then the following equation, which is
proved in Section K.11, establishes the basic relationship between RFS theory and
the conventional MTA approach:
RFS theory
?∫
??
?
fk+1 (Zk+1 |X) · f0 (X)δX =
MTA theory
?∑
??
?
ℓZk+1 |Xk+1|k (θ) .
(7.48)
θ
That is, the probability (density) that the measurement set Zk+1 will be collected
from the predicted tracks Xk+1|k is the same as the total likelihood of association
between Zk+1 and Xk+1|k , taken over all possible associations.
Remark 15 (MTA with track existence probabilities) Note that (7.47) is not the
only possible prior distribution for the predicted tracks. For example, suppose that
each track has an associated probability of existence qi . In this case, f0 (X)
should be chosen to be a multi-Bernoulli distribution as defined in (7.48) . In
this case, (7.48) would give rise to a significantly more complicated formula for
ℓZk+1 |Xk+1|k (θ). Even so, it should be recognized that any MTA approach that
presumes a fixed form for f0 (X) is inherently a heuristic approximation. The only
theoretically rigorous choice for f0 (X) is f0 (X) = fk+1|k (X|Z (k) ).
7.3
AN APPROXIMATE STANDARD LIKELIHOOD FUNCTION
The multitarget likelihood function in (7.21) is intended for use in the measurementupdate step of the multitarget Bayes filter, (5.2). Even in particle implementations—
see [179], Chapter 15—this step will be, in general, computationally challenging.
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For this reason Reuter and Dietmayer proposed a computational approximation of (7.21) for use with multitarget particle filters when targets are not too close
together [246], [247]. This approximation, which presumes that there is no clutter,
is:
(
)X
∑
∼
fk+1 (Z|X) = 1 − pD +
pD L z
(7.49)
z∈Z
X
where the power-functional notation h was defined in (3.5). The expression in
the parentheses on the right is not mathematically well defined, however. This is
because the term 1 − pD (x) is unitless, whereas the units of measurement of the
summation are inverse to those of z.
Nevertheless, the underlying idea can both be rendered valid and generalized.
Assume that there is clutter and that it is Poisson. If targets are not too close
together, an approximate multitarget likelihood for the standard measurement model
is:

fk+1 (Z|X) ∼
= κk+1 (Z) · 1 − pD +
∑
z∈Zk+1
X
pD L z 
κk+1 (z)
(7.50)
where the Poisson-clutter factor κk+1 (Z) is as in (7.21):
|Z|
κk+1 (Z) = e−λk+1 · λk+1
∏
ck+1 (z).
(7.51)
z∈Z
This approximate equality is established in Section K.12. Note that if κk+1 (z) is
constant on some bounded region (compact subset) of X, then (7.49) becomes a
special case of (7.50).
Remark 16 Note the right side of (7.50)
∫ does not define an actual multitarget
likelihood function. This is because
fk+1 (Z|X)δZ is not necessarily equal
to 1. Rather, (7.50) defines an approximation of a multitarget likelihood function.
7.4
STANDARD MULTITARGET MOTION MODEL
The standard multitarget motion model ([179], pp. 466-474) was introduced in Section 5.4. It is a direct mathematical analog of the standard multitarget measurement
Introduction to Part II
175
model. Recall from (5.4) that it has the form
Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n′ ) ∪ Bk+1|k
(7.52)
where
• x′1 , ..., x′n′ are the target states at the earlier time tk .
• Bk+1|k is the Poisson target appearance (“target birth”) RFS.
• The Bernoulli RFS Tk+1|k (x′i ) is the set of targets generated at time tk+1
by the ith target at time tk , with |Tk+1|k (x′i )| ≤ 1.
• Tk+1|k (x′1 ), ..., Tk+1|k (x′n′ ), Bk+1|k are independent.
Thus Ξk+1|k is the union of a multi-Bernoulli RFS (the existing targets) and
a Poisson RFS (the newly appearing or “birth” targets). Since |Tk+1 (x′i )| ≤ 1,
any existing target either survives into the next time-step, or it disappears (“target
death”).
From the RFS motion model we get the following model functions:
• Probability of target survival. This is the probability that a target with state
x′ at time tk will survive into time tk+1 :
pS (x′ ) abbr.
= pS,k+1|k (x′ ) = Pr(Tk+1|k (x′ ) ̸= ∅).
(7.53)
• Single-target Markov transition density. This is the probability density
function
δpk+1|k
fk+1|k (x|x′ ) =
(∅|x′ )
(7.54)
δz
of the probability measure
pk+1|k (S|x′ ) = Pr(Tk+1|k (x′ ) ⊆ S|Tk+1|k (x′ ) ̸= ∅).
It is the probability (density) that a target with state x′ at time
transition to a target with state x at time tk+1 .
(7.55)
tk will
• Target-appearance intensity function. The PHD (first-moment density) of the
target-birth RFS:
bk+1 (x) =
δβBk+1|k
(∅)
δx
where βBk+1|k (S) = Pr(Bk+1|k ⊆ S). (7.56)
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• Target-birth rate. The expected number of newly appearing targets:
∫
B
Nk+1|k = bk+1|k (x)dx.
(7.57)
• Target-birth spatial distribution. The spatial distribution of the newly appearing targets:
bk+1|k (x)
sB
.
(7.58)
k+1|k (x) =
B
Nk+1|k
• Target-appearance p.g.f. and cardinality distribution. These are, respectively,
GB
k+1|k (x)
=
Nk+1|k ·(x−1)
GB
k+1|k [x] = e
pB
k+1|k (n)
=
(Nk+1|k )
B
1 d Gk+1|k
(0) = e−Nk+1|k ·
n!
dx
n!
n
(7.59)
B
B
n
(7.60)
where
bk+1|k [h−1]
GB
k+1|k [h] = e
(7.61)
is the p.g.fl. of the Poisson target appearance RFS with PHD bk+1|k (x);
where
∫
bk+1|k [h − 1] = (h(x − 1) · bk+1|k (x)dx
(7.62)
and where it must be the case that
[
B
Nk+1|k
=
dGB
k+1|k
(x)
]
(7.63)
.
dx
x=1
Given this, the fundamental statistical descriptors for the standard motion
model are as follows. Let X ′ = {x′1 , ..., x′n′ } with |X ′ | = n′ be the targets
at time tk and and X = {x1 , ..., xn } with |X| = n be the targets at time tk+1 .
Then:
• p.g.fl. of the standard multitarget motion model ([179], Eq. (13.61)):
Gk+1 [h|X ′ ] = ebk+1 [h−1] · (1 − pS + pS Mh )X
where the notation hX was defined in (3.5) and where
∫
′
Mh (x ) = h(x) · fk+1|k (x|x′ )dx.
′
(7.64)
(7.65)
Introduction to Part II
177
• Multitarget Markov transition density for the standard multitarget motion
model ([179], Eqs. (113.42,13.43)):
fk+1|k (X|X ′ )
′
bk+1|k (X) · (1 − pS )X
(7.66)
′
∑ ∏
pS (xi ) · fk+1|k (xθ(i) |xi )
·
(1 − pS (x′i )) · bk+1|k (xθ(i) )
=
θ
i:θ(i)>0
where
B
bk+1|k (X) = e−Nk+1|k
∏
bk+1|k (x)
(7.67)
x∈X
∫
B
is the Poisson target appearance process; where Nk+1|k
= bk+1|k (x)dx;
and where the summation is taken over all functions θ : {1, ..., n′ } →
{0, 1, ..., n} such that θ(i) = θ(i′ ) > 0 implies i = i′ . For a
given θ, θ(i) = 0 indicates that the target with state xi disappeared;
whereas if θ(i) > 0 then it is hypothesized to have transitioned to a target
with state xθ(i) . By convention, the product in (7.66) equals 1 for the
unique association such that θ(i) = 0 identically (that is, all of the targets
disappear).
Three special cases should be pointed out:
B
• Multitarget Markov density—no target births, Nk+1|k
= 0 ([179], Eqs.
′
(13.38,13.39)): Then fk+1|k (X|X ) = 0 if n > n′ and, otherwise,
fk+1|k (X|X ′ )
=
(7.68)
X′
(1 − pS )
∑
·
n
∏
pS (x′ij ) · fk+1|k (xj |x′ij )
1≤i1 ̸=...̸=im ≤n′ j=1
=
(1 − pS )X
′
1 − pS (x′ij )
∑ ∏ pD (τ (x)) · fk+1|k (x|τ (x))
τ
x∈X
(7.69)
1 − pS (τ (x))
where the second summation is taken over all one-to-one functions τ : X →
X ′ . That is, for a given τ and X = {x1 , ..., xn } with |X| = n, the targets
with states {τ (x1 ), ..., τ (xn )} ⊆ X ′ at time tk+1 are hypothesized to have
transitioned to the targets x1 , ..., xn at time tk+1 ; and the other targets in
X ′ are hypothesized to have disappeared.
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• Multitarget Markov density—no target deaths, pS (x′ ) = 0:
fk+1|k (X|X ′ ) = bk+1 (X) ·
∑
∏
θ
i:θ(i)>0
fk+1|k (xθ(i) |x′i )
.
bk+1|k (xθ(i) )
(7.70)
B
• Multitarget Markov density—no target births or deaths, Nk+1
= 0, pS = 1:
In this case ([179], Eq. (12.35)):
∑
fk+1|k (X|X ′ ) = δn,n′
fk+1|k (x|x′π1 ) · · · fk+1|k (x|x′πn )
(7.71)
π
where the summation is taken over all permutations π on the numbers
1, ..., n.
7.5
STANDARD MOTION MODEL WITH TARGET SPAWNING
A variant of the standard motion model, (7.52), involves relaxing the assumption
that Tk+1|k (x′ ) is Bernoulli, allowing |Tk+1|k (x′ )| to have values other than 0
and 1. In this case, the target x′i at time tk is said to have spawned the targets
in Tk+1|k (x′i ) at time tk+1 . (A target can, of course, spawn itself.)
The formulas for the p.g.fl. and the Markov transition density for the standard
motion model with spawning can be found in [179], pp. 472-474.
7.6
ORGANIZATION OF PART II
Part II is organized as follows:
1. Chapter 8: The “classical” PHD and CPHD filters. This includes the general
PHD filter and a zero-false-alarms (ZFA) CPHD filter.
2. Chapter 9: Practical implementation of the classical PHD and CPHD filters.
3. Chapter 10: PHD and CPHD filters for multiple sensors.
4. Chapter 11: Jump-Markov versions of the PHD and CPHD filters—a method
for increasing the performance of the PHD and CPHD filters, given the
presence of rapidly maneuvering, “noncooperative” targets.
5. Chapter 12: Extension of the PHD filter to estimation of unknown spatial
biases in the sensors.
Introduction to Part II
179
6. Chapter 13: Multi-Bernoulli filters, including the Bernoulli and CBMeMBer
filters.
7. Chapter 14: RFS Bayes multitarget smoothers.
8. Chapter 15: The Vo-Vo exact closed-form solution of the multitarget Bayes
filter.
Chapter 8
Classical PHD and CPHD Filters
8.1
INTRODUCTION
The fundamental reasoning that leads to the “classical” PHD and CPHD filters was
described in Sections 5.10.4 and 5.10.5. The purpose of this chapter is to describe
these filters, as well as certain special cases and generalizations, in greater detail.
8.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• A general PHD filter, which allows general models for clutter and general
models for target-generated measurements. Its measurement-update equation
has the form
Dk+1|k+1 (x)
= 1 − p̃D (x) +
Dk+1|k (x)
∑
P⊟Zk+1
ωP
∑
LW (x)
κW + τW
(8.1)
W ∈P
where the summation is taken over all partitions P of the current measurement set Zk+1 . This filter is a consequence of Clark’s general chain rule
(3.91), and can be generalized to the multisensor case (see Section 10.3).
• Equation (8.1) is the underlying (but previously unrecognized) theoretical
basis for previously reported PHD filters—for example, the extended-target
PHD filter ([174], [226], Chapter 21) and the unresolved-target PHD filter of
[175], to be described in Chapter 21.
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• While (8.1) is combinatorial in nature, researchers are devising serviceable
approximations (as will be explained in more detail in Section 21.4.3.3).
• An immediate special case of (8.1) is a generalization of the classical PHD filter that is valid for arbitrary clutter processes and has the same computational
complexity as the classical PHD filter. Its measurement-update equation is
∑
Dk+1|k+1 (x)
= 1 − pD (x) +
Dk+1|k (x)
pD (x) · Lz (x)
κk+1 ({z})
z∈Zk+1 κk+1 (∅) + τk+1 (x)
(8.2)
where κk+1 (Z), the multiobject density function for the clutter RFS, is
evaluated at Z = ∅ and Z = {z}.
• An immediate and cautionary consequence of (8.2) is (see Remark 21 of
Section 8.3.3):
– Merely because a clutter-like term (such as κk+1 ({z})/κk+1 (∅)) occurs in a PHD filter-like formula (such as (8.2)), this does not necessarily mean that this term is actually the intensity function (PHD) of the
clutter RFS.
• The measurement-update equation for the classical PHD filter is a special
case of (8.2):
∑
Dk+1|k (x)
pD (x) · Lz (x)
= 1 − pD (x) +
.
Dk+1|k (x)
κk+1 (z) + τk+1 (x)
(8.3)
z∈Zk+1
• The time-update and measurement-update equations for the “classical” CPHD
filter (Section 8.5).
• If targets are assumed to be not too close together, the classical CPHD
filter can be replaced by an approximate CPHD filter that has the same
computational complexity as the classical PHD filter (Section 8.5.7).
• A useful special case of the classical CPHD filter, the zero false alarms
(ZFA) CPHD filter, is valid when there is no clutter. It has the same
Classical PHD and CPHD Filters
183
computational complexity as the classical PHD filter. It has the measurementupdate equations (Section 8.6):
(m+1)
Nk+1|k · Dk+1|k+1 (x)
Dk+1|k (x)
Gk+1|k (ϕk )
=
(8.4)
(1 − pD (x)) ·
(m)
Gk+1|k (ϕk )
+
∑ pD (x) · Lz (x)
τ̂k+1 (z)
z∈Zk+1
pk+1|k+1 (n)
8.1.2
∝
Cn,m · ϕn−m
. · pk+1|k (n).
k
(8.5)
Organization of the Chapter
The chapter is organized as follows:
1. Section 8.2: The general PHD filter for arbitrary clutter models and arbitrary
target-generated measurements.
2. Section 8.3: The generalization of the classical PHD filter to arbitrary clutter
processes.
3. Section 8.4: The classical PHD filter.
4. Section 8.5: The classical CPHD filter.
5. Section 8.6: The zero-false-alarms (ZFA) CPHD filter.
6. Section 8.7: A generalization of the PHD filter to the case when the Poisson
clutter depends on the states of the targets.
8.2
A GENERAL PHD FILTER
The classical PHD filter is based on the standard multitarget measurement model
described in Section 7.2. This model—targets generate at most single measurements
and the clutter RFS is Poisson—ensures that the classical PHD filter is computationally tractable.
However, there is a much more general PHD filter, originally reported in [47],
that is based on the generalized standard multitarget measurement model described
at the end of Section 5.5. It allows both the clutter and target measurementgeneration processes to be arbitrary.
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This filter has the following limitation: the PHD measurement-update equation involves a combinatorial sum over all partitions of the current measurement
set. Despite this fact, serviceable approximations are currently being devised for
measurement-update equations of this type.
The purpose of this section is to summarize this general PHD filter. A
multisensor version of the filter is described in Section 10.3.
The classical PHD filter is based on a multitarget measurement model of the
form
Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1
(8.6)
where the clutter RFS Ck+1 is Poisson, where Υk+1 (x) is a Bernoulli RFS for
all x, and where Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent.
In this section, we will allow both Ck+1 and Υk+1 (x) to be arbitrary.
The general PHD filter has the same motion model as the classical PHD filter,
namely
Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n ) ∪ Bk+1|k
(8.7)
where x′1 , ..., x′n are the target states at time tk ; where the target appearance RFS
Bk+1|k is Poisson; where Tk+1|k (x′ ) is Bernoulli, and where
Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k
are independent.
Remark 17 (Cluster processes) The RFS of target-generated measurements provides a specific instance of a cluster process, as briefly discussed in Example 2 of
Section 4.4.2. In the notation of that section, Ψ = Ξk+1|k is the parent process, ∆x = Υx abbr.
= Υk+1 (x) is the daughter process, and the target-generated
measurement RFS
∪
∆ = Σk+1 =
Υk+1 (x)
(8.8)
x∈Ξk+1|k
is the total cluster process. The joint measurement-target p.g.fl. is, from (4.141),
F [g, h] = GΞk+1|k ,Σk+1 [g, h] = GΞk+1|k [h · GΥ∗ [g]]
where
T [g](x) abbr.
= GΥx [g] =
∫
g Z · fΥx (Z)δZ
(8.9)
(8.10)
are the p.g.fl.’s of the daughter RFSs, and where GΞk+1|k [h · GΥ∗ [g]] is shorthand
for GΞk+1|k [h · T [g]]. Similar remarks apply to the RFS of surviving targets.
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The section is organized as follows:
1. Section 8.2.1: Motion modeling for the general PHD filter (as well as for the
classical PHD filter).
2. Section 8.2.2: The time-update equation for the general PHD filter (as well
as for the classical PHD filter).
3. Section 8.2.3: Measurement modeling for the general PHD filter.
4. Section 8.2.4: The measurement-update equation for the general PHD filter.
8.2.1
General PHD Filter: Motion Modeling
The following motion models apply not just to this filter, but also to the classical
PHD filter (Section 8.4.1) and its generalization to arbitrary clutter processes
(Section 8.3.1).
1. Single-target Markov transition density: fk+1|k (x|x′ ). This is the probability
(density) that a target with state x′ at time tk will have state x at time
tk+1 .
2. Target-survival probability: pS (x′ ) abbr.
= pS,k+1 (x′ ). This is the probability
′
that a target with state x at time tk will not disappear at time tk+1 .
3. Target-appearance PHD: bk+1|k (x). This is the PHD (intensity function) of
Birth
the multitarget distribution fk+1|k
(X)—that is, of the probability (density)
that a set X of new targets will appear in the scene at time tk+1 . The
quantity
∫
B
Nk+1|k
=
bk+1|k (x)dx
(8.11)
is the target-birth rate (the expected number of newly appearing targets); and
sB
k+1|k (x) =
bk+1|k (x)
B
Nk+1|k
(8.12)
is the spatial distribution of the appearing targets.
4. Target-spawning PHD: bk+1|k (x|x′ ). This is the PHD of the multitarget
Spawn
distribution fk+1|k
(X|x′ )—that is, of the probability (density) that a target
′
with state x at time tk will spawn a set X of new targets at time tk+1 .
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The quantity
B
Nk+1|k
(x′ ) =
∫
bk+1|k (x|x′ )dx
(8.13)
is the target-spawning rate (the expected number of targets spawned by the
target with state x′ ); and
′
sB
k+1|k (x|x ) =
bk+1|k (x|x′ )
B
Nk+1|k
(x′ )
(8.14)
is the spatial distribution of the targets spawned by the target with state x′ .
8.2.2
General PHD Filter: Predictor
The following time-update equations apply not only to the general PHD filter but
also to the classical PHD filter (Section 8.4.1) and its generalization to arbitrary
clutter processes (Section 8.3.1).
Suppose that we already have in hand the PHD Dk|k (x) and the expected
number of targets Nk|k . We are to determine the predicted PHD Dk+1|k (x) and
the predicted expected number of targets Nk+1|k . The predicted PHD is given by
the exact (not approximate) equation1
Dk+1|k (x) = bk+1|k (x) +
∫
Fk+1|k (x|x′ ) · Dk|k (x′ )dx′
(8.15)
where the PHD filter pseudo-Markov density is
Fk+1|k (x|x′ ) = pS (x′ ) · fk+1|k (x|x′ ) + bk+1|k (x|x′ ).
(8.16)
The expected number of predicted targets is, therefore,
Nk+1|k
=
=
1
∫
Dk+1|k (x)dx
(8.17)
∫ (
)
B
B
Nk+1|k
+
pS (x′ ) + Nk+1|k
(x′ ) · Dk|k (x′ )dx′ . (8.18)
Note: No special assumption is made regarding the nature of the prior multitarget distribution
fk|k (X|Z (k) ). In particular, it is not presumed to be Poisson.
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Remark 18 (Special cases of the time-update equation) For the special case in
which no target-spawning occurs, these formulas simplify to:
Dk+1|k (x)
=
Nk+1|k
=
∫
bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (8.19)
∫
B
Nk+1|k + pS (x′ ) · Dk|k (x′ )dx′ .
(8.20)
If in addition no targets appear at all, they further simplify to
Dk+1|k (x)
Nk+1|k
=
∫
pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
(8.21)
=
∫
pS (x′ ) · Dk|k (x′ )dx′ .
(8.22)
If in addition no targets disappear, they finally simplify to
Dk+1|k (x)
Nk+1|k
8.2.3
=
∫
fk+1|k (x|x′ ) · Dk|k (x′ )dx′
(8.23)
=
∫
Dk|k (x′ )dx′ = Nk|k .
(8.24)
General PHD Filter: Measurement Modeling
The time-update formulas for the general PHD filter require the following models:
• Multi-measurement, single-target likelihood function. This is the multitarget
probability density function of Υk+1 (x):
LZ (x) abbr.
= fk+1 (Z|x) =
δGxk+1
[0]
δZ
(8.25)
where Gxk+1 [g] is the p.g.fl. of Υk+1 (x).
• Generalized probability of detection. This is the probability that at least one
measurement is collected from a target with state x,
p̃D (x) abbr.
= p̃D,k+1 (x) = 1 − fk+1 (∅|x).
(8.26)
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• Log-clutter density. This is the multiobject density function of the clutter
log-p.g.fl. log Gκk+1 [g], where Gκk+1 [g] is the p.g.fl. of the clutter RFS
Ck+1 :
δ log Gκk+1
κZ =
[0].
(8.27)
δZ
• Nonubiquity of clutter. In order to ensure that log Gκk+1 [g] is well defined,
we must assume that it is possible for no clutter measurements to be collected:
pκk+1 (0) > 0.
(8.28)
(Note that this assumption is automatically true for Poisson clutter.) As a
consequence, Gκk+1 [g] > 0 for all g and thus log Gκk+1 [g] is well
defined.
8.2.4
General PHD Filter: Corrector
This equation is a consequence of Clark’s general chain rule, (3.91):
Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x)
(8.29)
where the PHD pseudolikelihood is given by ([47], Eqs. (27-29))
LZk+1 (x) = 1 − p̃D (x) +
∑
P⊟Zk+1
ωP
∑
LW (x)
.
κW + τW
(8.30)
W ∈P
Here, the summation is taken over all partitions P of the measurement set Zk+1
and
∫
τW =
LW (x) · Dk+1|k (x)dx
(8.31)
∏
W ∈P (κW + τW )
∏
ωP = ∑
.
(8.32)
Q⊟Zk+1
V ∈Q (κV + τV )
For a proof, see [47], Section IV. See Appendix D for an introduction to the theory
of partitions.
Remark 19 (Computational complexity of general PHD filter) The practical utility of (8.30) might be questioned because of the combinatorial sum. However, similar combinatorial sums occur in the corrector equation for the extended-target PHD
Classical PHD and CPHD Filters
189
filter of [174], to be described in Chapter 21. Yet serviceable approximations are
being developed despite this fact [226], [285]—see Section 21.4.3. It should be also
be remembered that the “ideal” multihypothesis tracker (MHT) is also inherently
combinatorial. Nevertheless, numerous techniques have been developed to make it
computationally practical.
8.3
ARBITRARY-CLUTTER PHD FILTER
An immediate consequence of (8.29) through (8.32) is that:
• The measurement-update for the classical PHD filter (Section 8.4.3) can
be generalized to include arbitrary clutter processes, but without increasing
computational complexity.
This filter is described in this section.
8.3.1
Time Update Equations for the Arbitrary-Clutter Classical PHD Filter
The motion models and time-update equations for this filter are the same as those
for the general PHD filter—see Sections 8.2.1 and 8.3.1.
8.3.2
Measurement Modeling for the Arbitrary-Clutter Classical PHD Filter
This filter employs the following models. Because of (5.42), the generalized
probability of detection p̃D (x) of (8.26) reduces to the conventional probability of
detection:
p̃D (x) = pD (x).
(8.33)
Similarly, because of (5.49) the general single-target likelihood function LZ (x),
(8.25), reduces to

if
Z=∅
 1 − pD (x)
δGxk+1
pD (x) · Lz (x) if Z = {z} .
LZ (x) =
[0] =
(8.34)

δZ
0
if |Z| > 1
Thus we end up with the following models:
1. Sensor probability of detection: pD (x) abbr.
= pD,k+1 (x). This is the probability
that a target with state x at time tk+1 will generate some measurement.2
2
The generalized probability of detection p̃D (x) in (8.26) reduces to this: p̃D (x) = pD (x).
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2. Sensor likelihood function: Lz (x) abbr.
= fk+1 (z|x). This is the probability
(density) that, if a target with state x at time tk+1 does generate a
measurement, it will generate measurement z.
3. Arbitrary clutter distribution κk+1 (Z), of which only two aspects—κk+1 ({z})
and κk+1 (∅) with κk+1 (∅) > 0—need be known a priori.
8.3.3
Arbitrary-Clutter PHD Filter: Corrector
Under these assumptions, (8.30) reduces to
∑
LZk+1 (x) = 1 − pD (x) +
pD (x) · Lz (x)
κ̃k+1 (z) + τk+1 (z)
(8.35)
z∈Zk+1
where (8.31) reduces to
τk+1 (z) =
∫
pD (x) · Lz (x) · Dk+1|k (x)dx
(8.36)
and where the clutter “pseudointensity” function is, because of (8.27),
κ̃k+1 (z) =
κk+1 ({z})
.
pκk+1 (0)
(8.37)
Remark 20 (Derivation of arbitrary-clutter PHD filter) Because of (8.34), LW =
0 whenever |W | > 1. Thus the only term in (8.30) that survives is the one
corresponding to that single partition P of Zk+1 whose cells are the |Zk+1 |
singleton subsets of Zk+1 . Thus (8.30) reduces to
∑
L{z} (x)
κ{z} + τ{z}
(8.38)
pD (x) · Lz (x) · Dk+1|k (x)dx
(8.39)
LZk+1 (x) = 1 − pD (x) +
z∈Zk+1
where from (8.31)
τ{z} =
∫
L{z} (x) · Dk+1|k (x)dx =
∫
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and where from (8.27)
κ{z}
=
=
[
]
δ log Gκk+1
δ log Gκk+1
[0] =
[g]
δz
δz
g=0
[
]
κ
δGk+1
1
[g]
Gκk+1 [g] δz
g=0
1
κk+1 ({z})
· κk+1 ({z}) = κ
.
Gκk+1 [0]
pk+1 (0)
=
(8.40)
(8.41)
(8.42)
Thus (8.38) becomes (8.35).
Remark 21 (Cautionary note) The form of (8.35), namely
LZk+1 (x) = 1 − pD (x) +
∑
pD (x) · Lz (x)
,
κ̃k+1 (z) + τk+1 (x)
(8.43)
z∈Zk+1
might lead one to assert that κ̃k+1 (z) is the PHD (intensity function)
κk+1 (z) =
δGκk+1
δ log Gκk+1
[1] =
[1]
δz
δz
(8.44)
of the clutter RFS Ck+1 . However, this is clearly not true since, in general,
δ log Gκk+1
δ log Gκk+1
[0] ̸=
[1].
δz
δz
(8.45)
Thus, just because one sees a clutter intensity-like function κ̃k+1 (z) in an equation
such as (8.43), this does not necessarily mean that κ̃k+1 (z) is actually a clutter
intensity function. A separate proof is required to demonstrate the fact. We will
have occasion to revisit this issue with regard to “clutter agnostic” PHD and CPHD
filters in Section 18.5.5.
8.4
CLASSICAL PHD FILTER
The classical PHD filter arises when we assume in (8.35) that the clutter is Poisson:
κk+1 (Z) = e−λk+1 · κZ
k+1 ,
(8.46)
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in which case (8.37) reduces to
κ̃k+1 (z) =
κk+1 ({z})
e−λk+1 · κk+1 (z)
=
= κk+1 (z).
κ
pk+1 (0)
e−λk+1
(8.47)
The purpose of this section is to describe the classical PHD filter and its major
characteristics. The section is organized as follows:
1. Section 8.4.1: The time-update equations for the classical PHD filter.
2. Section 8.4.2: Measurement-modeling assumptions for the classical PHD
filter.
3. Section 8.4.3: The measurement-update equations for the classical PHD
filter.
4. Section 8.4.4: Multitarget state estimation for the classical PHD filter.
5. Section 8.4.5: Multitarget uncertainty estimation for the classical PHD filter.
6. Section 8.4.6: Characteristics of the classical PHD filter.
8.4.1
Classical PHD Filter: Predictor
The motion models and time-update equations for the classical PHD filter are the
same as those for the general PHD filter (see Section 8.2.2).
8.4.2
Classical PHD Filter: Measurement Modeling
The PHD filter measurement-update formulas in Section 8.4.3 require the following
models (originally defined in Section 7.2):
1. Sensor probability of detection: pD (x) abbr.
= pD,k+1 (x). This is the probability
that a target with state x at time tk+1 will generate some measurement.
2. Sensor likelihood function: Lz (x) abbr.
= fk+1 (z|x). This is the probability
(density) that, if a target with state x at time tk+1 does generate a
measurement, it will generate measurement z.
3. Clutter intensity function (also known as clutter PHD) κk+1 (z). This is
the PHD of the multiobject distribution κk+1 (Z), which, in turn, is the
probability (density) that a set Z of clutter measurements will be generated
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at time tk+1 . The quantity
λk+1 =
∫
κk+1 (z)dz
(8.48)
is the clutter rate (expected number of clutter measurements); and
ck+1 (z) =
κk+1 (z)
λk+1
(8.49)
is the clutter spatial distribution.
8.4.3
Classical PHD Filter: Corrector
Suppose that we have:
• A new measurement set Zk+1 = {z1 , ..., zm } with |Zk+1 | = m.
• The predicted PHD Dk+1|k (x).
• The predicted expected number of targets Nk+1|k .
We are to determine the measurement-updated PHD Dk+1|k+1 (x) and the
measurement-updated expected number of targets Nk+1|k+1 . To achieve closedform formulas, the following assumption is required:
• The predicted multitarget distribution fk+1|k (X|Z (k) ) is Poisson.
The measurement-updated PHD is given by
Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x)
(8.50)
where the PHD filter pseudolikelihood function is
∑
pD (x) · Lz (x)
κk+1 (z) + τk+1 (z)
(8.51)
pD (x) · Lz (x) · Dk+1|k (x)dx.
(8.52)
LZ (x) = 1 − pD (x) +
z∈Z
and where
τk+1 (z) =
∫
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The expected number of target tracks is, therefore, found by integrating
Dk+1|k+1 (x):
∑
τk+1 (z)
κk+1 (z) + τk+1 (z)
(8.53)
(1 − pD (x)) · Dk+1|k (x)dx.
(8.54)
Remark 22 (A useful identity) Let
∫
fk+1 (Z|Z (k) ) = fk+1 (Z|X) · fk+1|k (X|Z (k) )δX
(8.55)
Nk+1|k+1 = Dk+1|k [1 − pD ] +
z∈Zk+1
where
Dk+1|k [1 − pD ] =
∫
be the Bayes normalization factor and assume that fk+1|k (X|Z (k) ) is Poisson.
Then the following identity is sometimes useful ([165], Eq. (116)):
∏
fk+1 (Z|Z (k) ) = e−λk+1 −Dk+1|k [pD ]
(κk+1 (z) + τk+1 (z))
(8.56)
z∈Z
where
Dk+1|k [pD ] =
8.4.4
∫
pD (x) · Dk+1|k (x|Z (k) )dx.
(8.57)
Classical PHD Filter: State Estimation
The following procedure is commonly employed to estimate the current number of
targets and their states.
1. The quantity Nk+1|k+1 in (8.53) is the expected number of targets. Round
it off to the nearest integer n.
2. Look for the n largest local suprema x1 , ..., xn of Dk+1|k+1 (x|Z (k+1) )—
that is, those x corresponding to its n largest “peaks.”
3. Take the x1 , ..., xn to be the estimates of the states of the target tracks.
4. In the event that there are fewer than n local suprema, the states corresponding to the actual number n′ of local suprema are taken to be the state
estimates. In this case, it is implicitly understood that some targets are so
close together that they correspond to a single peak of the PHD. Additionally, if the clutter rate is large then it is possible for the states corresponding
to clutter-induced peaks to be selected as estimated target states.
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8.4.5
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Classical PHD Filter: Uncertainty Estimation
The PHD filter is a “first order” approximation of the multitarget Bayes filter, in a
point process sense. This does not mean, however, that the PHD filter cannot supply
second-order information in the conventional sense—that is, track covariances. Let
x1 , ..., xn be the track state estimates. Then for a given xi , the basic idea is
to determine the covariance matrix Pi such that the Gaussian NPi (x − xi ) is,
according to some criterion, a best fit to Dk+1|k+1 (x) at x = xi .
For Gaussian mixture (GM) implementations of the PHD filter, determination
of Pi is simple. It is more difficult for sequential Monte Carlo (SMC) implementations, because data-clustering techniques are required. See Sections 9.5 and 9.6
for more details.
8.4.6
Classical PHD Filter: Characteristics
The purpose of this section is to summarize some of the more important points
regarding the classical PHD filter. These are:
1. Section 8.4.6.1: Consistency property of the PHD filter.
2. Section 8.4.6.2: Computational complexity of the PHD filter.
3. Section 8.4.6.3: “Target-like” measurements versus “clutter-like” measurements.
4. Section 8.4.6.4: The “self-gating” property of the PHD filter.
5. Section 8.4.6.5: The linearizing effect of the PHD filter.
6. Section 8.4.6.6: Window-averaging to achieve better target-number estimates.
7. Section 8.4.6.7: A generalization of the PHD filter in which the Poisson
approximation is relaxed to a Gauss-Poisson approximation.
8. Section 8.4.6.8: Alternative mathematical derivations of the PHD filter.
8.4.6.1
Classical PHD Filter: Consistency Property
Assume that the scenario is single-sensor, single-target. That is, suppose that: (1)
there is a single target, (2) there is a single sensor, (3) there are no missed detections,
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and (4) there are no false alarms/clutter. Given these assumptions, the PHD filter
reduces to the single-sensor, single-target Bayes filter.
To see why, write Dk+1|k (x) = fk+1|k (x) and Dk|k (x) = fk|k (x) where
fk+1|k (x) and fk|k (x) are probability density functions. Under our assumptions,
(8.23) reduces to:
∫
fk+1|k (x) = fk+1|k (x|x′ ) · fk|k (x′ )dx′ .
(8.58)
This is the time-update equation for the single-sensor, single-target Bayes filter,
(2.25).
Under our assumptions, the measurement-update for Zk+1 = {z1 }, (8.50)
and (8.51), reduce to:
fk+1|k+1 (x)

=
=
=
(8.59)

m
∑
pD (x) · Lzj (x)
 · fk+1|k (x)
κ
(z
)
+
τ
(z
)
k+1
j
k+1
j
j=1
(
)
1 · Lz1 (x)
1−1+
· fk+1|k (x)
fk+1|k [1 · Lz1 ]
1 − pD (x) +
Lz1 (x) · fk+1|k (x)
.
fk+1|k [Lz1 ]
(8.60)
(8.61)
This is the measurement-update equation for the single-sensor, single-target Bayes
filter—that is, Bayes’ rule, (2.26).
8.4.6.2
Classical PHD Filter: Computational Complexity
The PHD filter has favorable computational characteristics. Inspection of (8.51) reveals that the PHD filter measurement-update step has computational order O(mn),
where m is the current number of measurements and n is the current number of
tracks. For, suppose that Dk+1|k (x) has the form
Dk+1|k (x) =
n
∑
i=1
wi · fi (x)
(8.62)
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where fi (x) is the probability distribution of a target track. Then the targetdetection part of Dk+1|k+1 (x) has the form
n ∑
m
∑
pD (x) · Lzj (x) · wi · fi (x)
.
(8.63)
κk+1 (zj ) + τk+1 (zj )
i=1 j=1
Another way of looking at the computational complexity of the PHD filter is
as follows. Suppose that probability of detection pD is constant. If there are n
tracks currently,
then on average these tracks will generate pD n measurements.
∫
If λ = κk+1 (z)dz is the current clutter rate, then the total current number of
measurements is, on average, pD n + λ. Thus the computational order of the PHD
filter will be
O(n · (pD n + λ)) = O(pD n2 + λn).
(8.64)
Thus if pD and n are large, the computational load will also be large. For this
reason, in practice it may be advantageous, if possible, to partition the targets into
statistically noninteracting clusters, and then apply a separate PHD filter to each
cluster. This issue is discussed in more detail in Section 9.2.
8.4.6.3
“Target-Like” versus “Clutter-Like” Measurements
Equations (8.50) and (8.51) can be rewritten in the form
Dk+1|k+1 (x)
=
(1 − pD (x)) · Dk+1|k (x)
∑
+
ωk+1|k (z) · sk+1|k+1 (x|z)
(8.65)
z∈Zk+1
where
ωk+1|k (z)
=
sk+1|k+1 (x|z)
=
τk+1 (z)
κk+1 (z) + τk+1 (z)
pD (x) · Lz (x) · Dk+1|k (x)
.
τk+1 (z)
(8.66)
(8.67)
The first term of (8.65) is the PHD corresponding to undetected targets, and its
integral Dk+1|k [1 − pD ] is the expected number of undetected targets. The second
term—the part of the PHD corresponding to the detected targets—is a weighted
sum of probability distributions sk+1|k+1 (x|z).
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The weight 0 ≤ ω(z) ≤ 1 determines the degree to which sk+1|k+1 (x|z),
and therefore z, contributes to the detected-target PHD. For each z, if
ωk+1|k (z) >
then the measurement z
target. If
1
2
is target-like—that is, it was probably generated by a
1
2
then z is clutter-like—that is, it was probably not generated by a target.
ωk+1|k (z) <
8.4.6.4
(8.68)
(8.69)
Classical PHD Filter: Self-Gating Property
The common practice in conventional multitarget tracking algorithms is to eliminate
measurements that do not fall within the measurement-association gates of any
track. One drawback of this practice is that new targets can be difficult to detect,
since their measurements are likely to be discarded (since they typically will not
fall within the gate of any existing track). The PHD filter sidesteps this problem to
some extent. This is because it implicitly discounts nontrack measurements while
retaining those measurements that are generated by new targets.
As an example, suppose that Dk+1|k (x) has the form
Dk+1|k (x) =
n
∑
wi · fi (x)
(8.70)
i=1
where fi (x) is the track distribution of the ith track. Then (8.52) has the form
τk+1 (zj ) =
n
∑
wi
∫
pD (x) · Lzj (x) · fi (x)dx.
(8.71)
i=1
If the measurement zj does not associate with any track, then every integral in this
summation will be small. This means that τk+1 (zj ) will be small and that zj is
a clutter-like measurement in the sense of Section 8.4.6.3. Thus zj will tend to
have little influence on the sum in (8.71).
8.4.6.5
Classical PHD Filter: The Linearizing Effect
Suppose that at most a single target is present, that probability of detection is
constant, and that no measurements are collected at time tk+1 : Zk+1 = ∅. The
Classical PHD and CPHD Filters
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measurement-updated expected number of targets provided by the PHD filter is
Nk+1|k+1 = (1 − pD ) · Nk+1|k
(8.72)
where by assumption Nk+1|k ≤ 1. But as noted by Erdinc, Willett, and BarShalom [82], the exact formula for Nk+1|k+1 under such assumptions is
Nk+1|k+1
=
(1 − pD ) · Nk+1|k
1 − Nk+1|k · pD
(8.73)
=
2
(1 − pD )Nk+1|k + pD (1 − pD )Nk+1|k
(8.74)
3
+p2D (1 − pD )Nk+1|k
+ ...
where the second equation is the Taylor’s series expansion of Nk+1|k+1 in the
variable Nk+1|k around Nk+1|k = 0. In other words:
• The Poisson approximation assumed in the PHD filter corrector equation has
the effect of linearizing the target number.
This has little effect when pD ∼
= 1, but considerable effect otherwise. For
example, if Nk+1|k = 0.9 and pD = 0.9 then
exact— (8.73):
linearized— (8.72):
Nk+1|k+1 = 0.473 68
Nk+1|k+1 = 0.09.
(8.75)
(8.76)
See p. 632 of [179] for a more complete discussion of this issue.
8.4.6.6
Classical PHD Filter: Window-Averaging
One consequence of (8.73) is that the instantaneous estimate Nk+1|k+1 of target
number is unstable—that is, it has large variance. This, in turn, is a consequence
2
of the fact that the variance σk+1|k+1
of the Poisson distribution is equal to its
2
expected value: σk+1|k+1 = Nk+1|k+1 .
This limitation can be remedied by averaging nk+1|k+1 over some time
window, resulting in stable, low-variance estimates of target number. However,
averaging is not appropriate for scenarios in which targets rapidly appear and/or
disappear.
By way of contrast, the CPHD filter tends to produce accurate, small-variance
estimates of target number.
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8.4.6.7
Classical PHD Filter: Generalization to Gauss-Poisson Processes
In 2009, Singh, Vo, Baddeley, and Zuyev [272] employed virtuoso point process
reasoning to derive a generalization of the PHD filter. In this generalization, the
predicted multitarget distribution fk+1|k (X|Z (k) ) is assumed to be Gauss-Poisson
rather than Poisson. In a Gauss-Poisson target RFS, targets are not statistically
independent but, rather, are assumed to be pairwise-correlated ([55], pp. 247-249).
8.4.6.8
Classical PHD Filter: Alternative Derivations
The PHD and CPHD filters have attracted considerable research interest since they
were introduced. Some of this interest has taken the form of diverse efforts to
reverse-engineer their derivations (and that of the PHD filter especially) while
avoiding the p.g.fl. and multitarget calculus methodology of finite-set statistics. The
purpose of this section is to summarize some of these efforts.
• Brute-Force Approach (2000): Mahler’s original derivation of the PHD filter
[182] presumed constant pD and employed infinite power series expansions
of belief-mass functions.3 It was supplanted a year later by the p.g.fl.-based
approach [168].
• Physical-Space (Bin-Occupancy) Approach (2006): Erdinc, Willett, and
Bar-Shalom proposed derivations of the PHD and CPHD filters based on
a discretized single-target state space and physics-based intuition (see [81]
and [179], pp. 599-609). They described this discrete-space PHD filter as
a “bin occupancy filter” [80]. This approach provides valuable physically
intuitive insights. However, it would be unwise to regard it as a general,
systematic, and theoretically rigorous methodology for deriving RFS filter
equations. This is because physical intuition, while useful for many things,
can be deceiving.
• “Poisson Point Process” or “PPP” Approach (2008): Streit and Stone
[281] claimed to derive the PHD filter “in essentially elementary terms
[using]. . . PPP’s at an elementary level” ([281] Section 1, second paragraph).
3
Erratum: In this original derivation—because of an incorrectly summed power series—the misseddetection term 1 − pD in the measurement-update was erroneously derived as
pD · Nk+1|k
1 − Nk+1|k · pD
.
Classical PHD and CPHD Filters
201
However, this derivation contains serious mathematical errors and seriously
restrictive hidden assumptions.4
• Alternative Derivation of the CPHD Filter (2008): Svensson and Svennson
[284] proposed an alternative derivation of the measurement-update step of
the Gaussian mixture (GM) CPHD filter, based on “more common mathematical statistics.” Their derivation results in a different formula for the
measurement-update of the GM-PHD filter. This leads one to believe that
either additional assumptions were implicitly made, or that the derivation is
not mathematically correct.
• Measure-Theoretic Approach (2011): Caron, del Moral, Doucet, and Pace
[30] proposed an “elementary and self-contained random measure theoretic
approach” for deriving the equations for the classical PHD filter, using
virtuoso measure-theoretic reasoning.
Section VI-D of [181] demonstrates that the classical PHD filter can be
derived in a few lines using the p.g.fl. methodology.
8.5
CLASSICAL CARDINALIZED PHD (CPHD) FILTER
This section is organized as follows:
1. Section 8.5.1: Modeling assumptions for the CPHD filter time-update.
2. Section 8.5.2: The CPHD filter time-update equations.
3. Section 8.5.3: Modeling assumptions for the CPHD filter measurementupdate.
4. Section 8.5.4: The CPHD filter measurement-update equations.
5. Section 8.5.5: Multitarget state estimation for the CPHD filter.
4
The most serious mathematical error occurs following Eq. (27), where the authors attempt to use a
MAP estimator to show that the constant c is actually m (the current number of measurements).
In so doing, they implicitly confuse the random measurement-number Mk+1 with its current
instantiation Mk+1 = m. Since m is a nonrandom a priori constant and c = m, then so is c.
But nonrandom a priori constants cannot be estimated. In particular, since c is a constant equal to
m, its probability distribution has to be δm (c)—and not proportional to e−c · cm , Eq. (28). The
authors further claim that the correct value of c is the MAP estimate c = m of e−c · cm . But
why this and not—for example—the EAP estimate (which is not m)? See Appendix A of [163]
for further details.
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6. Section 8.5.6: Characteristics of the CPHD filter.
7. Section 8.5.7: Approximate forms of the classical CPHD and PHD filters,
assuming that targets are not too close together.
8.5.1
Classical CPHD Filter Motion Modeling
The CPHD filter time-update is based on the same modeling assumptions as the
classical PHD filter and the general PHD filter (Section 8.2.1) except that, unlike
the PHD filter, the CPHD filter does not include a formal model for target spawning.
Specifically:
• Probability of target survival: pS (x) abbr.
= pS,k+1|k (x).
• PHD of the target appearance RFS: bk+1|k (x) with
B
Nk+1|k
=
∫
(8.77)
bk+1|k (x)dx.
• Cardinality distribution of the target appearance RFS: pB
k+1|k (n) with
∑
B
n · pB
k+1|k (n) = Nk+1|k .
(8.78)
n≥0
• p.g.f. of the target appearance RFS:
GB
k+1|k (x) =
∑
n
pB
k+1|k (n) · x .
(8.79)
n≥0
The following abbreviations are employed in what follows:
=
pk|k (n|Z (k) ),
pk+1|k (n) = pk+1|k (n|Z (k) )
Gk|k (x)
=
Gk|k (x|Z
(k)
(k)
sk|k (x)
=
=
pk|k (n)
Dk|k (x)
8.5.2
(8.80)
)
(8.81)
sk|k (x|Z (k) ),
sk+1|k (x) = sk+1|k (x|Z (k) )
(8.82)
(k)
(k)
Dk|k (x|Z
),
),
Gk+1|k (x) = Gk+1|k (x|Z
Dk+1|k (x) = Dk+1|k (x|Z
Classical CPHD Filter: Predictor
Suppose that we have in hand:
).
(8.83)
Classical PHD and CPHD Filters
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• The target spatial distribution sk|k (x).
• The expected number of targets Nk|k .
• The cardinality distribution pk|k (n) or, equivalently, its p.g.f. Gk|k (x)—in
which case we must have
∑
(1)
Nk|k = Gk|k (1) =
n · pk|k (n).
(8.84)
n≥0
To achieve closed-form formulas, the following assumption is required:
• The multitarget distribution fk|k (X|Z (k) ) is i.i.d.c. (in the sense of Section
4.3.2).
We are to determine the predicted spatial distribution sk+1|k (x), the predicted
expected number of targets Nk+1|k , and either the predicted cardinality distribution
pk+1|k (n) or the predicted p.g.f. Gk+1|k (x). The predicted spatial distribution is
given by
∫
bk+1|k (x) + Nk|k pS (x′ ) · fk+1|k (x|x′ ) · sk|k (x′ )dx′
sk+1|k (x) =
(8.85)
B
Nk+1|k
+ Nk|k · ψk
or, equivalently, the PHD is given by
∫
Dk+1|k (x) = bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
where
ψk = sk|k [pS ] =
∫
pS (x′ ) · sk|k (x′ )dx′ .
(8.86)
(8.87)
The predicted cardinality distribution and its corresponding p.g.f. are given by5
Gk+1|k (x)
=
pk+1|k (n)
=
GB
k+1|k (x) · Gk|k (1 − ψk + ψk · x)
∑
pk+1|k (n|n′ ) · pk|k (n′ )
(8.88)
(8.89)
n′ ≥0
where the pseudo-Markov transition function for target number is6
pk+1|k (n|n′ ) =
n
∑
′
i
n −i
pB
.
k+1|k (n − i) · Cn′ ,i · ψk (1 − ψk )
i=0
5
6
Note: The expression in (8.89) is different than (but equivalent to) Eq. (16.313) in [179].
Note: As was specified in (2.1), Cn′ ,i = 0 if n′ < i.
(8.90)
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The predicted expected number of targets is
B
Nk+1|k = Nk+1|k
+ Nk|k · ψk .
(8.91)
Remark 23 (Special cases of time-update) Three special cases are worth pointB
ing out. First, if there are no target births, then bk+1|k (x) = 0, Nk+1|k
= 0, and
B
Gk+1|k (x) = 1. In this case,
Gk+1|k (x)
=
(8.92)
Gk|k (1 − ψk + ψk · x)
′
pk+1|k (n|n′ )
=
sk+1|k (x)
=
Dk+1|k (x)
=
Cn′ ,n · ψkn (1 − ψk )n −n
∫
1
pS (x′ ) · fk+1|k (x|x′ ) · sk|k (x′ )dx′
ψk
∫
pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ .
(8.93)
(8.94)
(8.95)
Second, if there are no target disappearances, then pS = 1. In this case ψk = 1
and so
Gk+1|k (x)
=
GB
k+1|k (x) · Gk|k (x)
pk+1|k (n|n′ )
=
′
pB
k|k (n − n )
sk+1|k (x)
=
Dk+1|k (x)
=
(8.96)
(8.97)
∫
′
′
bk+1|k (x) + Nk|k fk+1|k (x|x ) · sk|k (x )dx
Bk+1|k + Nk|k
∫
bk+1|k (x) + fk+1|k (x|x′ ) · Dk|k (x′ )dx′ .
′
(8.98)
(8.99)
Third, suppose that there are no target appearances or disappearances. Then
8.5.3
Gk+1|k (x)
=
Gk|k (x)
(8.100)
′
pk+1|k (n|n )
=
(8.101)
sk+1|k (x)
=
Dk+1|k (x)
=
δn,n′
∫
fk+1|k (x|x′ ) · sk|k (x′ )dx′
∫
fk+1|k (x|x′ ) · Dk|k (x′ )dx′ .
(8.102)
(8.103)
Classical CPHD Filter: Measurement Modeling
The CPHD filter measurement-update is based on the same modeling assumptions
as the PHD filter (Section 8.4.2), except that the clutter RFS is i.i.d.c. rather than
Poisson. Specifically:
Classical PHD and CPHD Filters
205
• Probability of detection: pD (x) abbr.
= pD,k+1 (x).
• Sensor likelihood function: Lz (x) abbr.
= fk+1 (z|x).
• Spatial distribution of the clutter RFS: ck+1 (z).
• Cardinality distribution of the clutter RFS: pκk+1 (m).
∑
• p.g.f. of the clutter RFS: Gκk+1 (z) = m≥0 pκk+1 (m) · z m .
8.5.4
Classical CPHD Filter: Corrector
Suppose that we have in hand:
• A new measurement set Zk+1 = {z1 , ..., zm } with |Zk+1 | = m.
• The predicted spatial distribution sk+1|k (x).
• The predicted ∑
cardinality distribution pk+1|k (n) or, equivalently, its p.g.f.
Gk+1|k (x) = n≥ pk+1|k (n) · xn , where the predicted expected number of
targets must satisfy
(1)
Nk+1|k = Gk+1|k (1) =
∑
n · pk+1|k (n).
(8.104)
n≥0
To achieve closed-form formulas, the following assumption is required:
• The predicted multitarget distribution fk+1|k (X|Z (k) ) is i.i.d.c. (in the sense
of Section 4.3.2).
We are to determine the posterior spatial distribution sk+1|k+1 (x), the
posterior expected number of targets Nk+1|k+1 , and either the posterior cardinality
distribution pk+1|k+1 (n) or, equivalently, the posterior p.g.f. Gk+1|k+1 (x). The
spatial distribution is given by
ˆ Z (x) · sk+1|k (x)
sk+1|k+1 (x) = L
k+1
(8.105)
or, equivalently, the PHD is given by
Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x).
(8.106)
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Here, the CPHD filter pseudolikelihood functions are
ˆ Z (x)
L
k+1
1
=
Nk+1|k+1

ND
· (1 − pD (x)) · L Zk+1 +
m
∑
pD (x) · Lzj (x)
ck+1 (zj )
j=1
D
(8.107)

· LZk+1 (zj )
1
LZk+1 (x)
=
Nk+1|k

ND
· (1 − pD (x)) · L Zk+1 +
where
m
∑
pD (x) · Lzj (x)
ck+1 (zj )
j=1
( ∑m
)
L Zk+1 = ( ∑m
)
κ
i=0 (m − i)! · pk+1 (m − i)
(i+1)
·σi (Zk+1 ) · Gk+1|k (ϕk )
ND
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
( ∑m−1
(m − i − 1)! · pκk+1 (m − i − 1)
(i+1)
·σi (Zk+1 − {zj }) · Gk+1|k (ϕk )
D
( ∑m
)
LZk+1 (zj ) =
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
∑
(l)
Gk+1|k (ϕk ) =
pk+1|k (n) · l! · Cn,l · ϕn−l
k
i=0
D
(8.108)

· LZk+1 (zj )
(8.109)
)
(8.110)
(8.111)
n≥l
(j+1)
Gk+1|k (ϕk ) =
∑
n≥j+1
pk+1|k (n) · (j + 1)! · Cn,j+1 · ϕn−j−1
k
(8.112)
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and where
ϕk
=
=
σi (Zk+1 )
=
σi (Zk+1 − {zj })
=
sk+1|k [1 − pD ]
∫
(1 − pD (x)) · sk+1|k (x)dx
(
)
τ̂k+1 (z1 )
τ̂k+1 (zm )
σm,i
, ...,
ck+1 (z1 )
ck+1 (zm )


τ̂?
τ̂k+1 (z1 )
k+1 (zj )
, ...,
,
σm−1,i  ck+1 (z1 ) τ̂ (zck+1) (zj ) 
m
k+1
..., ck+1 (zm )
(8.113)
(8.114)
(8.115)
and where
τ̂k+1 (z) = sk+1|k [pD Lz ] =
∫
pD (x) · Lz (x) · sk+1|k (x)dx
(8.116)
and where y1 , ..., y?j , ..., ym indicates that yj has been removed from the list
y1 , ..., ym —that is, the altered list is y1 , ..., yj−1 , yj+1 , ..., ym .
Also, the measurement-updated number of targets is
ND
Nk+1|k+1 = ϕk · L Zk+1 +
∑ τ̂k+1 (z) D
· LZk+1 (z).
ck+1 (z)
(8.117)
z∈Zk+1
On the other hand, the measurement-updated cardinality distribution and
p.g.f. are, respectively,
pk+1|k+1 (n)
=
Gk+1|k+1 (x)
=
ℓZ (n) · pk+1|k (n)
∑ k+1
l≥0 ℓZk+1 (l) · pk+1|k (l)
( ∑m j
)
κ
j=0 x · (m − j)! · pk+1 (m − j)
·G(j) (x · ϕk ) · σj (Zk+1 )
( ∑m
)
κ
i=0 (m − i)! · pk+1 (m − i)
·G(i) (ϕk ) · σi (Zk+1 )
(8.118)
(8.119)
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where the cardinality pseudolikelihood function is
( ∑
)
min{m,n}
(m − j)! · pκk+1 (m − j)
j=0
·j! · Cn,j · ϕn−j
· σj (Zk+1 )
k
( ∑m
)
ℓZk+1 (n) =
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
∑min{m,n} dm−j Gκk+1
n−j
· σj (Zk+1 )
dz m−j (0) · j! · Cn,j · ϕk
.
∑m dm−l Gκk+1
(l)
(0))
·
σ
(Z
)
·
G
(ϕ
)
l
k+1
k
m−l
l=0
k+1|k
dz
j=0
=
(8.120)
(8.121)
Note that (8.118) has the same form as Bayes’ rule.
8.5.5
Classical CPHD Filter: State Estimation
The state-estimation process for the CPHD filter is slightly different than that for
the PHD filter. The expected number of targets Nk+1|k+1 is no longer used. In its
place, we use the MAP estimate
n̂ = arg sup pk+1|k+1 (n).
(8.122)
n
That is, we look for the n̂ largest local suprema x1 , ..., xn̂ of the density
Dk+1|k+1 (x)—those x corresponding to the n̂ largest “peaks” of Dk+1|k+1 (x)
or sk+1|k+1 (x). These are the estimates of the states of the target tracks. In the
event that there are fewer than n̂ local suprema, these suprema are taken to be the
state estimates. In this case, it is implicitly understood that some targets are so close
together that they correspond to a single peak of the PHD.
8.5.6
Classical CPHD Filter: Characteristics
The purpose of this section is to summarize some of the more important properties
of the classical CPHD filter:
1. Section 8.5.6.1: Computational complexity of the CPHD filter.
2. Section 8.5.6.2: The PHD filter is a special case of the CPHD filter.
3. Section 8.5.6.3: Stability of the CPHD filter’s estimate of target number.
4. Section 8.5.6.4: The CPHD filter and target-spawning models.
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5. Section 8.5.6.5: The relationship between the CPHD filter and the Bernoulli
filter
8.5.6.1
Classical CPHD Filter: Computational Complexity
Inspection of (8.109) and (8.110) reveals that the CPHD filter measurement-update
step has computational order O(m3 n), where m is the current number of
measurements and n is the current number of tracks. A numerical balancing
technique due to Guern can be used to reduce this to O(m2 n) [103]. However,
care must be taken to avoid numerical instability.
As with the PHD filter and (8.64), there is another way of looking at the
CPHD filter’s computational load. Suppose that probability of detection pD is
constant and that the current clutter rate is λ. Then the total current number of
measurements will be, on average, pD n + λ. Thus the computational order of the
CPHD filter will be
O(m3 n) = O((pD n + λ)3 n)
(8.123)
which is of the fourth order in n. Thus when pD and n are large, it will
be necessary (if possible) to partition the targets into statistically noninteracting
clusters, and then apply a separate CPHD filter to each cluster. This issue is
discussed in more detail in Section 9.2.
8.5.6.2
The PHD Filter is a Special Case of the CPHD Filter
Suppose that
pk+1|k (n)
pκk+1 (m)
=
=
e−Nk+1|k ·
e−λk+1 ·
n
Nk+1|k
(8.124)
n!
λm
k+1
.
(8.125)
m!
That is, both the predicted-target RFS and the clutter RFS are Poisson. Then with
a bit of algebraic manipulation, it can be shown that (8.106) reduces to (8.50)—that
is, to the measurement-update equation for the PHD filter.
8.5.6.3
Classical CPHD Filter: Stability of the Target-Number Estimate
The CPHD filter produces accurate, small-variance estimates of target number. This
is because the CPHD filter produces an ongoing estimate pk+1|k+1 (n) of the
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entire probability distribution on target number. One limitation of the PHD filter,
as described in Section 8.4.6.6, is thereby avoided.
8.5.6.4
Classical CPHD Filter: Spawning Models?
Unlike the PHD filter, the CPHD filter does not have an explicit model for targets
that are spawned by existing targets. This is because target-spawning models are
not mathematically consistent with the i.i.d.c. RFS approximation that underlies
the CPHD filter.
This lack of an explicit spawning model might lead to the conclusion that the
CPHD filter cannot be used to address problems in which spawning targets occur.
Such an inference would be mistaken, as has already been noted in Section 1.1.2.
8.5.6.5
Relationship Between CPHD Filter and Bernoulli Filter
The Bernoulli filter (which will be discussed in Section 13.2) is the Bayes-optimal
approach to detecting, tracking, and classifying a single target in an arbitrary clutter
and detection background. Suppose that:
• Target number is no larger than 1, in which case the predicted p.g.f. is
Gk+1|k (x) = 1 − pk+1|k + pk+1|k · x where pk+1|k is the probability
of existence of the target.
• The clutter process is i.i.d.c., in which case κk+1 (Z) = |Z|!·pκk+1 (|Z|)·cZ
k+1 .
Then it is easy to show that the time-update and measurement-update equations for the Bernoulli filter and the CPHD filter are identical.
8.5.7
Approximate Classical CPHD Filter
Under the assumption that the targets are not too close together, the following
approximation of the multitarget likelihood for the standard measurement model
was given in (7.50):
fk+1 (Z|X) ∼
= κk+1 (Z) ·
(
∑ pD L z
1 − pD +
κk+1 (z)
)X
(8.126)
z∈Z
where
κk+1 (Z) = e−λk+1
∏
z∈Z
κk+1 (z)
(8.127)
Classical PHD and CPHD Filters
211
∫
is the distribution of the Poisson clutter RFS and where λk+1 = κk+1 (z)dz.
This equation leads to an approximate form of the classical CPHD filter. The
time-update equations for this filter are the same as for the classical CPHD filter.
Thus only the measurement-update equations need to be considered.
We are given the predicted cardinality distribution pk+1|k (n) or, equivalently,
the predicted p.g.f. Gk+1|k (x). We are also given the predicted spatial distribution
sk+1|k (x) or, equivalently, the predicted PHD Dk+1|k (x). Given this, the corrector
equations for the approximate CPHD filter are:
• Measurement update for the cardinality distribution and p.g.f.:
pk+1|k+1 (n)
ϕnk
=
Gk+1|k (ϕk )
Gk+1|k+1 (x)
=
· pk+1|k (n)
Gk+1|k (x · ϕk )
Gk+1|k (ϕk )
(8.128)
(8.129)
where
ϕk =
∫


∑ pD (x) · Lz (x)
1 − pD (x) +
 · sk+1|k (x)dx. (8.130)
κk+1 (z)
z∈Zk+1
• Measurement update for the spatial distribution and PHD:
sk+1|k+1 (x)
sk+1|k (x)
1 − pD (x) +
=
∑
z∈Zk+1
pD (x)·Lz (x)
κk+1 (z)
(8.131)
ϕk
(1)
Dk+1|k+1 (x)
Dk+1|k (x)
Gk+1|k (ϕk )
=
(8.132)
Nk+1|k · Gk+1|k (ϕk )


∑ pD (x) · Lz (x)
.
· 1 − pD (x) +
κk+1 (z)
z∈Zk+1
These formulas are proved in Section K.13. Note that the computational
complexity of this filter is the same as that of the classical PHD filter: O(mn),
where m = |Zk+1 | is the current number of measurements and n is the current
number of tracks.
Remark 24 (Approximate classical PHD filter) The PHD filter special case of
the approximate CPHD filter is uninteresting. It results if we set Gk+1|k (x) =
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eNk+1|k (x−1) . Equation (8.132) then reduces to a truncated form of the measurementupdate equation for the classical PHD filter:
∑ pD (x) · Lz (x)
Dk+1|k+1 (x)
= 1 − pD (x) +
.
Dk+1|k (x)
κk+1 (z)
(8.133)
z∈Zk+1
8.6
ZERO FALSE ALARMS (ZFA) CPHD FILTER
The ZFA-CPHD filter is a special case of the CPHD filter. It results from the
assumption that there is no clutter: κk+1 (z) = 0 and Gκk+1 (z) = 1 (or,
equivalently, that pκk+1 (m) = δm,0 ). The ZFA-CPHD filter can be useful in those
situations in which the clutter background is negligible. It is also of interest for the
following reasons:
• Unlike the CPHD filter, which has computational complexity O(m3 n), the
computational complexity of the ZFA-CPHD filter is the same as that of the
PHD filter: O(mn).
• The ZFA-CPHD filter is fundamental to the derivation of the “clutteragnostic” CPHD filters described in Chapter 18.
Since the time-update equations for the CPHD filter remain unchanged, only
the measurement-update equations for the ZFA-CPHD filter need be specified. As
before, suppose that we have:
• A new measurement set Zk+1 = {z1 , ..., zm } with |Zk+1 | = m.
• The predicted spatial distribution sk+1|k (x).
• The predicted expected number of targets Nk+1|k .
• The predicted cardinality distribution pk+1|k (n) or, equivalently, its p.g.f.
Gk+1|k (x).
Since there are no false alarms, m is no larger than the actual number n of
targets. Given this,
• Measurement update for the ZFA-CPHD filter spatial distribution:
sk+1|k+1 (x) = LZk+1 (x) · sk+1|k (x)
(8.134)
Classical PHD and CPHD Filters
213
where the ZFA-CPHD filter pseudolikelihood function is
1
LZk+1 (x)
=
(8.135)

(m+1)
m
Gk+1|k (ϕk ) ∑
pD (x) · Lzj (x)

· (1 − pD (x)) · (m)
+
τ̂k+1 (zj )
G
(ϕk ) j=1
Nk+1|k+1

k+1|k
and where
ϕk
=
∫
(1 − pD (x)) · sk+1|k (x)dx
(8.136)
(m+1)
Gk+1|k (ϕk )
Nk+1|k+1
=
ϕk ·
+m
(8.137)
(m)
Gk+1|k (ϕk )
τ̂k+1 (z)
=
∫
pD (x) · Lz (x) · sk+1|k (x)dx.
(8.138)
• Measurement update for the ZFA-CPHD filter cardinality distribution and
p.g.f.:
pk+1|k+1 (n)
=
∑
ℓZk+1 (n) · pk+1|k (n)
l≥0 ℓZk+1 (n) · pk+1|k (n)
(8.139)
(m)
xm · Gk+1|k (x · ϕk )
Gk+1|k+1 (x)
=
(8.140)
(m)
Gk+1|k (ϕk )
where the ZFA-CPHD filter pseudolikelihood for target number is
ℓZk+1 (n) = Cn,m · ϕn−m
.
k
(8.141)
Since Cn,m = 0 if m > n—see (2.1)—it follows that pk+1|k+1 (n) = 0
for all n < m.
8.6.1
Comparison of the PHD and ZFA-CPHD Filters
Multiplying both sides of (8.134) by Nk+1|k+1 and using (8.52) to note that
τk+1 (z) = Nk+1|k · τ̂k+1 (z), we get
(m+1)
m
Dk+1|k+1 (x)
1 − pD (x) Gk+1|k (ϕk ) ∑ pD (x) · Lzj (x)
=
· (m)
+
.
Dk+1|k (x)
Nk+1|k
τk+1 (zj )
G
(ϕk ) j=1
k+1|k
(8.142)
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Note that the PHD filter with no clutter has the form (see (8.50) and (8.51))
m
∑
Dk+1|k+1 (x)
pD (x) · Lzj (x)
= 1 − pD (x) +
.
Dk+1|k (x)
τk+1 (zj )
j=1
(8.143)
From this we may conclude that the ZFA-CPHD filter will behave much like the
PHD filter—except that the nondetection term will be more accurate because of the
factor
(m+1)
Gk+1|k (ϕk )
1
· (m)
.
Nk+1|k G
(ϕk )
k+1|k
This in part should help produce more accurate and stable instantaneous estimates
of target number—see the example that follows.
Nevertheless, if pD (x) = 1 identically then the ZFA-CPHD filter reduces to
the PHD filter.
Example 5 (ZFA-CPHD filter one target) Assume that the scene contains at most
a single target, in which case
Gk+1|k (x) = 1 − Nk+1|k + Nk+1|k · x
with Nk+1|k ≤ 1. Assume also that
ϕk = 1 − pD and (8.142) becomes
(8.144)
pD is constant and that m = 0. Then
(1)
1 − pD Gk+1|k (ϕk )
·
· Dk+1|k (x)
Nk+1|k Gk+1|k (ϕk )
Nk+1|k
1 − pD
·
· Dk+1|k (x)
Nk+1|k 1 − Nk+1|k + Nk+1|k · ϕk
1 − pD
· Dk+1|k (x)
1 − Nk+1|k · pD
Dk+1|k+1 (x) =
=
=
(8.145)
(8.146)
(8.147)
from which we get
Nk+1|k+1 =
(1 − pD ) · Nk+1|k
.
1 − Nk+1|k · pD
(8.148)
Comparing this to (8.73), we conclude: When there is no clutter, the ZFA-CPHD
filter should have better performance than the PHD filter.
Classical PHD and CPHD Filters
8.7
215
PHD FILTER FOR STATE-DEPENDENT POISSON CLUTTER
The standard multitarget measurement model was given in (7.7):
Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 ,
(8.149)
where the target-measurement RFS Υk+1 (x) is Bernoulli for every x; where
the clutter RFS Ck+1 is Poisson; and where Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are
independent.
In many applications, clutter Ck+1 is not independent of the targets x1 , ..., xn .
One example is multipath propagation, in which a radar or sonar pulse is reflected
from various surfaces, thus creating multiple returns at the receiver.
Another example is radio frequency (RF) spoofing by aircraft. In RF spoofing, electronic countermeasure (ECM) devices on the aircraft receive impinging
radar pulses from an air defense system. These devices then retransmit delayed
versions of these pulses back to the radar. The effect is to create one or more false
targets.
Vo, Vo, and Cantoni have devised a generalization of the Bernoulli filter
that detects and tracks a single target in Poisson state-dependent clutter [309].
The purpose of this section, however, is to propose a PHD filter for detecting
and tracking multiple targets in target-dependent Poisson clutter. It employs a
measurement model first considered in [179], pp. 424-426.
Remark 25 (Erratum) In [173], Mahler proposed a PHD filter for target-dependent
Poisson clutter. The reader is warned that the derivation of this filter appears to
be incorrect. The PHD filter proposed in what follows therefore replaces the one
proposed in [173].
Consider a clutter RFS that has the form
0
Ck+1 = Ck+1 (x1 ) ∪ ... ∪ Ck+1 (xn ) ∪ Ck+1
(8.150)
where
0
• Ck+1
is independent Poisson background
clutter, with intensity function
∫
0
κk+1 (z) and clutter rate λ0k+1 = κ0k+1 (z)dz.
• Ck+1 (x) is Poisson clutter associated with a target with
∫ state x, with
intensity function κk+1 (z|x) and clutter rate λk+1 (x) = κk+1 (z|x)dz.
0
• Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 (x1 ), ..., Ck+1 (xn ), Ck+1
are independent.
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Then the total measurement RFS is
0
˜ k+1 (x1 ) ∪ ... ∪ Υ
˜ k+1 (xn ) ∪ Ck+1
Σk+1 = Υ
(8.151)
˜ k+1 (x) = Υk+1 (x) ∪ Ck+1 (x).
Υ
(8.152)
where
We can then apply the measurement-update formula for the general PHD filter
of Section 8.2 to get the following:
• Measurement update equations for PHD filter for target-dependent Poisson
clutter: This is
(8.153)
Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x)
where the PHD pseudolikelihood is
∑
LZk+1 (x) = e−λk+1 (x) ·(1− pD (x))+
ωP
P⊟Zk+1
∑
LW (x)
(8.154)
κW + τW
W ∈P
where the summation is taken over all partitions P of Zk+1 , and where
LW (x)
=
e
−λk+1 (x)
·
(
∏
κk+1 (z|x)
)
(8.155)
z∈W
[
· 1 − pD (x) +
κW
=
τW
=
∑ pD (x) · Lz (x)
κk+1 (z|x)
z∈W

0
 e−λk+1 if
W =∅
κ0k+1 (z) if W = {z}

0
if |W | ≥ 2
∫
LW (x) · Dk+1|k (x)dx
and where
ωP = ∑
∏
W ∈P (κW + τW )
Q⊟Zk+1
∏
V ∈Q (κV + τV )
The derivation can be found in Section K.18.
.
]
(8.156)
(8.157)
(8.158)
Chapter 9
Implementing Classical PHD/CPHD
Filters
9.1
INTRODUCTION
The purpose of this chapter is to describe the major methods and issues associated
with practical implementation of the classical PHD and CPHD filters.
The filtering equations for both filters involve multidimensional integrals,
and so further approximation is necessary to create computationally tractable algorithms. Two approaches have gained currency: Gaussian-mixture (GM) implementation and sequential Monte Carlo (SMC, also known as “particle”) implementation. SMC implementation was independently proposed in 2003 by Sidenbladh
[271], Zajic and Mahler [330], and Vo, Singh, and Doucet [306]; while GM implementation was introduced in 2005 by Vo and Ma [299]. Both approaches have subsequently been extended and refined in several directions, to be described shortly.
A second major development in the field has been an increasingly deeper understanding of the practical behavior of both filters. The most notable development
has been analysis of the “spooky action at a distance” phenomenon, first pointed out
by Fränken and Ulmke [293], [88].
9.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• PHD and CPHD filters can be implemented using Gaussian mixture methods,
including the following variants: extended Kalman filter (EKF), unscented
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Kalman filter (UKF), cubature Kalman filter (CKF), and Gaussian particle
filter (GPF). See Sections 9.5.4 and 9.5.5.
• PHD and CPHD filters can be implemented using sequential Monte Carlo
(SMC, also known as “particle”) methods. See Section 9.6.
• Because of the phenomenon known as “spooky action at a distance,” PHD
and CPHD filters tend to shift probability mass from undetected tracks to
detected ones. See Section 9.2.
• Because of this “spookiness,” a scenario should be partitioned into statistically noninteracting target-clusters, and a separate PHD or CPHD filter applied to each cluster. See Section 9.2.
• Typical SMC implementations of PHD and CPHD filters require a complicated and computationally demanding “clustering” step in order to estimate
the number and states of the targets. A new “measurement-driven” implementation approach, due to Ristic, Clark, Vo, and Vo, avoids this step as well
as other difficulties. See Section 9.6.4.
• A similar approach results in efficient selection of the target-birth process in
Gaussian-mixture implementations of PHD and CPHD filters. See Section
9.5.7.
• Gaussian mixture implementations of PHD and CPHD filters can be extended
to include target type and therefore target classification. See Section 9.5.6.
• The same is true of sequential Monte Carlo implementations of these filters.
See Section 9.6.5.
9.1.2
Organization of the Chapter
The chapter is organized as follows:
1. Section 9.2: “Spooky action at a distance”—referring to the tendency of the
PHD and CPHD filters to shift probability mass from undetected tracks to
detected ones.
2. Section 9.3: Cluster-merging and cluster-splitting for PHD filters.
3. Section 9.4: Cluster-merging and cluster-splitting for CPHD filters.
4. Section 9.5: Gaussian-mixture implementation of PHD and CPHD filters.
Implementing Classical PHD/CPHD Filters
219
5. Section 9.6: Sequential Monte Carlo (SMC, also known as “particle”)
implementation of PHD and CPHD filters.
9.2
“SPOOKY ACTION AT A DISTANCE”
The PHD and CPHD filters are, at their core, cluster trackers. That is, when
confronted with a multiple-target scenario, both filters first try to interpret it as a
target cluster. Subsequently, they attempt to resolve individual targets only as the
quantity and quality of measurements permit this to be accomplished.
In [293] and [88], Fränken and Ulmke noted that the cluster-tracker nature of
PHD and CPHD filters produces the following two counterintuitive behaviors:
1. “Spooky action at a distance”: The classical PHD and CPHD filters both
shift PHD mass away from undetected tracks to detected tracks—even if these
tracks are so distant from each another (with respect to sensor resolution) that
they are statistically noninteracting.
2. Violation of superposition: Applying CPHD filters separately to widelyseparated target-clusters does not produce the same result as applying a single
CPHD filter to the entire scene. The same is not true for the PHD filter,
however.
Consequently, for the purpose of practical implementation:
• A multitarget scene should first be partitioned into statistically noninteracting
target-clusters, with CPHD filters applied separately to each of the clusters.
• As was discussed in Section 8.4.6.2, this approach is routinely applied in
multihypothesis trackers (MHTs) to reduce computational complexity [245],
[23], [24]. It must be applied to PHD and CPHD filters for both theoretical
and practical reasons.
Points 1 and 2 above can be demonstrated analytically using a simple example. Assume a scenario with the following assumptions:
1. There are at most two tracks, which are well separated with respect to sensor
resolution.
2. Both tracks have track probability 0 < a ≤ 1.
3. Both targets are static, and their initial track distributions are f1 (x) and
f2 (x).
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4. There are no false alarms and the probability of detection pD is constant.
5. A measurement z1 is collected from the first track, but the second track is
not detected.
Under these assumptions, the prior multitarget distribution is a multi-Bernoulli
distribution of degree two:

2
(1 − a)
if
X=∅




a(1 − a) · (f1 (x) + f2 (x))
if
X = {x}

f0|0 (X) =
a2 · (f1 (x1 ) · f2 (x2 ) + f1 (x2 ) · f2 (x1 )) if X = {x1 , x2 },


|X| = 2



0
if
otherwise
(9.1)
The prior p.g.fl., p.g.f., cardinality distribution, PHD, and expected number of
targets are, respectively,
G0|0 [h]
=
(1 − a + a · f1 [h]) · (1 − a + a · f2 [h])
2
=
(1 − a + a · x)
p0|0 (n)
n
D0|0 (x)
N0|0
G0|0 (x)
(9.2)
(9.3)
2−n
=
C2,n · a (1 − a)
(9.4)
=
=
a · f1 (x) + a · f2 (x)
2a
(9.5)
(9.6)
where C2,n was defined in (2.1).
In Section K.14, formulas for the measurement-updated PHD D1|1 (x) are
derived using three filters—the PHD filter, the CPHD filter, and the multitarget
recursive Bayes (MRB) filter. The results are as follows:
PHD filter:
CPHD filter:
MRB filter:
D1|1 (x) = [1 + a(1 − pD )] · f1 (x)
+a(1 − pD ) · f2 (x)
(
)
(1 − pD ) a
D1|1 (x) = 1 +
· f1 (x)
2 (1 − apD )
(1 − pD ) a
+
· f2 (x)
2 (1 − apD )
(1 − pD )a
D1|1 (x) = f1 (x) +
· f2 (x).
1 − apD
These results can be interpreted as follows:
(9.7)
(9.8)
(9.9)
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221
• Multitarget Bayes filter: The result for this filter is exact. As one would
expect, it is “superpositional,” in the sense that the two-target PHD is what
would result if one applied the multitarget Bayes filter to the two tracks
separately and then summed their respective PHDs (see Section K.14.3).
Since the first track was detected, it definitely exists and so its track weight
must be unity. The weight of the second track has decreased since it was not
detected:
(1 − pD ) · a
a ?→
≤ a.
1 − apD
• CPHD filter: The result for the CPHD filter is not superpositional, since the
two-target PHD is not the sum of the two single-target PHDs. Moreover, the
weight of the undetected track is half of what it should be. The “missing
half” has been shifted to the weight of the detected track, making that weight
larger than it should be. This seeming “entanglement” between the weights of
arbitrarily distant tracks is what Fränken and Ulmke dubbed “spooky action
at a distance.”
• PHD filter: The result for the PHD filter is superpositional, since the twotarget PHD is the sum of the two single-target PHDs (see Section K.14.1).
However, this result is even “spookier” than the CPHD filter result. That is,
the weight of the undetected track is even smaller than it should be, and this
missing weight has been shifted to the detected track.
9.3
MERGING AND SPLITTING FOR PHD FILTERS
As was noted in Section 8.4.6.2, it will often be desirable to partition the targets in a
scene into statistically noninteracting clusters, and then apply a separate PHD filter
to each cluster. In this case it will often become necessary to merge the PHDs (if
multiple clusters join together) or split the PHDs (if a cluster separates into multiple
clusters). The purpose of this section is to describe how this is accomplished.
9.3.1
Merging for PHD Filters
Suppose that two PHD filters are tracking two target groups, and that these groups
move close enough that we should apply a single PHD filter to both of them. Let
1
2
D k|k (x),
D k|k (x)
(9.10)
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be the PHDs produced by the two PHD filters at time tk . Assuming that the
two target groups are approximately independent, the two PHDs are merged via
superposition:
1
2
(9.11)
Dk|k (x) = D k|k (x) + D k|k (x).
9.3.2
Splitting for PHD Filters
Suppose that a single PHD filter is tracking a target group that splits into two
subgroups. How do we split the PHD into two new ones, one for each of the
subgroups? The intuitively obvious approach is as follows. Suppose that we can
partition the PHD into two separated groups of weighted track distributions
Group 2
Group 1
??
?
?
??
? ?
Dk|k (x) = w1 · f1 (x) + ... + wn · fn (x) + w̃1 · f˜1 (x) + ... + w̃ñ · f˜ñ (x) (9.12)
where 0 < wi , w̃j ≤ 1 and where Group 1 is in some region T and thus Group 2
is in region T c . Then the PHDs for the separating groups are as follows:
1
D k|k (x)
(9.13)
=
1T (x) · Dk|k (x) = w1 · f1 (x) + ... + wn · fn (x)
=
1T c (x) · Dk|k (x) = w̃1 · f˜1 (x) + ... + w̃ñ · f˜ñ (x). (9.14)
2
D k|k (x)
Remark 26 These heuristic approaches for merging and splitting are actually
1
2
theoretically justifiable. In the case of merging, let Ξk|k and Ξk|k be the
statistically independent multitarget RFSs that are to be merged, and let their
1
2
1
respective PHDs be D k|k (x), D k|k (x). Then the merged RFS is Ξk|k = Ξk|k ∪
2
1
2
Ξk|k and it is easily shown that its PHD is Dk|k (x) = D k|k (x) + D k|k (x). In
1
2
the case of splitting, it is being assumed that Ξk|k = Ξk|k ∪ Ξk|k and that there
1
2
is a region T such that Ξk|k = Ξk|k ∩ T and Ξk|k = Ξk|k ∩ T c . Since Ξk|k
1
2
is assumed to be Poisson, Ξk|k and Ξk|k are independent—see Remark 12 of
Section 4.3.1. Thus one can conclude that Dk|k (x) can be split into the sum
1
2
1
2
1
D k|k (x) + D k|k (x) where D k|k (x), D k|k (x) are the respective PHDs of Ξk|k
2
and Ξk|k .
Implementing Classical PHD/CPHD Filters
9.4
223
MERGING AND SPLITTING FOR CPHD FILTERS
CPHD filtering of target clusters is not superpositional. That is, if we apply a CPHD
filter to each group and then add the respective PHDs, this will not result in the same
PHD that would result if a single CPHD filter were applied to the entire scene.
Thus, as already noted, CPHD filters must be applied separately to noninteracting
target groups. Such prepartitioning is already common practice for conventional
multitarget trackers such as MHT, in order to reduce computational complexity.
Pre-partitioning would similarly be necessary for CPHD filters even if it were not a
theoretical necessity.
The purpose of this section is to address the issues associated with merging
and splitting CPHD filters.
9.4.1
Merging for CPHD Filters
Suppose that two CPHD filters are tracking two target groups, and that these groups
move close enough that we should apply a single CPHD filter to both of them. How
do we merge the two CPHD filters? Let
1
2
1
2
D k|k (x), pk|k (n),
(9.15)
D k|k (x), pk|k (n)
be the PHDs and cardinality distributions produced by the two CPHD filters at time
tk . Assuming that the two target groups are approximately independent, the two
filters can be merged as follows:
1
Dk|k (x)
=
2
1
pk|k (n)
=
(9.16)
D k|k (x) + D k|k (x)
2
(9.17)
(pk|k ∗ pk|k )(n)
where
∑
(p1 ∗ p2 )(n) =
(9.18)
p1 (i) · p2 (j)
i+j=n
denotes convolution of the discrete probability distributions p1 (n) and p2 (n).
1
This approach for merging CPHD filters is not heuristic. If
Ξk|k and
2
Ξk|k are the multitarget RFSs that are to be merged, then the merged RFS is
1
2
1
2
Ξk|k = Ξk|k ∪ Ξk|k . Because Ξk|k , Ξk|k are independent, the p.g.fl. of Ξk|k
factors as
1
2
Gk|k [h] = Gk|k [h] · Gk|k [h]
(9.19)
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1
2
1
2
where Gk|k [h], Gk|k [h] are the p.g.fl.’s of Ξk|k , Ξk|k . Thus
1
Dk|k (x)
2
D k|k (x) + D k|k (x)
=
∑ 1
2
Gk|k (x) · Gk|k (x) =
(pk|k ∗ pk|k )(n) · xn .
1
Gk|k (x)
(9.20)
=
2
(9.21)
n≥0
9.4.2
Splitting for CPHD Filters
Suppose that a single CPHD filter is tracking a target group that splits into two
subgroups. How do we split the CPHD filter into two new ones, one for each of the
subgroups? The theoretical approach described in Section 9.3.2 cannot be applied,
because the partitioned RFSs Ξk|k ∩ T and Ξk|k ∩ T c of Remark 26 in Section
9.3.2 are not statistically independent (since, in general, Ξk|k is not Poisson).
From a purely theoretical point of view, Clark’s deconvolution formula,
(4.20), could be applied—but the result would not be computationally feasible in
general.
It therefore appears that only heuristic approaches are possible. The approach
described here was proposed by Petetin, Clark, Ristic, and Maltese [237], [238] and
appears to work reasonably well.1
Consider the PHD Dk|k (x) first. As with the PHD filter in (9.12), partition
it into two separated groups of weighted track distributions
Group 1
Group 2
??
?
?
??
? ?
Dk|k (x) = w1 · f1 (x) + ... + wn · fn (x) + w̃1 · f˜1 (x) + ... + w̃ñ · f˜ñ (x) (9.22)
where 0 < wi , w̃j ≤ 1. Then as with the PHD filter, specify PHDs for the
separating groups:
1
D k|k (x)
=
w1 · f1 (x) + ... + wn · fn (x)
(9.23)
=
w̃1 · f˜1 (x) + ... + w̃ñ · f˜ñ (x).
(9.24)
2
D k|k (x)
Now consider the cardinality distribution pk|k (i). We must specify distribu1
1
2
2
tions pk|k (i) and pk|k (i), with corresponding expected values N k|k and N k|k ,
1
Note that these authors employed it not with a classical CPHD filter, but with a CPHD filter that
was hybridized with data association techniques.
Implementing Classical PHD/CPHD Filters
225
which have the following properties:
1
2
(pk|k ∗ pk|k )(i)
=
pk|k (i)
(9.25)
=
w1 + ... + wn
(9.26)
=
w̃1 + ... + w̃ñ .
(9.27)
1
N k|k
2
N k|k
Assume that the target RFS is approximately multi-Bernoulli—that is, its
p.g.fl. approximately has the form
Gk|k [h]
(1 − w1 + w1 · f1 [h]) · · · (1 − wn1 + wn · fn [h])
·(1 − w̃1 + w̃1 · f˜1 [h]) · · · (1 − w̃ñ + w̃ñ · f˜ñ [h]).
=
(9.28)
2
Note that this assumption is not valid unless the variance σk|k
of the cardinality
distribution pk|k (n) is smaller than its expected value Nk|k —see Section 4.3.1.
Given this and given the properties of multi-Bernoulli RFSs (see Section 4.3.4), it
follows that the p.g.f., cardinality distribution, and PHD of the target RFS are
Gk|k (x)
=
pk|k (i)
=
(9.29)
(1 − w1 + w1 · x) · · · (1 − wn + wn · x)
·(1 − w̃1 + w̃1 · x) · · · (1 − w̃ñ + w̃ñ · x)
( n
) ( ñ
)
∏
∏
(1 − wi )
(1 − w̃i )
i=1
(9.30)
i=1
(
Dk|k (x)
=
w1
wn
w̃1
w̃ñ
, ...,
,
, ...,
·σn+ñ,i
1 − w1
1 − wn 1 − w̃1
1 − w̃ñ
w1 · f1 (x) + ... + wn · fn (x)
+w̃1 · f˜1 (x) + ... + w̃ñ · f˜ñ (x)
)
(9.31)
where σN,n (x1 , ..., xN ) is the elementary symmetric function of degree n in N
variables. (When i > n + ñ, pk|k (i) = 0.) Define
1
pk|k (i)
=
( n
∏
(1 − wi )
)
· σn,i
(
· σñ,i
(
i=1
2
pk|k (i)
=
( ñ
∏
i=1
(1 − w̃i )
)
w1
wn
, ...,
1 − w1
1 − wn
w̃1
w̃ñ
, ...,
1 − w̃1
1 − w̃ñ
)
(9.32)
)
(9.33)
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with corresponding expected values and p.g.f.’s
1
=
w1 + ... + wn
(9.34)
N k|k
=
w̃1 + ... + w̃ñ
(9.35)
Gk|k (x)
=
(1 − w1 + w1 · x) · · · (1 − wn + wn · x)
(9.36)
=
(1 − w̃1 + w̃1 · x) · · · (1 − w̃ñ + w̃ñ · x).
(9.37)
N k|k
2
1
2
Gk|k (x)
Then
1
2
Gk|k (x) · Gk|k (x) = Gk|k (x)
(9.38)
from which (9.25) through (9.27) follow.
One potential limitation of this approach is its relative computational com1
plexity. Because of the elementary symmetric functions, the distributions pk|k (i)
2
and pk|k (i) have computational order O(n2 ) and O(ñ2 ), respectively.
9.5
GAUSSIAN MIXTURE (GM) IMPLEMENTATION
The exact closed-form Gaussian mixture implementation of PHD filters was introduced in 2005 in a seminal paper by Vo and Ma [299], [300]. It was extended by
Clark, Panta, and Vo to include track labels, and its convergence properties were
established in [46], [48]. In 2007 it was generalized to the CPHD filter by Vo, Vo,
and Cantoni [308] and by Ulmke, Erdinc, and Willett [292].
The GM approximation depends on the assumption that the probability of
detection is constant. In [292], Ulmke et al. proposed an approximation that relaxes
this by presuming that the probability of detection is nonconstant but spatially
slowly varying (see Section 9.5.6).
Because the GM-PHD and GM-CPHD filters approximate the PHD using
Gaussian sums, they can be implemented as a bank of extended Kalman filters
(EKFs). Since EKFs are appropriate only for mild nonlinearities, Vo and Ma
proposed that the bank of EKFs be replaced by a bank of unscented Kalman
filters (UKFs) [300]. UKF implementations are usually sufficient for the degree
of nonlinearity associated with range-bearing measurement models.
For scenarios with still greater nonlinearity, Macagnano and de Abreu [149]
proposed the cubature Kalman filter (CKF) of Arasaratnam and Haykin [9] as a
replacement for the UKF in the GM-PHD filter. Clark, Vo, and Vo [49] have
Implementing Classical PHD/CPHD Filters
227
similarly proposed the Gaussian particle filter (GPF) of Kotecha and Djuric [138]
as a replacement.
The purpose of this section is to summarize the basic concepts underlying
GM approximation. It is organized as follows:
1. Section 9.5.1: The standard GM approximation.
2. Section 9.5.2: Pruning Gaussian mixtures.
3. Section 9.5.3: Merging Gaussian mixtures.
4. Section 9.5.4: The GM-PHD filter.
5. Section 9.5.5: The GM-CPHD filter.
6. Section 9.5.6: GM approximation with nonconstant probability of detection.
7. Section 9.5.7: GM approximation with partially uniform target appearances.
8. Section 9.5.8: GM approximation with target identity.
9.5.1
Standard GM Implementation
The Gaussian mixture approximation exploits the fact that Gaussian distributions
are algebraically closed under multiplication (see (2.3) and [179], pp. 699-703):
NP1 (x − x1 ) · NP2 (x − x2 )
E −1
=
=
NP1 +P2 (x2 − x1 ) · NE (x − e) (9.39)
P1−1 + P2−1
(9.40)
E −1 e
=
P1−1 x1 + P2−1 x2 .
(9.41)
It follows that if the PHD is approximated as a Gaussian mixture, the time-update
and measurement-update equations for the PHD and CPHD filters can be evaluated
in closed form, provided that a few restrictions are imposed (see Sections 9.5.4.1
and 9.5.5.1). The most severe restriction is that the probability of detection is
constant, pD (x) = pD (although one must also assume that the probability of
target survival is constant as well, pS (x′ ) = pS ).
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In GM approximation, the prior and predicted PHDs are assumed to be, at
least approximately, Gaussian mixtures
νk|k
Dk|k (x)
∑
=
k|k
k|k
(9.42)
· NP k|k (x − xi )
wi
i
i=1
νk+1|k
Dk+1|k (x)
∑
=
k+1|k
k+1|k
· NP k+1|k (x − xi
wi
)
(9.43)
i
i=1
where the expected numbers of targets are given by
νk|k
Nk|k =
∑
νk+1|k
k|k
wi ,
Nk+1|k =
i=1
∑
k+1|k
wi
.
(9.44)
i=1
It follows that the Time propagation of the PHDs is equivalent to the Time
propagation of families of the form
k|k
k|k
k|k
(ℓi , wi , Pi
k|k ν
k|k
, xi )i=1
k|k
where, in addition to the other items in the family, ℓi
GM component.
is the track label of the ith
Remark 27 (Track management in GM-PHD/CPHD filters) The track management scheme described in the following sections is based on simple rules for propagating the track labels of Gaussian components. It is easily implemented, but performance tends to suffer when targets cross or when the clutter rate is high. More
effective label management requires track-to-track association techniques, such as
those described in [146], [229], [231], [230].
9.5.2
Pruning Gaussian Components
As time progresses, the number of components in the GM approximation of a PHD
tends to increase without bound. Thus techniques must be employed to merge
similar components and prune less important components ([179], p. 630).
From a purely logical point of view, it would seem that components should
be merged before they are pruned. However, it is better to prune before merging.
Doing so avoids the computational cost associated with merging GM components
that will end up being pruned anyway.
Implementing Classical PHD/CPHD Filters
229
Suppose that we are to prune components from the measurement-updated GM
system
k+1|k+1
k+1|k+1
k+1|k+1 νk+1|k+1
(wi
, Pi
, xi
)i=1
∑νk+1|k+1 k+1|k+1
with Nk+1|k+1 = i=1
wi
. Set a pruning threshold τprune , identify
those components for which
k+1|k+1
wi
< τprune
(9.45)
and then eliminate them. This results in a pruned system
k+1|k+1
(w̌i
k+1|k+1 ν̌k+1|k+1
)i=1
k+1|k+1
, Pˇi
, x̌i
with ν̌k+1|k+1 components. Let
ν̌k+1|k+1
w̌ k+1|k+1 =
∑
k+1|k+1
w̌i
(9.46)
i=1
be the combined weight of all components that remain. Define the renormalized
weights
k+1|k+1
w̌
k+1|k+1
ŵi
(9.47)
= Nk+1|k+1 · ik+1|k+1
w̌
for all i = 1, ..., ν̌k+1|k+1 . Then
k+1|k+1
(ŵi
k+1|k+1
, Pˇi
k+1|k+1 ν̌k+1|k+1
)i=1
, x̌i
is the pruned GM system.
9.5.3
Merging Gaussian Components
Suppose that two Gaussian components
w1 · NP1 (x − x1 ) + w2 · NP2 (x − x2 )
(9.48)
are to be merged into a single component
w0 · NP0 (x − x0 ).
(9.49)
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Advances in Statistical Multisource-Multitarget Information Fusion
We must first have some criterion for determining whether or not they should
be merged. The Hellinger distance, (6.35), leads to the following measure of
overlap between NP1 (x − x1 ) and NP2 (x − x2 ):
√
det P1 P2
I = −2 log
+ (x1 − x2 )T (P1 + P2 )−1 (x1 − x2 ).
(9.50)
1
det 2 (P1 + P2 )
Note that when the two track distributions are identical, I = 0. For computational
purposes, (9.50) can be approximated as the Mahalanobis distance
I˜ = (x1 − x2 )T (P1 + P2 )−1 (x1 − x2 ).
(9.51)
When I˜ < τmerge for some merging threshold τmerge , we conclude that the two
components are sufficiently similar that they should be merged.
Suppose, then, that we have determined that n components
D(x) =
n
∑
wi · NPi (x − xi )
(9.52)
i=1
should be merged into a single component
w0 · NP0 (x − x0 ).
(9.53)
In Section K.16 it is shown that the Gaussian component that has the same mean
and covariance as the sum in (9.52) is
w0
x0
P0
=
=
=
=
n
∑
wi
(9.54)
i=1
n
∑
ŵi · xi
(9.55)
i=1
n
∑
ŵi · Pi +
∑
ŵi · ŵj · (xi − xj )(xi − xj )T
(9.56)
n
∑
ŵi · xi xTi
(9.57)
(i = 1, ..., n).
(9.58)
i=1
1≤i<j≤n
n
∑
ŵi · Pi − x0 xT0 +
i=1
i=1
where
ŵi =
wi
w0
Implementing Classical PHD/CPHD Filters
231
Remark 28 These formulas generalize those given on p. 630 of [179], which apply
only to the case n = 2. Equation (9.51) should be regarded as more theoretically
grounded than Eq. (16.286) in [179].
Remark 29 (Merging and track labels) When components are merged, the merged
component is assigned the label of the premerged component with the largest
weight. Also, because of the measurement-update step, it will often be the case
that many components will have the same label even after merging and pruning. In
this case the component with the largest weight keeps the common label and new
labels are assigned to the others.
Remark 30 (Other approaches for merging) The proposed approach for merging has been suggested because of its conceptual simplicity. However, it should
be noted that there is an extensive literature devoted to the problem of reduction
(merging) techniques for Gaussian mixtures. See [51] for a discussion of many of
these approaches.
9.5.4
GM-PHD Filter
This section summarizes the exact closed-form Gaussian mixture (GM) implementation of the PHD filter ([179], pp. 623-630). It is organized as follows:
1. Section 9.5.4.1: Modeling assumptions for the GM-PHD filter.
2. Section 9.5.4.2: Time update equations for the GM-PHD filter.
3. Section 9.5.4.3: Measurement update equations for the GM-PHD filter.
4. Section 9.5.4.4: Multitarget state estimation for the GM-PHD filter.
5. Section 9.5.4.5: The unscented Kalman filter (UKF) variant of the GM-PHD
filter.
6. Section 9.5.4.6: The cubature Kalman filter (CKF) variant of the GM-PHD
filter.
7. Section 9.5.4.7: The Gaussian particle filter (GPF) variant of the GM-PHD
filter.
9.5.4.1
GM-PHD Filter Models
The GM implementation of the PHD filter requires the following models:
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• Probability of target survival: does not depend on target state, pS,k+1 (x) =
pS,k+1 abbr.
= pS .
• Single-target Markov transition density—is linear-Gaussian:2
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(9.59)
• Target-appearance PHD—is a Gaussian mixture:
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
bi
k+1|k
· NB k+1|k (x − bi
)
(9.60)
i
i=1
and so the expected number of birth targets is
B
νk+1|k
B
Nk+1|k
=
∑
k+1|k
bi
(9.61)
.
i=1
• Target-spawning PHD—is a Gaussian mixture:
S
νk+1|k
bk+1|k (x|x′ ) =
∑
k+1|k
ej
k+1|k ′
· NGk+1|k (x − Ej
x)
(9.62)
j
j=1
and so the expected number of targets spawned by a target with state x′ is
independent of x′ :
B
νk+1|k
S
Nk+1|k
=
∑
k+1|k
ej
.
(9.63)
j=1
• Probability of detection—is independent of target state, pD,k+1 (x) =
pD,k+1 abbr.
= pD (this assumption can be removed using the approximation
described in Section 9.5.6).
• Sensor likelihood function: is linear-Gaussian:3
Lz (x) = fk+1 (z|x) = NRk+1 (z − Hk+1 x).
2
3
(9.64)
This assumption can be relaxed to allow fk+1|k (x|x′ ) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
This assumption can be relaxed to allow fk+1 (z|x) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
Implementing Classical PHD/CPHD Filters
233
• Clutter intensity function: κk+1 (z) = λk+1 · ck+1 (z) where λk+1 is the
clutter rate and ck+1 (z) is the clutter spatial distribution.
9.5.4.2
GM-PHD Filter Time Update
We are given a system of Gaussian components
k|k
k|k
k|k
(ℓi , wi , Pi
k|k ν
k|k
, xi )i=1
with
νk|k
∑
Nk|k =
k|k
(9.65)
wi
i=0
being the expected number of targets. We are to determine formulas for the
predicted system of Gaussian components
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
k+1|k νk+1|k
)i=1 .
, xi
These will actually have the structure
B
k+1|k νk|k +νk+1|k
)i=1
,
S
ν
,v
k+1|k
k+1|k
k+1|k
k+1|k k|k k+1|k
.
(ℓi,j , wi,j , Pi,j , xi,j )i=1;j=1
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
, xi
Given this, the time-update for the GM-PHD filter is defined by the following:
• Time updated number of GM components:
B
S
νk+1|k = νk|k + νk+1|k
+ νk|k · vk+1|k
.
(9.66)
B
Here, there are νk|k components corresponding to persisting targets, νk+1|k
S
components corresponding to newly appearing targets, and νk|k ·vk+1|k
components corresponding to spawned targets. The time-updated components are
indexed as follows:
=
1, ..., νk|k
(persisting)
(9.67)
i
=
B
νk|k + 1, ..., νk|k + νk+1|k
(appearing)
(9.68)
i
=
1, ..., νk|k ;
S
j = 1, ..., vk+1|k
(spawned).
(9.69)
i
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• Persisting-target GM components, for i = 1, ..., νk|k :
k+1|k
k|k
ℓi
=
k+1|k
wi
k+1|k
xi
k+1|k
Pi
=
=
=
ℓi
(9.70)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk .
(9.71)
(9.72)
(9.73)
B
• Appearing-target GM components, for i = νk|k + 1, ..., νk|k + νk+1|k
:
k+1|k
ℓi
k+1|k
wi
k+1|k
xi
k+1|k
Pi
=
new label
(9.74)
=
k+1|k
bi−νk|k
(9.75)
=
k+1|k
bi−νk|k
(9.76)
=
k+1|k
Bi−νk|k .
(9.77)
S
• Spawned-target GM components, for i = 1, ..., νk|k and j = 1, ..., vk+1|k
:
k+1|k
ℓi,j
=
new label
=
ej
· wi
(9.79)
=
k+1|k k|k
Ej
xi
(9.80)
=
Ej
k+1|k
wi,j
k+1|k
xi
k+1|k
k+1|k
Pi,j
9.5.4.3
(9.78)
k|k
k+1|k
k|k
Pi
k+1|k T
(Ej
k+1|k
) + Gj
.
(9.81)
GM-PHD Filter Measurement Update
We are given the predicted system of Gaussian components:
k+1|k
(ℓi
k+1|k
, wi
k+1|k νk+1|k
)i=1
k+1|k
, Pi
, xi
with
νk+1|k
Nk+1|k =
∑
k+1|k
wi
.
(9.82)
n=0
We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | =
mk+1 . We are to determine formulas for the measurement-updated system of
Implementing Classical PHD/CPHD Filters
235
Gaussian components
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1 vk+1|k+1
)i=1
.
k+1|k+1
, Pi
, xi
This will actually have the structure
k+1|k+1 νk+1|k
)i=1
k+1|k+1
k+1|k+1
k+1|k+1
k+1|k+1 νk+1|k ,mk+1
(ℓi,j
, wi,j
, Pi,j
, xi,j
)i=1;j=1
.
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1
, Pi
, xi
Given this, the measurement-update for the GM-PHD filter is given by the following:
• Measurement updated number of GM components for the PHD:
(9.83)
νk+1|k+1 = νk+1|k + mk+1 · νk+1|k
where there are νk+1|k components for undetected tracks and mk+1 · νk+1|k
components for detected tracks. The measurement-update components are
indexed as follows:
i
i
=
=
1, ..., νk+1|k
1, ..., νk+1|k ; j = 1, ..., mk+1
(undetected)
(detected).
(9.84)
(9.85)
• Measurement updated nondetection components: for i = 1, ..., νk+1|k ,
k+1|k+1
ℓi
k+1|k
=
k+1|k+1
wi
k+1|k+1
xi
k+1|k+1
Pi
=
=
=
ℓi
(9.86)
k+1|k
(1 − pD ) · wi
k+1|k
xi
k+1|k
Pi
.
(9.87)
• Measurement updated detection components:
j = 1, ..., mk+1 ,
k+1|k+1
ℓi,j
(9.88)
(9.89)
for i = 1, ..., νk+1|k and
k+1|k
=
(9.90)
ℓi
νk+1|k
τk+1 (zj )
=
pD
∑
k+1|k
(9.91)
wi
i=1
k+1|k
·NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
(zj − Hk+1 xi
)
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k+1|k+1
(9.92)
wi,j
pD · N R
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
k+1|k
(zj − Hk+1 xi
)
=
k+1|k+1
xi,j
k+1|k
k+1|k+1
=
Kik+1
9.5.4.4
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
)
(9.93)
(
) k+1|k
k+1
I − Ki Hk+1 Pi
(9.94)
(
)−1
k+1|k T
k+1|k T
Pi
Hk+1 Hk+1 Pi
Hk+1 + Rk+1
.(9.95)
=
Pi,j
k+1|k
wi
κk+1 (zj ) + τk+1 (zj )
=
GM-PHD Filter Multitarget State Estimation
State estimation for the GM-PHD filter is accomplished as follows. We are given
the measurement-updated system
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1 νk+1|k+1
)i=1
.
k+1|k+1
, Pi
, xi
Let n be the integer nearest to
νk+1|k+1
Nk+1|k+1 =
∑
k+1|k+1
wi
.
(9.96)
i=1
k+1|k+1
Determine those n Gaussian components for which wi
are largest. Then
k+1|k+1
k+1|k+1
the associated xi
are the track state estimates, and the associated Pi
are their track covariances.
9.5.4.5
GM-PHD Filter—Unscented Kalman Filter Variant
From (9.72), (9.73), (9.81), (9.93),(9.95), it is clear that, as an algorithm, the GMPHD filter consists of a bank of Kalman filters or extended Kalman filters, EKFs,
with each filter being separately applied to each Gaussian component. Since the
EKF can address only slight nonlinearities, Vo and Ma [300] proposed that the
bank of EKFs be replaced by a bank of unscented Kalman filters (UKFs). In this
manner, the GM-PHD filter can be extended to deal with sensors with moderately
severe nonlinearities, such as range-bearing sensors. For purposes of conceptual
completeness, the UKF is summarized in this section, based on the concise tutorial
introduction in Terejanu [289].
Implementing Classical PHD/CPHD Filters
237
The UKF was introduced by Julier and Uhlmann as a means of dealing with
nonlinearities that are severe enough that the EKF is no longer effective [129],
[130], [131]. It utilizes their “unscented transform,” which in turn is a means of
approximating the mean and covariance of the random vector ϕ(X), where ϕ is a
nonlinear transformation and X is a Gaussian-distributed random vector.4
Sigma-points: Suppose that we are given the mean x and covariance matrix
P of a Gaussian random vector X. Let N be the dimension of the Euclidean state
space. The basic idea behind the unscented transform is to choose deterministic
“sigma points” x0 , x1 , ..., x2N and associated fixed weights w0 , w1 , ..., w2N that
obey the following constraints [130]:
1=
2N
∑
wi ,
x=
i=0
2N
∑
wi · xi ,
P =
i=0
2N
∑
wi · (xi − x)T (xi − x).
(9.97)
i=0
The mean xϕ and covariance Cϕ of the transformed random variable ϕ(X) are
approximated as
∼
=
xϕ
2N
∑
wi · ϕ(xi )
(9.98)
wi · (ϕ(xi ) − xϕ ) · (ϕ(xi ) − xϕ )T
(9.99)
i=0
∼
=
Cϕ
2N
∑
i=0
while the cross-covariance between X and ϕ(X) is approximated as
C˜ϕ ∼
=
2N
∑
wi · (xi − x) · (ϕ(xi ) − xϕ )T .
(9.100)
i=0
The unscented transform: The most common approach for selecting the
sigma-points and their weights is as follows, for i = 1, ..., N :
w0
wi
wi+n
4
=
=
=
s
,
N +s
1
,
2(N + s)
1
,
2(N + s)
(9.101)
x0 = x
xi = x0 +
(√
xi+N = x0 −
(N + s)P
(√
)
(9.102)
i
(N + s)P
)
.
(9.103)
i
The unscented transform is often referred to as a “sigma-point transform” and the UKF as a “sigmapoint Kalman filter” or a “linear regression Kalman filter.”
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Advances in Statistical Multisource-Multitarget Information Fusion
(√ )
√
Here,
C
denotes the ith column vector of the matrix square root C of
i
the covariance matrix C; and the scaling factor s determines the degree of the
spread of the x1 , ..., x2N around x0 .
The effectiveness of (9.101) through (9.103) depends on the proper choice
of s. In the special case s = 3 − n < 0, the estimated covariance can become
nonpositive definite. The “scaled unscented transform” was devised to address this
special case [128].
The unscented Kalman filter: The UKF employs the unscented transform as
follows.
For the time-update step, let φk (x) be the state transition function and let
k|k
xk|k , Pk|k be the mean and covariance at time tk . Set x0 = xk|k . Select
k|k
k|k
k|k
k|k
k|k
sigma-points x1 , ..., x2N and weights w0 , w1 , ..., w2N as in (9.101) through
k|k
k|k
k|k
(9.103). Propagate x0 , x1 , ..., x2N through φk to get
k|k
k|k
k|k
φk (x0 ), φk (x1 ), ..., φk (x2N ).
Estimate xk+1|k and Pk+1|k using (9.98) and (9.99), while accounting for the
uncertainty due to plant noise:
xk+1|k
=
2N
∑
k|k
wi
k+1|k
· φk (xi
)
(9.104)
i=0
Pk+1|k
=
Qk +
2N
∑
k|k
wi
(
)
k|k
· φk (xi ) − xk+1|k
(9.105)
i=0
(
)T
k|k
· φk (xi ) − xk+1|k .
For the measurement-update step, let ηk+1 (x) be the measurement funck+1|k
k+1|k
k+1|k
, ..., x2N
and
tion and set x0
= xk+1|k . Select sigma-points x1
k+1|k
k+1|k
k+1|k
weights w0
, w1
, ..., w2N
as in (9.101) through (9.103). Propagate
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k
x0
, x1
,..., x2N
through ηk+1 to get ηk+1 (x0
), ηk+1 (x1
),...,
k+1|k
ηk+1 (x2N ). Estimate the expected predicted observation zk+1|k+1 , the innovation covariance matrix Sk+1|k+1 , and the cross-covariance matrix Ck+1|k+1 , using
Implementing Classical PHD/CPHD Filters
239
(9.98) and (9.99), while accounting for measurement noise:
zk+1|k+1
=
2N
∑
k+1|k
k+1|k
· ηk+1 (xi
wi
(9.106)
)
i=0
Sk+1
=
Rk+1 +
2N
∑
k+1|k
wi
(
)
k+1|k
· ηk+1 (xi
) − zk+1|k+1 (9.107)
i=0
(
k+1|k
· ηk+1 (xi
Ck+1|k+1
=
2N
∑
k+1|k
wi
) − zk+1|k+1
)T
(
)
k|k
· φk (xi ) − xk+1|k
(9.108)
i=0
(
k+1|k
ηk+1 (xi
) − zk+1|k+1
)T
.
Then given a new measurement zk+1 , estimate the posterior state and covariance
as
xk+1|k+1
Kk
Pk+1|k+1
=
xk+1|k + Kk (zk+1 − zk+1|k+1 )
(9.109)
=
−1
Ck+1 Sk+1
(9.110)
=
Pk+1|k − Kk Sk+1 KkT .
(9.111)
Finally, repeat recursively.
9.5.4.6
GM-PHD Filter—Cubature Kalman Filter Variant
Arasaratnam and Haykin introduced the cubature Kalman filter (CKF) in 2009, as a
replacement for the EKF and UKF when nonlinearities are too severe for either to
be effective [9]. In 2010, Macagnano and de Abreu [149], [148] proposed the CKF
as a replacement for the EKF or UKF in the GM-PHD filter. The purpose of this
section is to briefly describe the CKF.
The CKF has a structure similar to that of the UKF ([9], p. 1267, Appendix
A). Arasaratnam and Haykin argue that, despite this fact, it has the following
advantages over the UKF ([9], p. 1262): it is (1) more mathematically principled,
(2) numerically more accurate, and (3) always results in positive-definite covariance
matrices.
For the CKF time-update, let xk|k and Pk|k be the mean and covariance at
time tk and let φk be the state transition function. Construct the cubature points,
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Advances in Statistical Multisource-Multitarget Information Fusion
for i = 1, ..., N :
xk|k +
(√
xk|k −
(√
k|k
xi
=
k|k
xi+N
=
N · Pk|k
N · Pk|k
)
(9.112)
)i
(9.113)
i
(√
)
where
N · Pk|k i denotes the ith column vector of the matrix square root.
Use these cubature points to estimate the predicted state and covariance:
N
xk+1|k
=
)
1 ∑(
k|k
k|k
φκ (xi ) + φκ (xi+n )
2N i=1
Pk+1|k
=
Qk − xTk+1|k xk+1|k
+
(9.114)
(9.115)
N (
∑
)
1
k|k
k|k
k|k
k|k
φκ (xi )T φκ (xi ) + φκ (xi+n )T φκ (xi+n ) .
2N i=1
For the CKF measurement-update, let ηk+1 be the measurement function.
Evaluate the cubature points for i = 1, ..., N :
xk+1|k +
(√
xk+1|k −
(√
k+1|k
xi
=
k+1|k
xi+N
=
N · Pk+1|k
N · Pk+1|k
)
(9.116)
)i
(9.117)
i
and use them to estimate the expected predicted measurement, innovation covariance matrix, and cross-covariance matrix:
N
)
1 ∑(
k+1|k
k+1|k
ηκ+1 (xi
) + ηκ+1 (xi+n )
2N i=1
(9.118)
zk+1|k+1
=
Sk+1
=
Rk+1 − zTk+1|k+1 zk+1|k+1
(9.119)
)
(
N
k+1|k T
1 ∑ ηκ+1 (xk+1|k
) · ηκ+1 (xi
)
i
+
k+1|k
k+1|k T
2N i=1
+ηκ+1 (xi+n ) · φκ (xi+n )
Ck+1
=
−xTk+1|k zk+1|k+1
(
)
N
k+1|k
k+1|k T
1 ∑
(xi
) · ηκ+1 (xi
)
+
.
k+1|k
k+1|k
2N i=1
+(xi+n ) · ηκ+1 (xi+n )T
(9.120)
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241
Then given a new measurement zk+1 , estimate the measurement-updated mean
and covariance:
xk+1|k+1
Kk
Pk+1|k+1
=
xk+1|k + Kk (zk+1 − zk+1|k+1 )
(9.121)
=
−1
Ck+1 Sk+1
(9.122)
=
Pk+1|k − Kk Sk+1 KkT .
(9.123)
CKF-PHD Filter Performance Results: In [149], Macagnano and de Abreu
compared the performance of the CKF-PHD filter with that of the EKF-PHD filter in
two simple two-dimensional simulations. In the first simulation, two targets appear,
approach each other, and then veer off, with the targets spawning a third target at
roughly mid-scenario. In the second simulation, the two targets cross and the third
target appears during the simulation.
The authors performed two sets of simulations to assess the accuracy of targetnumber estimation. In the first set, the clutter rate λ increased from 1.5 to 39
while pD = 0.98 was held constant. In the second, pD was increased from 0.8
to 1.0 while λ = 10 was held constant. For varying λ, the CKF-PHD filter
showed significant improvement in target-number accuracy, compared to the EKFPHD filter, for λ ≥ 12. For varying pD , the CKF-PHD filter was significantly
better for pD ≤ 0.98. The two filters were also compared for increasing λ using
the OSPA metric (Section 6.2.2). The EKF-PHD filter outperformed the CKF-PHD
filter for λ ≤ 6, whereas the reverse was true for λ > 6.
In [148], Macagnano and de Abreu modified the CKF-PHD filter to include
an adaptive measurement-gating scheme. The authors concluded that their adaptive gating approach achieved significant performance improvements, compared to
standard elliptical gating.
9.5.4.7
GM-PHD Filter—Gaussian Particle Filter Variant
In this approach, proposed by Clark, Vo, and Vo in 2007 [49], the EKF or UKF in
a GM-PHD filter is replaced by the Gaussian particle filter (GPF) of Kotecha and
Djuric [138]. The Markov transition density and likelihood function are allowed to
be nonlinear-Gaussian
fk+1|k (x|x′ )
=
NQk (x − φk (x′ ))
(9.124)
fk+1 (z|x)
=
NRk+1 (z − ηk+1 (x))
(9.125)
where the functions φk (x′ ) and ηk+1 (x) are arbitrary. As usual, the PHDs
are approximated as Gaussian mixtures. However, the filtering equations will now
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Advances in Statistical Multisource-Multitarget Information Fusion
involve products of the form
k|k
NQk (x − φk (x′ )) · NP k|k (x − xi )
i
k+1|k
NRk+1 (x − ηk+1 (x)) · NP k+1|k (x − xi
).
i
form
Applying (9.124) to persisting targets, we will end up with integrals of the
∫
k|k
NQk (x − φk (x′ )) · NP k|k (x − xi )dx.
i
These can be numerically evaluated via Monte Carlo integration, as follows. First,
k|k
for fixed i draw ν samples ui,1 , ..., ui,ν from NP k|k (∗ − xi ). Second and
i
again for fixed i, draw a single sample vi,j from NRk+1 (∗ − ηk+1 (ui,j )). Then
by the law of large numbers,
−→
1∑
NRk+1 (x − ηk+1 (ui,j )
ν j=1
∫
k|k
NRk+1 (x − ηk+1 (x′ )) · NP k|k (x − xi )dx
i
almost surely as ν → ∞. So the left side can be taken as an approximation of the
right side. Third, for fixed i compute the sample mean and covariance
ν
=
1∑
ui,j
ν j=1
=
1∑
k+1|k
k+1|k T
(ui,j − xi
)(ui,j − xi
) .
ν j=1
k+1|k
xi
(9.126)
ν
k+1|k
Pi
(9.127)
Fourth, the predicted PHD for persisting targets can be approximated as
νk|k
pS
∑
k+1|k
k|k
wi
· NP k+1|k (x − xi
).
i
i=1
Applying (9.125) to detected targets, we must evaluate terms of the form
k+1|k
pD · NRk+1 (zj − ηk+1 (x)) · NP k+1|k (x − xi
i
i
κk+1 (zj ) + τk+1
(zj )
)
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243
where
i
τk+1
(zj ) = pD
∫
k+1|k
NRk+1 (zj − ηk+1 (x)) · NP k+1|k (x − xi
)dx.
(9.128)
i
The following procedure is employed. First, for fixed i, j, draw samples
wi,j,1 , ..., wi,j,ν
from some importance-sampling function π(x). (Clark suggests that this function
k+1|k
could be NP k+1|k (x − xi
) or the EKF/UKF measurement-update of this
i
density using zj .) Second, compute the weight corresponding to each sample for
l = 1, ..., ν:
k+1|k
NRk+1 (zj − ηk+1 (wi,j,l )) · NP k+1|k (wi,j,l − xi
)
i
wi,j,l =
.
(9.129)
π(wi,j,l )
Third, note that
ν
1∑
i
wi,j,l −→ τk+1
(zj )
ν
l=1
as ν → ∞, and so the left side can be taken as an approximation of the right side.
Fourth, compute the sample means and covariances and weights
k+1|k+1
xi,j
=
∑ν
w
· wi,j,l
l=1
∑ν i,j,l
w
′
l =1 i,j,l′
(9.130)
k+1|k+1
=
k+1|k+1
wi,j
=
(9.131)
Pi,j
∑ν
k+1|k+1
k+1|k+1 T
)(wi,j,l − xi,j
)
l=1 wi,j,l · (wi,j,l − xi,j
∑ν
′
l′ =1 wi,j,l
∑ν
k+1|k
1
wi
· pD · ν l=1 wi,j,l
.
(9.132)
∑νk+1|k k+1|k 1 ∑ν
κk+1 (zj ) + pD i=1 wi
· ν l=1 wi,j,l
Fifth, approximate the detected-target PHD as
mk+1 νk+1|k
∑ ∑
j=1
i=1
k+1|k+1
wi,j
k+1|k+1
· NP k+1|k+1 (x − xi,j
i,j
).
(9.133)
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GPF-PHD/CPHD Filter Performance Results: Clark et al. implemented
the GPF approach for both GM-PHD and GM-CPHD filters. They tested it in
simulations involving a range-bearing sensor and five appearing and disappearing
targets [49]. The target state was assumed to be (x, y, vx , vy , ω) where ω is the
turn rate of a nonlinear coordinated-turn motion model. Clutter was assumed to be
spatially uniform and Poisson with clutter rate λ = 25; and probability of detection
was pD = 0.98. The GPF-PHD and GPF-CPHD filters both accurately tracked
the targets, even when they crossed, with the GPF-CPHD filter more accurately
estimating target number than the GPF-PHD filter.
9.5.5
GM-CPHD Filter
This section summarizes the exact closed-form Gaussian mixture (GM) implementation of the CPHD filter ([179], pp. 646-649). It is organized as follows:
1. Section 9.5.5.1: Modeling assumptions for the GM-CPHD filter.
2. Section 9.5.5.2: Time update equations for the GM-CPHD filter.
3. Section 9.5.5.3: Measurement update equations for the GM-CPHD filter.
4. Section 9.5.5.4: Multitarget state estimation for the GM-CPHD filter.
5. Section 9.5.5.5: The unscented Kalman filter (UKF) variant of the GMCPHD filter.
Remark 31 The time-update and measurement-update formulas presented here for
the GM-CPHD filter are slightly different than those given in [179], pp. 646-649.
They are nevertheless equivalent to them.
9.5.5.1
GM-CPHD Filter Models
The GM implementation of the CPHD filter requires the following models:
• Cardinality distributions—are finite, that is, pk|k (n) = 0 for n ≥ nmax .
• Probability of target survival—is constant, pS,k+1|k (x) = pS,k+1|k abbr.
= pS .
• Single-target Markov density—is linear-Gaussian:5
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
5
(9.134)
This assumption can be relaxed to allow fk+1|k (x|x′ ) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
Implementing Classical PHD/CPHD Filters
245
• Target–appearance PHD—is a Gaussian mixture:
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
k+1|k
· NB k+1|k (x − bi
bi
)
(9.135)
i
i=1
and so the expected number of appearing targets is
B
νk+1|k
B
Nk+1|k
=
∑
k+1|k
bi
.
(9.136)
i=1
• Target-appearance cardinality distribution—is finite, that is, pB
k+1|k (n) = 0
for sufficiently large n; and it must be the case that
∑
B
Nk+1|k
=
n · pB
k+1|k (n).
(9.137)
n≥1
• Probability of detection—is constant, pD,k+1 (x) = pD,k+1 abbr.
= pD (this
assumption can be removed using the approximation described in Section
9.5.6).
• Sensor likelihood function—is linear-Gaussian:6
Lz (x) = fk+1 (z|x) = NRk+1 (z − Hk+1 x).
(9.138)
• Clutter cardinality distribution—pκk+1 (m) is arbitrary; or equivalently, the
clutter p.g.f. Gκk+1 (z) is arbitrary.
• Clutter spatial distribution: ck+1 (z) is arbitrary.
9.5.5.2
GM-CPHD Filter Time Update
We are given a cardinality distribution and system of Gaussian components for the
PHD,
k|k
k|k
k|k
k|k νk|k
pk|k (n),
(ℓi , wi , Pi , xi )i=1
,
6
This assumption can be relaxed to allow fk+1 (z|x) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
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with
νk|k
∑
Nk|k =
k|k
wi
=
n=0
n∑
max
(9.139)
n · pk|k (n).
n=0
We are to determine formulas for the predicted cardinality distribution and the
predicted system of Gaussian components,
k+1|k
pk+1|k (n),
(ℓi
k+1|k
, wi
k+1|k
, Pi
k+1|k νk+1|k
)i=1 .
, xi
These are determined as follows:
• Time updated cardinality distribution:
pk+1|k (n) =
n∑
max
pk+1|k (n|n′ ) · pk|k (n′ )
(9.140)
n′ =0
where
min{n,n′ }
′
pk+1|k (n|n ) =
∑
′
i
n −i
pB
. (9.141)
k+1|k (n − i) · Cn′ ,i · pS,k (1 − pS,k )
i=0
and where Cn′ ,i was defined in (2.1).
• Time updated number of GM components for the PHD:
B
νk+1|k = νk|k + νk+1|k
.
(9.142)
Here, there are νk|k components corresponding to persisting targets, and
B
νk+1|k
components corresponding to newly appearing targets. The timeupdated components are indexed as follows:
i
i
=
1, ..., νk|k
(persisting)
(9.143)
=
B
νk+1 + 1, ..., νk+1 + νk+1|k
(appearing).
(9.144)
• Persisting-target GM components, for i = 1, ..., νk|k :
k+1|k
ℓi
k+1|k
wi
k+1|k
xi
k+1|k
Pi
k|k
=
=
=
=
ℓi
(9.145)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk .
(9.146)
(9.147)
(9.148)
Implementing Classical PHD/CPHD Filters
247
B
• Appearing-target GM components, for i = νk|k + 1, ..., νk|k + νk+1|k
:
k+1|k
ℓi
k+1|k
wi
=
new label
(9.149)
=
k+1|k
bi−νk|k
(9.150)
k+1|k
=
bi−νk|k
(9.151)
k+1|k
=
Bi−νk|k .
(9.152)
k+1|k
xi
k+1|k
Pi
9.5.5.3
GM-CPHD Filter Measurement Update
We are given the predicted cardinality distribution and the predicted system of
Gaussian components,
k+1|k
pk+1|k (n),
(ℓi
k+1|k
, wi
k+1|k
, Pi
k+1|k νk+1|k
)i=1 ,
, xi
with
νk+1|k
∑
Nk+1|k =
k+1|k
wi
=
n=0
n∑
max
n · pk+1|k (n).
(9.153)
n=0
We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | =
mk+1 . We are to determine formulas for the measurement-updated cardinality
distribution and the measurement-updated system of Gaussian components:
k+1|k+1
pk+1|k+1 (n),
(ℓi
k+1|k+1
, wi
k+1|k+1
, Pi
k+1|k+1 νk+1|k+1
)i=1
.
, xi
The system will actually have the structure
k+1|k+1 νk+1|k
)i=1 ,
k+1|k+1
k+1|k+1
k+1|k+1
k+1|k+1 νk+1|k ,mk+1
(ℓi,j
, wi,j
, Pi,j
, xi,j
)i=1;j=1
.
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1
, Pi
, xi
These are determined as follows:
• Measurement updated number of GM components for the PHD:
νk+1|k+1 = νk+1|k + mk+1 · νk+1|k
(9.154)
where, as with the GM-PHD filter, there are νk+1|k components for undetected tracks and mk+1 · νk+1|k components for detected tracks. The
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Advances in Statistical Multisource-Multitarget Information Fusion
measurement-updated components are indexed as follows:
i
i
=
=
(undetected)
(detected).
1, ..., νk+1|k
1, ..., νk+1|k ; j = 1, ..., mk+1
(9.155)
(9.156)
• Measurement updated cardinality distribution:
ℓZ (n) · pk+1|k (n)
pk+1|k+1 (n) = ∑ k+1
l≥0 ℓZk+1 (l) · pk+1|k (l)
(9.157)
where
( ∑
ℓZk+1 (n) =
min{mk+1 ,n}
(mk+1 − j)! · pκk+1 (mk+1 − j)
j=0
·j! · Cn,j · ϕn−j
· σj (Zk+1 )
k
( ∑mk+1
)
κ
l=0 (mk+1 − l)! · pk+1 (mk+1 − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
)
(9.158)
where Cn,j was defined in (2.1), and where
ϕk
=
Gk+1|k (ϕk )
=
(l)
(9.159)
1 − pD,k+1
nmax
∑
pk+1|k (n) · l!Cn,l · ϕn−l
k
(9.160)
n=l
σi (Zk+1 )
=
τ̂k+1 (zj )
=
σmk+1 ,i
(
τ̂k+1 (zmk+1 )
τ̂k+1 (z1 )
, ...,
ck+1 (z1 )
ck+1 (zmk+1 )
)
(9.161)
νk+1|k
∑
pD
Nk+1|k
·NR
k+1|k
(9.162)
wi
l=1
k+1|k
T
Hk+1
k+1 +Hk+1 Pl
(zj − Hk+1 xk+1
).
l
• Measurement updated undetected-target GM components for the PHD: for
i = 1, ..., vk+1|k ,
k+1|k+1
ℓi
k+1|k
=
ℓi
=
k+1|k
(1 − pD ) · wi
k+1|k+1
wi
Nk+1|k
k+1|k+1
xi
k+1|k+1
Pi
k+1|k
(9.163)
ND
· L Zk+1
(9.164)
=
xi
(9.165)
=
k+1|k
Pi
(9.166)
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249
where
ND
L Zk+1
(9.167)
)
=
( ∑mk+1
(
n∑
max
(j+1)
Gk+1|k (ϕk )
κ
j=0 (mk+1 − j)! · pk+1 (mk+1 − j)
(j+1)
·σj (Zk+1 ) · Gk+1|k (ϕk )
)
∑mk+1
κ
l=0 (mk+1 − l)! · pk+1 (mk+1 − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
=
(9.168)
pk+1|k (n) · (j + 1)! · Cn,j+1
n=j+1
·ϕn−j−1
.
k
• Measurement updated detected-target GM components for the PHD: for
i = 1, ..., vk+1|k and j = 1, ..., mk+1 ,
k+1|k+1
ℓi,j
k+1|k
=
(9.169)
ℓi
D
k+1|k
pD · w i
Nk+1|k
k+1|k+1
wi,j
=
·
LZk+1 (zj )
ck+1 (zj )
(9.170)
k+1|k
·NR
k+1|k
xi,j
k+1|k
=
k+1|k
Pi,j
=
Kik+1
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
=
(zj − Hk+1 xi
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
(
) k+1|k
I − Kik+1 Hk+1 Pi
(9.171)
)
(9.172)
k+1|k T
Pi
Hk+1
(
k+1|k
· Hk+1 Pi
)
(9.173)
T
Hk+1
+ Rk+1
)−1
where
D
LZk+1 (zj )
( ∑m −1
k+1
i=0
=
(mk+1 − i − 1)! · pκk+1 (mk+1 − i − 1)
(i+1)
·σi (Zk+1 − {zj }) · Gk+1|k (ϕk )
( ∑mk+1
l=0
(mk+1 − l)! · pκk+1 (mk+1 − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
)
(9.174)
)
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Advances in Statistical Multisource-Multitarget Information Fusion
and
=
σi (Zk+1 − {zj })
(9.175)
(
)
?
τ̂k+1 (zmk+1 )
τ̂k+1 (z1 )
τ̂k+1
(zj )
σmk+1 −1,i
, ...,
, ...,
ck+1 (z1 )
ck+1 (zj )
ck+1 (zmk+1 )
and where x1 , ..., x?j , ..., xm indicates that the jth item
removed from the list x1 , ..., xm : x1 , ..., xj−1 , xj+1 , ..., xm .
9.5.5.4
xj is to be
GM-CPHD Filter Multitarget State Estimation
State estimation is the same as in Section 9.5.4.4, except that the nearest integer n
to Nk+1|k+1 is replaced by the MAP estimate
(9.176)
n̂k+1|k+1 = arg sup pk+1|k+1 (n).
n≥0
9.5.5.5
GM-CPHD Filter: UKF, CKF, and GPF Variants
As with the GM-PHD filter, the GM-CPHD filter can be regarded as a bank of
extended Kalman filters (EKFs), which are applied separately to each Gaussian
component of the PHD. The EKFs can be replaced by unscented Kalman filters
(UKFs) in the same manner as described in Section 9.5.4.5. Likewise for cubature
Kalman filters (Section 9.5.4.6) and Gaussian particle filters (Section 9.5.4.7).
9.5.6
Implementation with Nonconstant pD
As already noted, GM implementation requires that probability of detection be
constant. The reason is that the measurement-update equations for the PHD and
CPHD filter require multiplication of the components of the Gaussian mixtures by
the factors pD (x) and 1 − pD (x). This would prevent closed-form solution in
terms of Gaussian mixtures if pD (x) were not constant.
This challenge can be addressed using an approximation first proposed by
Ulmke, Erdinc, and Willett [292]. Assume that the probability of detection is approximately constant, as compared with the covariances of the target track distributions. Then we can write
k+1|k
pD (x) · NP k+1|k (x − xi
(9.177)
)
i
∼
=
k+1|k
pD (xi
k+1|k
) · NP k+1|k (x − xi
i
)
Implementing Classical PHD/CPHD Filters
251
and
k+1|k
(1 − pD (x)) · NP k+1|k (x − xi
(9.178)
)
i
∼
=
k+1|k
(1 − pD (xi
k+1|k
)) · NP k+1|k (x − xi
)
i
for all i = 1, ..., vk+1|k . Thus
νk+1|k
pD (x) · Dk+1|k (x)
∑
∼
=
k+1|k
wi
k+1|k
· pD (xi
)
(9.179)
i=1
k+1|k
·NP k+1|k (x − xi
)
i
and
νk+1|k
(1 − pD (x)) · Dk+1|k (x)
∑
∼
=
k+1|k
wi
k+1|k
· (1 − pD (xi
)) (9.180)
i=1
k+1|k
·NP k+1|k (x − xi
).
i
Remark 32 (A different approach) The “pD -agnostic” beta-Gaussian mixture
(BGM) approach described in Section 17.3, provides a more theoretically satisfying
way of addressing nonconstant pD . In this case, pD (x) is regarded as an unknown
state variable 0 ≤ a ≤ 1 and the usual state x is replaced by an augmented state
x̊ = (a, x).
9.5.7
Implementation with Partially Uniform Target Births
The PHD for the target appearance process was given in (9.135):
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
bi
k+1|k
· NB k+1|k (x − bi
).
(9.181)
i
i=1
The most obvious way to construct this PHD is to use whatever a priori information
k+1|k
k+1|k
is available to choose the respective bi
and Bi
, with the matrix norm
k+1|k
∥Bi
∥ typically chosen to be large. However, doing so will tend to create a
large number of birth-target components.
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Advances in Statistical Multisource-Multitarget Information Fusion
k+1|k
One therefore might place the bi
so as to correspond to the measurements collected at time tk+1 . This “measurement driven” approach will also
tend to create a large number of birth-target components if the clutter rate is large.
Moreover, it can produce a statistical bias in the target-number estimate when target
number is small, similar to that identified by Ristic, Clark, and Vo in regard to SMCPHD filters (see Section 9.6.4 and [256]).
A potentially more efficient approach, proposed by Beard, Vo, Vo, and Arulampalam [16], [17], is based on two innovations.
The first innovation, proposed by Ristic et al., provides a means of avoiding
this statistical bias [256], [257]; and will be described using a somewhat different
formulation in Section 9.6.4. It consists of augmenting the state x with a binary
variable o = 1, 2 where o = 0 indicates that (0, x) is the state of a persisting
target and o = 1 indicates that (1, x) is the state of a newly appearing target.
In this approach, the likelihood function and probability of detection for the
augmented state are defined to be, respectively,
Lz (o, x)
=
pD (o, x)
=
Lz (x)
{
pD (x)
1
(9.182)
if
if
o=0
.
o=1
(9.183)
That is, newly appearing targets are always detected, and they generate measurements in the same way as persisting targets. This is in accordance with the intuition
that no target can be asserted to exist, unless it first generates a measurement.
Similarly, the Markov transition is defined to be:

 fk+1|k (x|x′ )
′
′
f
(x|x′ )
fk+1|k (o, x|o , x ) =
 k+1|k
0
if
if
if
o = o′ = 0
o = 0, o′ = 1 .
otherwise
(9.184)
That is, birth targets and persisting targets can transition only to persisting targets.
Finally, the probability of target persistence and the target-birth PHD are
given by
pS (o′ , x′ )
=
bk+1|k (o, x)
=
pS (x′ )
{
0
bk+1|k (x)
(9.185)
if
if
o=0
.
o=1
(9.186)
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253
The second innovation, proposed by Beard, Vo, Vo, and Arulampalam, is the
partially uniform target appearance process.7 It is assumed that the state-vector x
can be decomposed into two parts,
x = (o, u)
(9.187)
with a corresponding decomposition of state spaces X = O × U with o ∈ O and
u ∈ U, such that the measurement function ηk+1 (x) has the following property:
ηk+1 (o, u) = ηk+1 (o).
(9.188)
That is,
• The state variables in o ∈ O are at least partially observed; whereas
• The state variables in u ∈ U are completely unobserved.
Given this, the basic idea is to replace (9.181) by
B
bk+1|k (o, u) = wk+1|k
·
1O′k+1|k (o)
|O′k+1|k |
2
· Nσk+1|k
I (u − uk+1|k )
(9.189)
where:
• O′k+1|k is an arbitrarily large but bounded region of the observed-state space
O, and |O′k+1|k | is its hypervolume.
• uk+1|k ∈ U and I is the identity matrix on U.
B
• wk+1|k
is the weight of the target appearance component.
That is, target appearances are:
• uniformly distributed with respect to the observed state variables, but
• Gaussian-distributed with respect to the unobserved ones.
Because of (9.189), the target appearance PHD will no longer be a Gaussian
mixture, and thus will not result in a time-updated PHD that is a Gaussian mixture.
However, Beard et al. show that one can still devise PHD and CPHD filters
7
This approach is similar to one proposed by Houssineau and Laneuville [116], who proposed
B
making the birth PHD uniform: bk+1|k (x) = wk+1|k
· 1X0 (x).
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Advances in Statistical Multisource-Multitarget Information Fusion
that are approximately Gaussian-mixture filters. This is because of the following
approximations:
∫ 1 ′
Ok+1|k (o)
|O′k+1|k |
∫
· NO (o − o0 )do
∼
=
1O′k+1|k (o)
|O′k+1|k |
· NO (o − o0 )
O′k+1|k
=
∼
=
NO (o − o0 )do
|O′k+1|k |
(9.190)
1
(9.191)
|O′k+1|k |
NO (o − o0 )
.
|O′k+1|k |
(9.192)
Because of these approximations it follows that, after each execution of a timeupdate followed by a measurement-update, the PHD will be a Gaussian mixture.
This fact is explained in detail in the following subsections:
1. Section 9.5.7.1: The PHD filter with a partially uniform target-birth PHD.
2. Section 9.5.7.2: The CPHD filter with a partially uniform target-birth PHD.
3. Section 9.5.7.3: Implementations of these PHD and CPHD filters.
9.5.7.1
PHD Filter with a Partially Uniform Target-Birth PHD
Time update: Assume that the single-target Markov density has the form
fk+1|k (x|x′ ) = NQk (x − Fk x′ )
(9.193)
and that target survival probability is constant: pS (x′ ) = pS . Also assume that the
measurement-updated PHD at time tk is a Gaussian mixture
o
νk|k
Dk|k (o, o, u) =
∑
k|k
k|k
k|k
wo,i · NP k|k ((o, u) − (oo,i , uo,i ))
(9.194)
o,i
i=1
k|k
where Po,i is a covariance matrix, expressed in terms of the coordinates associated
with the state representation x = (o, u). Given this, the time-updated PHD is given
Implementing Classical PHD/CPHD Filters
255
by (see Section K.19)
o
νk|k
Dk+1|k (0, o, u)
=
pS
∑
k|k
k+1|k
w0,i · NP k+1|k ((o, u) − (o0,i
k+1|k
, u0,i
)) (9.195)
0,i
i=1
o
νk|k
+pS
∑
k|k
k+1|k
w1,i · NR +F P k|k F T ((o, u) − (o1,i
k
k
1,i
k+1|k
, u1,i
))
k
i=1
Dk+1|k (1, o, u)
=
B
wk+1|k
·
1O′ (o)
2
· Nσk+1|k
I (u − uk+1|k )
|O′ |
(9.196)
where
k+1|k
(oo,i
k+1|k
, uo,i
k|k
)
Fk (oo,i , uo,i )
(9.197)
k|k
=
Qk + Fk Po,i FkT .
(9.198)
k+1|k
Po,i
k|k
=
Measurement update: Because of (9.188), the likelihood function has the
form
(9.199)
Lz (o, u) = Lz (o) = NRk+1 (z − Hk+1 o).
Also, assume that the (conventional) probability of detection is constant: pD (x) =
pD and that the predicted PHD has the form of (9.195) and (9.196):
νk+1|k
Dk+1|k (0, o, u)
∑
=
k+1|k
(9.200)
wi
i=1
k+1|k
·NP k+1|k ((o, u) − (oi
k+1|k
, ui
))
i
Dk+1|k (1, o, u)
=
B
wk+1|k
·
1O′ (o)
2
· Nσk+1|k
I (u − uk+1|k ). (9.201)
|O′ |
Let
νk+1|k
B
wk+1|k
τk+1 (z)
=
|O′ |
+ pD
∑
k+1|k
(9.202)
wi
i=1
k+1|k
T
·NRk+1 +Hk+1 Pk+1|k Hk+1
(z − Hk+1 oi
).
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Advances in Statistical Multisource-Multitarget Information Fusion
Then in Section K.19 it is shown that the measurement-updated PHD for persisting
targets is:
(9.203)
Dk+1|k+1 (0, o, u)
νk+1|k
= (1 − pD )
∑
k+1|k
wi
i=1
k+1|k
k+1|k
·NP k+1|k ((o, u) − (oi
, ui
i
(
∑ν
))
k+1|k
k+1|k
mk+1
+
∑
pD · w i
i=1
k+1|k
·NRk+1 +Hk+1 Pk+1|k Hk+1 (zj − Hk+1 oi
)
)
κk+1 (zj ) + τk+1 (zj )
j=1
k+1|k+1
·NP k+1|k+1 ((o, u) − (oi
k+1|k+1
, ui,j
))
i
where
k+1|k+1 −1
(Pi
)
=
k+1|k −1
(Pi
)
(9.204)
−1
T
+Hk+1
Rk+1
Hk+1
k+1|k+1 −1
(Pi
)
k+1|k+1
(oi
k+1|k+1
, ui,j
)
=
k+1|k −1
(Pi
)
(9.205)
k+1|k
k+1|k
·(oi
, ui
)
−1
T
+Hk+1
Rk+1
zj .
On the other hand, the measurement-updated PHD for appearing targets is
Dk+1|k+1 (1, o, u)
∑
=
z∈Zk+1
B
wk+1|k
|O′k+1|k |
(9.206)
2
Nσk+1|k
I (u − uk+1|k ) · NRk+1 (z − Hk+1 o)
·
9.5.7.2
.
κk+1 (z) + τk+1 (z)
CPHD Filter with a Partially Uniform Target-Birth PHD
While the underlying concept remains the same as for the PHD filter, the filtering
equations for the CPHD filter are correspondingly more complicated and will not
be further considered here. The interested reader is referred to the original papers
by Beard, Vo, Vo, and Arulampalam [16], [17].
Implementing Classical PHD/CPHD Filters
9.5.7.3
257
Implementations of PHD/CPHD Filters with Partially Uniform Target
Births
Beard et al. have implemented their PHD and CPHD filters, and compared them
with conventional GM-PHD and GM-CPHD filters whose birth PHDs consist of
varying numbers of Gaussian components [16], [17].
The authors tested the approach in a challenging application: a single bearingonly sensor carried on a platform making a sinusoidal maneuver, which observes six
appearing and disappearing targets in heavy clutter (clutter rate λ = 40).
Despite the low-observability conditions, the authors reported good tracking
performance, in comparison to conventional PHD/CPHD filters that employ a small
number of Gaussian components for the target-birth PHD. They also reported that
the conventional PHD/CPHD filters required significantly more computation time
than the new PHD/CPHD filters—largely because they tended to cause the number
of components to increase over time.
The authors further reported that a large number of birth target components—
64—are required before conventional PHD/CPHD filters perform as well as the
new filters. Still larger numbers of components (larger than 64) did not improve
performance.
9.5.8
Implementation with Target Identity
Suppose that the target state has the form x̃ = (τ, x) where x is the kinematic
state and τ is a discrete identity variable (class, type) belonging to a finite set
˜ = X×T .
T = {τ1 , ..., τN } of identity types. Thus the total target state space is X
Also assume that measurements have the form z̃ = (ϕ, z) where z is the kinematic
measurement and ϕ is a feature associated with target identity.8 Then the Gaussian
mixture approximation can be extended as follows:
νk|k
Dk|k (τ, x) =
∑
k|k
wi
k|k
k|k
· pi (τ ) · NP k|k (x − xi )
(9.207)
i
i=1
=
8
Dk+1|k (τ, x)
νk+1|k
∑ k+1|k k+1|k
k+1|k
wi
· pi
(τ ) · NP k+1|k (x − xi
)
i
i=1
(9.208)
The notation ‘ϕ’ for a feature should not be confused with the symbol ‘ϕk+1 ’ used in the
measurement-update formulas for CPHD filters. Likewise, the notation ‘τ ’ should not be confused
with the notations ‘τk+1 (z)’ and ‘τ̂k+1 (z)’ used in PHD and CPHD filters, respectively.
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k|k
k+1|k
where pi (τ ) and pi
(τ ) are probability distributions on T .9
In what follows, for the sake of conceptual clarity only the GM-PHD filter
will be considered in detail, and target spawning will be neglected. In this case the
PHD filter equations become
∑∫
Dk+1|k (τ, x) = bk+1|k (τ, x) +
pS (τ ′ , x′ )
(9.209)
τ′
·fk+1|k (τ, x|τ ′ , x′ ) · Dk|k (τ, x)dx
Dk+1|k+1 (τ, x)
Dk+1|k (τ, x)
=
(9.210)
1 − pD (τ, x)
+
∑
pD (τ, x) · L(ϕ,z) (τ, x)
κk+1 (ϕ, z) + τk+1 (ϕ, z)
(ϕ,z)∈Z
where
τk+1 (ϕ, z) =
∑∫
pD (τ, x) · L(ϕ,z) (τ, x) · Dk+1|k (τ, x)dx.
(9.211)
τ
Now turn to Gaussian-mixture implementation of this PHD filter.
9.5.8.1
GM-PHD Filter Time Update with Target ID
Set
′
pS (τ ′ , x′ )
fk+1|k (τ, x|τ ′ , x′ )
=
=
pτS
pk+1|k (τ |τ ′ ) · NQτk (x − Fkτ x′ )
(9.212)
(9.213)
(9.214)
bk+1|k (τ, x)
B
νk+1|k
=
∑
B,k+1|k
bτi · pi
k+1|k
(τ ) · NB k+1|k (x − bi
).
i
i=1
9
This approach uses a “flat” or single-layer target–identity taxonomy (also known as “ontology”)—
that is, one in which an attempt is made to directly identify target identity. In practical application it
is often preferable to employ a multilayer taxonomy—that is, one in which coarser determinations
of identity are made prior to finer ones. For example, one might first determine if a ground target is
a truck versus a tank, before attempting to determine what sort of truck or what sort of tank it is.
Implementing Classical PHD/CPHD Filters
259
Substituting (9.207) into (9.209), we get
B
νk+1|k
Dk+1|k (τ, x)
∑
=
B,k+1|k
bτi · pi
i=1
νk|k
+
∑
) (9.215)
i
∑
k|k
wi
k+1|k
(τ ) · NB k+1|k (x − bi
′
k|k
pτS · pi (τ ′ ) · pk+1|k (τ |τ ′ )
τ′
i=1
′
k|k
·NQτ ′ +F τ ′ P k|k (F τ ′ )T (x − Fkτ xi ).
k
i
k
k
For most applications we can set pk+1|k (τ |τ ′ ) = δτ,τ ′ since, typically (but not
invariably—see Remark 33 below), targets do not change identity. In this case we
get
B
νk+1|k
Dk+1|k (τ, x)
∑
=
B,k+1|k
bτi · pi
k+1|k
(τ ) · NB k+1|k (x − bi
) (9.216)
i
i=1
νk|k
+pS
∑
k|k
wi
k|k
· pi (τ )
i=1
k|k
·NQτ +F τ P k|k (F τ )T (x − Fkτ xi ).
k
k
i
k
Remark 33 (Dynamically changing target identity) It is not always the case that
fk+1|k (τ |τ ′ ) = δτ,τ ′ . Possibly the most extreme example is a diesel-electric
submarine. It has very different passive-acoustic phenomenologies, depending on
whether it is snorkeling (and thus using its diesel engines) or submerged (and thus
using its electric engines). Other examples include variable swept-wing aircraft
(extended-wing versus delta-wing modes) and mobile missile launchers (launch
mode versus transit mode). In such cases, a classification algorithm that presumes
fk+1|k (τ |τ ′ ) = δτ,τ ′ will typically exhibit degraded performance.
9.5.8.2
GM-PHD Filter Measurement Update with Target ID
Set
pD (τ, x)
=
pτD
Lϕ,z (τ, x)
=
Lϕ (τ ) · N
κk+1 (ϕ, z)
=
κk+1 (z)
(9.217)
τ
Rk+1
τ
(z − Hk+1
x)
(9.218)
(9.219)
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Advances in Statistical Multisource-Multitarget Information Fusion
where Lϕ (τ ) is the likelihood that a target of type τ will generate an identityfeature ϕ. Substituting (9.209) into (9.211), we get
νk+1|k
τk+1 (ϕ, z)
∑
=
k+1|k
wi
∑
k+1|k
pτD · Lϕ (τ ) · pi
(9.220)
(τ )
τ
i=1
k+1|k
·NRτ
k+1|k
τ
τ
(Hk+1
)T
k+1 +Hk+1 Pi
τ
(z − Hk+1
xi
).
Substituting (9.209) into (9.210) we get:
νk+1|k
Dk+1|k+1 (τ, x)
=
(1 − pτD )
∑
k+1|k
wi
k+1|k
· pi
(τ )
(9.221)
i=1
k+1|k
·NP k+1|k (x − xi
)
(
i
νk+1|k
+
∑
mk+1
k+1|k
wi
i=1
∑
k+1|k
pi
(τ ) · pτD · Lϕj (τ )
k+1|k+1
·NP k+1|k+1 (x − xi,j
)
)
i,j
κk+1 (zj ) + τk+1 (ϕj , zj )
j=1
where
k+1|k+1
xi,j
k+1|k
=
k+1|k+1
Pi,j
=
Kik+1
9.5.8.3
=
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
)
(9.222)
(
) k+1|k
k+1
I − Ki Hk+1 Pi
(9.223)
(
)−1
k+1|k T
k+1|k T
Pi
Hk+1 Hk+1 Pi
Hk+1 + Rk+1
. (9.224)
GM-CPHD Filter Measurement Update with Target ID
The case for the CPHD filter is more complicated but straightforward, and so will
not be considered in detail here. One innovation should be pointed out, however:
• It is possible to construct the cardinality distribution pτk|k (n) for the number
n of targets of type τ , for any τ .
Let
τ
Nk|k
=
τ
rk|k
=
∫
Dk|k (τ, x)dx
τ
Nk|k
∑
τ′
τ ′ Nk|k
(9.225)
(9.226)
Implementing Classical PHD/CPHD Filters
261
be, respectively, the expected number of, and the fraction of targets of, type τ . Then
the probability that there are n targets of type τ is
pτk|k (n) =
τ
(rk|k
)n
n!
(n)
τ
· Gk|k (1 − rk|k
).
(9.227)
Here, Gk|k (x) is the p.g.f. corresponding to the cardinality distribution pk|k (n)
on the total number n of targets (that is, regardless of target type).
˜ = X × T can be
Equation (9.227) is proved by noting that the joint space X
rewritten as
τN
τ1
˜ = X ⊎ ... ⊎ X
X
(9.228)
τi
where X = X × {τi } is the space of targets of type τ and where ‘⊎’ denotes
disjoint union (topological sum). The discussion to be presented in Section 11.6.4
then applies, with target identity τ playing the same role as the mode variable o .
Given this, (9.227) immediately follows from (11.129).
9.6
SEQUENTIAL MONTE CARLO (SMC) IMPLEMENTATION
SMC approximation, better known as particle approximation or particle-system
approximation, has become a standard tool for implementing RFS filters (in [179],
see Section 2.5.3, Chapter 15, Section 16.5.3, and Section 16.9.2). Standard
references and tutorials describing SMC methods include [11], [29], [61], [252].
For a more detailed discussion of particle implementation of PHD and CPHD filters,
see the book Particle Filters for Random Set Models by Ristic [250].
The first algorithmic implementations of the PHD filter were based on SMC
techniques, independently proposed in 2003 by Sidenbladh [271]; by Zajic and
Mahler [330]; and by Vo, Singh, and Doucet [306]. Subsequently, the convergence
properties of SMC implementations of the PHD filter were established, by Clark
and Bell [44], Johansen, S. Singh, Doucet, and Vo [126]; and (for an auxiliary SMC
implementation) by Whiteley, Singh, and Godsill [320], [321].
The purpose of this section is to summarize the major aspects of SMC implementation of PHD and CPHD filters, including some recent conceptual advances.
For conceptual clarity, the emphasis will be on the simplest, or “bootstrap,” implementation approach, in which the Markov density (“dynamic prior”) fk+1|k (x|x′ )
is used as the importance-sampling density.
The section is organized as follows:
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1. Section 9.6.1: Sequential Monte Carlo (SMC) approximation of PHD and
CPHD filters.
2. Section 9.6.2: The “bootstrap” SMC-PHD filter.
3. Section 9.6.3: The “bootstrap” SMC-CPHD filter.
4. Section 9.6.4: Using measurements to select new particles: the approach of
Ristic, Clark, and Vo.
5. Section 9.6.5: Particle implementation with target identity.
9.6.1
SMC Approximation
The particle approach for implementing PHD and CPHD filters is a direct generalization of the approach employed for the single-target particle filter. Intuitively
speaking, it is based on approximation of the PHD as a Dirac sum:
νk|k
Dk|k (x) ∼
=
∑
k|k
wi
(9.229)
· δxk|k (x)
i
i=1
k|k
k|k
k|k
k|k
where x1 , ..., xνk|k are the particles and w1 , ..., wνk|k are their respective
k|k
k|k
k|k
k|k
weights. More rigorously speaking, x1 , ..., xνk|k and w1 , ..., wνk|k form a
particle approximation of Dk|k (x) if
∫
νk|k
θ(x) · Dk|k (x|Z k )dx ∼
=
∑
k|k
wi
k|k
· θ(xi )
(9.230)
i=1
for any unitless function θ(x) of x. Furthermore, it must be the case that, for
particle approximations of arbitrarily large size,
∫
νk|k
θ(x) · Dk|k (x|Z k )dx =
lim
νk|k →∞
∑
k|k
wi
k|k
· θ(xi )
(9.231)
i=1
for any unitless function θ(x) of x.
The major difference between PHD particle approximation and single-target
particle approximation is that the sum of weights,
ν
∑
i=1
k|k ∼
wi
= Nk|k ,
(9.232)
Implementing Classical PHD/CPHD Filters
263
approximates the expected number of targets rather than being equal to 1.
Another difference is that, once targets have been adequately localized, a
k|k
k|k
single-target particle system x1 , ..., xνk|k consists of a single particle-cluster.
The corresponding PHD-particle system, by way of contrast, consists of several
particle-clusters—one for each detected target. This causes multitarget state estimation to be computationally challenging—see however, Section 9.6.4.
It is common practice to assume that the particles are equally weighted:
k|k
wi = 1/ν for all i = 1, ..., νk|k . In this case, a particle system is conceptually
like a statistical sample, with more particles located at large values of Dk|k (x)
and fewer particles located at smaller ones. This convention will be followed in the
sequel.
9.6.2
SMC-PHD Filter
The discussion in this section is a variant of that in [179], pp. 615-623.
9.6.2.1
SMC-PHD Filter: Time Update
For conceptual clarity, target-spawning will be neglected. Assume that the PHD
Dk|k (x) at time tk has been approximated by the particle system
k|k
k|k
{(w1 , x1 ), ..., (wνk|k
, xνk|k
)}
k|k
k|k
with
νk|k
Nk|k =
∑
k|k
(9.233)
wi
i=1
being the expected number of targets at time tk . Then we are to determine the timek+1|k
k+1|k
k+1|k
k+1|k
updated particle system {(w1
, x1
), ..., (wνk+1|k , xνk+1|k )}. Substituting
νk|k
Dk|k (x) =
∑
k|k
wi
(9.234)
· δxk|k (x)
i
i=1
into the PHD filter time-update equation, (8.15) and (8.16), yields
Dk+1|k (x)
=
(9.235)
bk+1|k (x)
νk|k
+
∑
i=1
k|k
wi
k|k
k|k
· pS (xi ) · fk+1|k (x|xi ).
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The first and second terms correspond, respectively, to appearing-target particles
and persisting-target particles. Consider each of these in turn.
• Appearing-target particles: Let
∫
B
Nk+1|k
=
(9.236)
bk+1|k (x)dx
B
B
and let νk+1|k
be the nearest integer to Nk+1|k
. Define the probability
density
ˆbk+1|k (x) = bk+1|k (x)
(9.237)
B
Nk+1|k
B
and draw νk+1|k
samples from ˆbk+1|k (x),
k+1|k
k+1|k
x1
, ..., xν B
∼ ˆbk+1|k (·),
(9.238)
k+1|k
with particles concentrated around locations where new targets are expected
k+1|k
k+1|k
to appear. Then the particles x1
, ..., xν B
represent the appearing
k+1|k
targets, with corresponding weights
k+1|k
wi
B
= Nk+1|k
·∑
ˆbk+1|k (xk+1|k )
i
.
ˆbk+1|k (xk+1|k )
l=1
(9.239)
l
Remark 34 A naı̈ve approach to birth-particle selection is to place target
-appearance particles in regions not currently containing targets. This approach
typically requires a prohibitively large number of particles. A more sophisticated
approach is to use measurements to guide the placement of new particles. However,
Ristic, Clark, and Vo have shown that naı̈ve implementations of this approach lead
to biased estimates of target number when the number of targets is small. They have
proposed an alternative method [256], which will be described in Section 9.6.4.
• Persisting-target particles: From (9.235) we see that the expected number of
surviving (persisting) targets is, approximately,
νk|k
S
Nk+1|k
=
∑
i=1
k|k
wi
k|k
· pS (xi ).
(9.240)
Implementing Classical PHD/CPHD Filters
265
Define the discrete probability distribution p̃S (i) on i ∈ {1, ..., νk|k } by
k|k
p̃S (i) =
k|k
· pS (xi )
.
S
Nk+1|k
wi
(9.241)
S
S
S
Let νk+1|k
be the nearest integer to Nk+1|k
and draw νk+1|k
samples
from pS (i):
i1 , ..., iν S
∼ pS (·).
(9.242)
k+1|k
k|k
k|k
Then the particles xi1 , ..., xν S
are chosen to represent targets that persist
k+1|k
into time tk+1 . For each of these persisting particles, draw a single sample
from the dynamic prior (the “bootstrap” approach):
k+1|k
k|k
k+1|k
,....,
∼ fk+1|k (·|x1 )
x1
k|k
∼ fk+1|k (·|xν S
xν S
k+1|k
k+1|k
Then the predicted particles x1
targets.
9.6.2.2
). (9.243)
k+1|k
k+1|k
, ..., xν S
represent the persisting
k+1|k
SMC-PHD Filter: Measurement Update
Assume that the predicted PHD Dk+1|k (x) has been approximated by the particle
k+1|k
system {(w1
k+1|k
, x1
k+1|k
k+1|k
), ..., (wνk+1|k , xνk+1|k )} with
νk+1|k
Nk+1|k =
∑
k+1|k
wi
.
(9.244)
i=1
Then we are to determine the time-updated particle system
k+1|k+1
{(w1
k+1|k+1
, x1
), ..., (wνk+1|k+1
, xk+1|k+1
νk+1|k+1 )}.
k+1|k
Substituting
νk+1|k
Dk+1|k (x) =
∑
i=1
k+1|k
wi
· δxk+1|k (x)
i
(9.245)
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Advances in Statistical Multisource-Multitarget Information Fusion
into the PHD filter measurement-update equation, (8.50) and (8.51), yields
νk+1|k
Dk+1|k+1 (x)
∑
=
k+1|k
k+1|k
· (1 − pD (xi
wi
)) · δxk+1|k (x)
(9.246)
i
i=1
mk+1 νk+1|k
+
∑ ∑ wik+1|k · pD (xik+1|k ) · Lzj (xik+1|k )
κk+1 (zj ) + τk+1 (zj )
j=1 i=1
·δxk+1|k (x)
i
where
νk+1|k
τk+1 (zj ) =
∑
k+1|k
k+1|k
· pD (xi
wi
k+1|k
) · Lzj (xi
).
(9.247)
i=1
The first and second terms correspond to the particles representing the undetected
targets and the detected targets. Consider each in turn:
• Undetected-target particles—for i = 1, ..., νk+1|k :
k+1|k+1
xi
k+1|k+1
wi
k+1|k
=
xi
(9.248)
=
k|k
k|k
(1 − pD (xi )) · wi .
(9.249)
• Detected-target particles—for i = 1, ..., νk+1|k and j = 1, ..., mk+1 :
k+1|k+1
xi,j
k+1|k
xi
=
pD (xi ) · Lzj (xi )
k|k
· wi .
κk+1 (zj ) + τk+1 (zj )
k|k
k+1|k+1
wi,j
9.6.2.3
(9.250)
=
k|k
SMC-PHD Filter: Multitarget State Estimation
State estimation for single-target particle filters is relatively simple. If
k+1|k+1
{(w1
k+1|k+1
, x1
), ..., (wνk+1|k+1
, xk+1|k+1
νk+1|k+1 )}
k+1|k
(9.251)
Implementing Classical PHD/CPHD Filters
267
is the measurement-updated single-target particle system, then the mean and covariance of the target can be computed as
νk+1|k+1
x̂k+1|k+1
=
∑
k+1|k+1
wi
k+1|k+1
(9.252)
· xi
i=1
νk+1|k+1
Pˆk+1|k+1
=
∑
k+1|k+1
wi
k+1|k+1
· (xi
− x̂k+1|k+1 )
(9.253)
i=1
k+1|k+1
·(xi
− x̂k+1|k+1 )T .
State estimation with SMC-PHD and SMC-CPHD filters, however, is typically complicated and computationally expensive. This is because a clustering
algorithm (the EM algorithm, k-means, and so on) must be used to partition the
multitarget particle system into separate single-target particle systems, with each
partition corresponding to a hypothesized target track. Once this has been accomplished, (9.252) and (9.253) can be used to determine the means and covariances of
the tracks.
Fortunately, Ristic, Clark, and Vo have devised an alternative formulation of
the SMC-PHD filter that does not require clustering [256]. Their approach will be
described in Section 9.6.4.
9.6.3
SMC-CPHD Filter
This section is a condensation of [179], pp. 644-649. The primary point of interest
is that the PHD, not the spatial distribution sk|k (x), is approximated using particle
methods.
9.6.3.1
SMC-CPHD Filter: Time Update
The time-update formulas for the CPHD filter were given in (8.86) through (8.90).
The particle implementation of (8.86), which is to say of
Dk+1|k (x) = bk+1|k (x) +
∫
pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ ,
(9.254)
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Advances in Statistical Multisource-Multitarget Information Fusion
is identical to the time-update for the SMC-PHD filter, (9.235), except that there is
no term for spawned targets:
νk|k
Dk+1|k (x) = bk+1|k (x) +
∑
k|k
k|k
k|k
· pS (xi ) · fk+1|k (x|xi ).
wi
(9.255)
i=1
The update of the cardinality distribution and p.g.f. remain unchanged from (8.86)
through (8.90), except that
ψk =
9.6.3.2
νk|k
1 ∑ k|k
k|k
w · pS (xi ).
Nk|k i=1 i
(9.256)
SMC-CPHD Filter: Measurement Update
The measurement-update equations for the CPHD filter were given in (8.121)
through (8.108). The equations for the cardinality distribution and p.g.f., (8.118)
through (8.121), are unchanged, except that (8.113) becomes
νk+1|k
1
ϕk =
Nk+1|k
∑
k+1|k
k+1|k
· (1 − pD (xi
wi
)).
(9.257)
i=1
The measurement-update equation for the PHD, (8.121), becomes
k+1|k
Dk+1|k+1 (x)
=
wi
k+1|k
· (1 − pD (xi
))
Nk+1|k
(9.258)
ND
· L Zk+1 · δxk+1|k (x)
i
+
k+1|k
m
k+1|k ∑
pD (xi
) · Lzj (xk+1|k )
wi
Nk+1|k j=1
ck+1 (zj )
D
·LZk+1 (zj ) · δxk+1|k (x).
i
Thus the particle representation for undetected targets is (for i = 1, ..., νk+1|k )
k+1|k+1
xi
k+1|k
=
k+1|k+1
wi
=
(9.259)
xi
k+1|k
wi
Nk+1|k
k+1|k
· (1 − pD (xi
ND
)) · L Zk+1
(9.260)
Implementing Classical PHD/CPHD Filters
269
whereas for detected targets it is (for i = 1, ..., νk+1|k and j = 1, ..., mk+1 )
k+1|k+1
xi,j
k+1|k
=
k+1|k+1
wi,j
9.6.3.3
=
(9.261)
xi
k+1|k
wi
Nk+1|k
·
k+1|k
pD (xi
) · Lzj (xk+1|k )
ck+1 (zj )
D
· LZk+1 (zj ). (9.262)
SMC-CPHD Filter: Multitarget State Estimation
State estimation is accomplished in the same manner as in Section 9.6.2.3, except
that the MAP estimate
νk+1|k+1 = arg sup pk+1|k+1 (n)
(9.263)
n
is used instead of Nk+1|k+1 =
9.6.4
∑νk+1|k+1
i=1
wk+1|k+1 .
Using Measurements to Choose New Particles
Since SMC-PHD and SMC-CPHD filters have target appearance models, it is
important to have a good methodology for choosing new particles. Since newly
appearing targets will generate unanticipated measurements, one obvious approach
is to use the measurements in Zk+1 at the next time tk+1 to choose the locations
of the new particles. This is often described as “measurement-driven” particle
placement.
However, Ristic, Clark, Vo, and Vo have shown that, if this approach is applied
naı̈vely to the PHD filter, then target-number estimates will be biased downward
(see [256], p. 3 and [257], p. 1659). The cause of this bias is not understood. It
also appears to be significant only for smaller numbers of targets, but can have a
pronounced effect when this is the case.
As a remedy, in [256], [257] Ristic et al. proposed a reformulation of the
SMC-PHD filter that:
• Avoids this bias.
• Is faster and more accurate than the conventional SMC-PHD filter.
• Avoids the necessity of using ad hoc and computationally expensive clustering algorithms during the multitarget state-estimation step.
Because of the final two points, the approach is advantageous even when
the number of targets is not small. The purpose of this section is to describe the
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technique, which can also be applied to the SMC-CPHD filter—see [257] for more
details. For the sake of conceptual simplicity, only the SMC-PHD filter version is
considered here.
The key idea is to replace the single-target state space X with a new state
space
˜ =X⊎B
X
(9.264)
where ‘⊎’ denotes disjoint union, and where B = X is a copy of X.10 Intuitively
speaking, X is the space of “persisting targets” whereas B is the space of “birth
˜ is defined as
targets.” Integration on X
∫
∫
∫
˜
˜
f (x̃)dx̃ =
f (x)dx +
f˜(b)db.
(9.265)
X
9.6.4.1
B
Unbiased SMC-PHD Filter: Models
Given this model, any motion or measurement model that involves the original
state variable x must be redefined using the new state variable x̃, where either
x̃ = x ∈ X or x̃ = b ∈ B. Thus Ristic et al. make the following definitions:
• Probability of target survival ([256], Eq. (14))—is the same for new and
persistent targets:
{
pS (x) if x̃ = x
p̃S (x̃) =
.
(9.266)
pS (b) if x̃ = b
• Birth-target PHD ([256], Eq. (11))—persisting targets cannot be new targets:
{
0
if x̃ = x
˜bk+1|k (x̃) =
.
(9.267)
bk+1|k (b) if x̃ = b
• Markov transition density ([256], Eq. (12))—a birth target can transition to a
persisting target, but not vice versa:

 fk+1|k (x|x′ ) if x̃ = x, x̃′ = x′
′
˜
f
(x|b′ ) if x̃ = x, x̃′ = b′ .
fk+1|k (x̃|x̃ ) =
(9.268)
 k+1|k
0
if
otherwise
10 In [256], Ristić et al. actually defined
˜ = X × {0, 1},
X
˜ = X × B. The alternative notation has been chosen for conceptual clarity.
which is equivalent to X
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271
• Probability of detection ([256], Eq. (17))—new targets are always detected:
{
pD (x) if x̃ = x
p̃D (x̃) =
.
(9.269)
1
if x̃ = b
• Likelihood function ([256], Eq. (18))—new targets and persisting targets have
the same measurement statistics:
{
Lz (x) if x̃ = x
˜
Lz (x̃) =
.
(9.270)
Lz (b) if x̃ = b
˜ can be equivalently expressed
˜ k|k (x̃) defined on X
It follows that a PHD D
as two PHDs
Dk|k (x)
=
˜ k|k (x)
D
(9.271)
=
˜ k|k (b)
D
(9.272)
B
D k|k (b)
defined on X and B, respectively. Consequently, the revised PHD filter will
actually consist of two coupled PHD filters, one for persisting targets and one for
birth targets:
... →
Dk|k (x)
... →
D k|k (b)
→
↑
Dk+1|k (x)
→
D k+1|k (b)
B
9.6.4.2
→
↑↓
Dk+1|k+1 (x)
→
D k+1|k+1 (b)
B
→ ...
B
→ ...
Unbiased SMC-PHD Filter: Time Update
In the following, the target-spawning model will be ignored for conceptual clarity.
The time-update equation results from substituting the models in Section 9.6.4.1
into the classical PHD filter time-update equation, (8.15), and using (9.265):
∫
˜
˜
˜ k+1|k (x̃′ )dx̃′ . (9.273)
Dk+1|k (x̃) = bk+1|k (x̃) + p̃S (x̃) · f˜k+1|k (x̃|x̃′ ) · D
That is,
• Persisting-target time-update ([256], Eq. (16)):
∫
(
)
B
Dk+1|k (x) = pS (x) · fk+1|k (x|x′ ) · Dk+1|k (x′ ) + D k+1|k (x′ ) dx′ .
(9.274)
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• Birth-target time-update ([256], Eq. (16)):
B
D k+1|k (b) = bk+1|k (b).
9.6.4.3
(9.275)
Unbiased SMC-PHD Filter: Measurement Update
The measurement-update equation results from substituting the models in Section
9.6.4.1 into the classical PHD filter measurement-update equation, (8.50) and
(8.51), and using (9.265):
˜ k+1|k+1 (x̃)
D
˜ k+1|k (x̃)
D
τk+1 (z)
=
∑
1 − p̃D (x̃) +
˜ z (x̃)
p̃D (x̃) · L
κk+1 (z) + τk+1 (z)
(9.276)
z∈Zk+1
=
∫
˜ z (x̃) · D
˜ k+1|k (x̃)dx̃.
p̃D (x̃) · L
(9.277)
That is,
• Persisting-target measurement-update ([256], Eq. (20)):
Dk+1|k+1 (x)
Dk+1|k (x)
=
1 − pD (x)
+
∑
(9.278)
pD (x) · Lz (x)
κk+1 (z) + τk+1 (z)
z∈Zk+1
τk+1 (z)
=
∫
pD (x) · Lz (x) · Dk+1|k (x)dx
∫
+ Lz (x) · bk+1|k (x)dx.
(9.279)
• Birth-target measurement-update ([256], Eq. (21)):
B
D k+1|k+1 (b) =
∑
z∈Zk+1
Lz (b) · bk+1|k (b)
.
κk+1 (z) + τk+1 (z)
(9.280)
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9.6.4.4
273
Unbiased SMC-PHD Filter: SMC Implementation
SMC implementation consists of the following steps. Suppose that the predicted
PHD is approximated as a particle system:
νk+1|k
Dk+1|k (x) ∼
=
∑
k+1|k
(9.281)
· δxk+1|k (x).
wi
i
i=1
Then:
1. Step 1: Creation of birth particles: For each zj ∈ Zk+1 , generate ρ new
k+1|k
k+1|k
particles b1,j , ..., bρ,j
in such a manner that zj can be considered
k+1|k
k+1|k
to be a random sample drawn from Lz (bi,j ) · bk+1|k (bi,j ). Their
weights are all set to the same value for i = 1, ..., ρ · mk+1 ([256], Eq. (22))
bk+1|k
k+1|k
=
(9.282)
bi
mk+1
=
ρ
∑ ∑
1
ρ · mk+1 j=1
k+1|k
(9.283)
Lzj (bl+(j−1)mk+1 ,j )
l=1
k+1|k
·bk+1|k (bl+(j−1)mk+1 ,j ).
2. Step 2: Write the approximate birth-target PHD as a particle system using
these weights and these particles:
bk+1|k (x) ∼
= bk+1|k
ρ m
k+1
∑
∑
(9.284)
δbk+1|k (x).
i,j
i=1 i=1
3. Step 3: Update the persisting-target PHD weights ([256], Eq. (23)):
k+1|k+1
wi
k+1|k
=
(1 − pD (xi
mk+1
+
∑
j=1
k+1|k
(9.285)
)) · wi
k+1|k
k+1|k
k+1|k
) · wi
pD (xi
) · Lzj (xi
L(zj )
.
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where ([256], Eq. (25)):
ρ·mk+1
L(zj )
=
κk+1 (zj ) +
∑
k+1|k
(9.286)
bi
i=1
νk+1|k
+
∑
k+1|k
k+1|k
) · Lzj (xi
pD (xi
k+1|k
) · wi
.
i=1
4. Step 4: Update the birth-target PHD weights ([256], Eq. (24)):
mk+1
k+1|k+1
bi
=
∑ bk+1|k
i
j=1
(9.287)
.
L(zj )
5. Step 5: Separately resample the birth-target particle set and persisting-target
particle set.
6. Step 6: Define the weights j = 0, 1, ..., mk+1 ([256], Eq. (26)):
k+1|k+1
wi,j
=

 (1 − pD (xk+1|k )) · w k+1|k
i
i
k+1|k

pD (xi
k+1|k
)·Lzj (xi
L(zj )
if
j=0
if
j>0
(9.288)
k+1|k
)·wi
νk+1|k
∑
k+1|k+1
Wj
=
k+1|k+1
wi,j
(9.289)
.
i=1
7. Step 7: Define the state estimates and their covariances, without resort to
clustering, as ([256], Eqs. (27,28)), for j = 1, ..., mk+1 :
νk+1|k
k+1|k+1
x̂j
=
∑
k+1|k+1
wi,j
k+1|k
(9.290)
· xi
i=1
νk+1|k
k+1|k+1
Pˆj
=
∑
k+1|k+1
wi,j
k+1|k
· (xi
k+1|k+1
− x̂j
)
(9.291)
i=1
k+1|k
k+1|k+1 T
·(xi
− x̂j
) .
k+1|k+1
8. Step 8: Eliminate those estimates in Step 7 for which Wj
a suitable threshold value.
is less than
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275
Ristic et al. compared this approach with a more conventional approach employing k-means clustering. It greatly outperformed the k-means approach,
achieving much better localization performance, while cardinality-estimation performance was essentially the same (see [256], Figure 4).
9.6.5
Implementation with Target Identity
The SMC-PHD and SMC-CPHD filters can be extended, at least in a naı̈ve fashion,
to include target identity. The single-target state will have the form (x, c) where
x is the kinematic state and c is a discrete ID parameter. Consequently, a
k|k
k|k
k|k
k|k
particle system will have the form (c1 , x1 ), ..., (cνk|k , xνk|k ) with weights
k|k
k|k
w1 , ...., wνk|k . The resulting particle-PHD filter has computational complexity
O(mnC) where m is the current number of measurements, n is the current
number of tracks, and C is the number of target types.
Chapter 10
Multisensor PHD and CPHD Filters
10.1
INTRODUCTION
The measurement-update equations for the classical PHD and CPHD filters, Sections 8.4.3 and 8.5.4, apply only to the single-sensor case. The subject of this
chapter is PHD and CPHD filters for multiple, independent sensors.
10.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• The general single-sensor PHD filter of Section 8.2 can be generalized to the
multisensor case (Section 10.3). The measurement-update for this filter is
combinatorial and thus computationally problematic in general.
• The “iterated-corrector” PHD and CPHD filters are the most commonly used
approximate multisensor PHD and CPHD filters. However, they are not
theoretically satisfactory, because they depend on the order of the sensor.
This order-dependence can result in degraded performance when the sensors’
probabilities of detection are significantly different. This is particularly
noticeable in the case of the iterated-corrector PHD filter (Section 10.5).
• Computationally tractable and theoretically satisfactory approximate multisensor PHD and CPHD filters exist: the parallel-combination approximate
multisensor (PCAM) PHD and CPHD filters. There are three such filters: the
PCAM-CPHD filter (Section 10.6.1), the PCAM-PHD filter (Section 10.6.2),
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and the simplified PCAM-PHD filter (Section 10.6.3). The simplified PCAMPHD filter is the conceptually and computationally simplest, but is not appropriate for scenarios containing a small number of targets.
• A proposed approximate multisensor PHD filter, based on the averaging of
PHD pseudolikelihoods, is conceptually and theoretically erroneous (Section
10.7).
• All of these filters have been implemented and compared in simulations.
Among the approximate filters, the PCAM-CPHD and PCAM-PHD filters
perform the best and the averaged-pseudolikelihood PHD filter performs the
worst (Section 10.8).
10.1.2
Organization of the Chapter
The chapter is organized as follows:
1. Section 10.2: The multisensor-multitarget recursive Bayes filter.
2. Section 10.3: The multisensor generalization of the general single-sensor
PHD filter of Section 8.2.
3. Section 10.4: The multisensor generalization of the single-sensor classical
PHD filter of Section 8.4.
4. Section 10.5: The iterated-corrector approximation for multisensor application of the classical PHD and CPHD filters.
5. Section 10.6: Principled but computationally tractable “parallel combination
approximate multisensor” (PCAM) approximations for multisensor application of the classical PHD and CPHD filters.
6. Section 10.7: An erroneous approximate multisensor PHD filter based on an
averaged-pseudolikelihood approach.
7. Section 10.8: Performance comparisons of the multisensor PHD and CPHD
filters described in this chapter.
Multisensor PHD and CPHD Filters
10.2
279
THE MULTISENSOR-MULTITARGET BAYES FILTER
Targets have state vectors. However, sensors also have state vectors. For example,
∗
the state x of a sensor could have the form
∗
˙ α̇, φ̇, µ, χ)
x = (x, y, z, ẋ, ẏ, ż, ℓ, θ, α, φ, θ,
(10.1)
where x, y, z and ẋ, ẏ, ż are the position and velocity coordinates of the sensor˙ α̇, φ̇ are the sensor’s bodycarrying platform, ℓ is its fuel level, θ, α, ϕ and θ,
frame coordinates and their rates, µ is the sensor mode, and χ is the current
communications transmission path employed by the sensor.
Consequently, a single-target measurement model actually has the form
∗
Zk+1 = ηk+1 (x) + Vk+1 abbr.
= ηk+1 (x, x) + Vk+1
(10.2)
∗
where x is the current state of the sensor. Similarly, the sensor likelihood function
actually has the form
∗
∗
∗
Lz (x, x) = fk+1 (z|x, x) = fVk+1 (z − ηk+1 (x, x))
(10.3)
and the multiobject probability distribution of the clutter process actually has the
form
∗
κk+1 (Z) abbr.
= κk+1 (Z|x).
(10.4)
Suppose that there are s sensors and that the jth sensor collects meaj
j
surements z drawn from the measurement space Z. At any given time, let the
j
measurement set collected by the jth sensor be denoted by Z. Then the joint
multisensor measurement space is
1
s
(10.5)
Z = Z ⊎ ... ⊎ Z
where ‘⊎’ denotes disjoint union. At any given time, the total measurement set
has the form
1
s
Z = Z ⊎ ... ⊎ Z.
(10.6)
Let
j
j
j
j
∗j
L j (X) abbr.
= f k+1 (Z|X) abbr.
= fk+1 (Z|X, x)
(10.7)
Z
∗j
be the multitarget likelihood function for the jth sensor, where x is the current
state of the jth sensor.
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According to the discussions in Section 3.5.3 and Section 4.2.5, we can write
the distribution of Z as a joint distribution:
1
s
1
s
fk+1 (Z|X) = fk+1 (Z ⊎ ... ⊎ Z|X) = fk+1 (Z, .., Z|X).
(10.8)
The sensors are conditionally independent of the multitarget state if
1
1
s
s
1
s
(10.9)
fk+1 (Z, .., Z|X) = f k+1 (Z|X) · · · f k+1 (Z|X).
j
j
j
Given this, let Z (k) : Z 1 , ..., Z k be the time sequence of measurements
collected by the jth sensor at time tk . Then the Bayes-optimal multisensor
measurement-update is accomplished using Bayes’ rule:
1
s
fk+1|k+1 (X|Z (k+1) , ..., Z (k+1) )
1
=
(10.10)
s
1
s
fk+1 (Z k+1 , .., Z k+1 |X) · fk+1|k (X|Z (k) , ..., Z (k) )
1
s
1
s
fk+1 (Z k+1 , ..., Z k+1 |Z (k) , ..., Z (k) )
( 1
)
s
1
s
f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X)
1
s
·fk+1|k (X|Z (k) , ..., Z (k) )
=
1
s
1
(10.11)
s
fk+1 (Z k+1 , ..., Z k+1 |Z (k) , ..., Z (k) )
where
1
s
1
fk+1 (Z, ..., Z|Z
∫
s
(k)
, ..., Z
(k)
)
=
1
s
1
s
f k+1 (Z|X) · · · f k+1 (Z|X) (10.12)
1
s
·fk+1|k (X|Z (k) , ..., Z (k) )δX.
In abbreviated form,
1
fk+1|k+1 (X) =
s
fk+1 (Z k+1 , .., Z k+1 |X) · fk+1|k (X)
1
s
.
(10.13)
fk+1 (Z k+1 , ..., Z k+1 )
As with the single-sensor, multitarget Bayes filter, the multisensor-multitarget
Bayes filter must be approximated. This is the purpose of the filters described in
the remainder of the chapter.
Multisensor PHD and CPHD Filters
10.3
281
THE GENERAL MULTISENSOR PHD FILTER
The general PHD filter described in Section 8.2 can be extended to a general
multisensor PHD filter, as was shown in [47], Section V. This filter is the subject of
this section.
10.3.1
General Multisensor PHD Filter: Modeling
For the sake of conceptual clarity, the two-sensor case will be considered. The general multisensor case follows by extrapolation (see Remark 35 in Section 10.3.2).
The measurement-update equations for the two-sensor PHD filter requires the
following models, which correspond to the single-sensor models in Section 8.2:
• A “undotted sensor” (first sensor) and a “dotted sensor” (second sensor),
where the nomenclature refers to the fact that the models for the latter will
be dotted (as in ‘ṗD ’) whereas those for the former will not be dotted (as in
‘pD ’).
• Joint measurement set:
Z˜k+1 = Zk+1 ⊎ Z˙ k+1
(10.14)
where Zk+1 is the measurement set collected by the undotted sensor, and
Z˙ k+1 is the measurement set collected by the dotted sensor.
• Single-target likelihood functions. These are:
LZ (x) =
δGx
[0],
δZ
δ G˙ x
L˙ Z˙ (x) =
[0]
δ Z˙
(10.15)
where Gx [g] is the p.g.fl. of the undotted target-measurement RFS Υk+1 (x);
and where G˙ x [ġ] is the p.g.fl. of the dotted target-measurement RFS
˙ k+1 (x).
Υ
• Generalized probabilities of detection. These are the probabilities that the
sensors will, respectively, collect at least one measurement:
πD (x) = 1 − L∅ (x),
π̇D (x) = 1 − L˙ ∅ (x).
˙ ⊆ Z˙ k+1 ,
• Joint likelihood integral. For any W ⊆ Zk+1 and W
∫
τW,W˙ = LW (x) · L˙ W˙ (x) · Dk+1|k (x)dx.
(10.16)
(10.17)
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• Joint generalized probability of detection. The probability that at least one of
the sensors will collect at least one measurement:
π̃D (x) = 1 − (1 − πD (x)) · (1 − π̇D (x)) .
(10.18)
• Non-ubiquity of clutter. We must assume the following: for both sensors, it
is possible that no clutter measurements will be collected at all:
pκk+1 (0) > 0,
ṗκk+1 (0) > 0
(10.19)
where pκk+1 (m) is the cardinality distribution of the undotted clutter RFS
Ck+1 and ṗκk+1 (m) is the cardinality distribution of the dotted clutter RFS
C˙ k+1 .
• Clutter log-distributions. These are the multiobject density functions of the
respective clutter log-p.g.fl.’s:
κZ =
δ log Gκk+1
[0],
δZ
κ̇Z˙ =
δ log G˙ κk+1
[0]
δ Z˙
(10.20)
where Gκk+1 [g] denotes the p.g.fl. of the undotted-sensor clutter process, and
G˙ κk+1 [ġ] denotes the p.g.fl. of the dotted-sensor clutter process.
10.3.2
General Multisensor PHD Filter: Update
This is
˜
Dk+1|k+1 (x) = L
Zk+1 ,Z˙ k+1 (x) · Dk+1|k (x)
(10.21)
where
˜
L
Zk+1 ,Z˙ k+1 (x)
=
1 − π̃D (x) +
∑
ωP
(10.22)
˜k+1
P⊟Z
·
∑
˙ ∈P
W ⊎W
LW (x) · L˙ W˙ (x)
.
δ|W˙ |,0 · κW + δ|W |,0 · κ̇W˙ + τW,W˙
Here, the first summation is taken over all partitions P of the joint measurement
set Z˜k+1 = Zk+1 ⊎ Z˙ k+1 ; and
(
)
∏
δ
·
κ
+
δ
·
κ̇
+
τ
˙
˙
˙
˙
W
|W
|,0
W ⊎W ∈P
|W |,0
W
W,W
(
).
(10.23)
ωP = ∑
∏
δ
·
κ
+
δ
·
κ̇
+
τ
˜
˙
˙
˙
˙
V
|V |,0
Q⊟Zk+1
V ⊎V ∈Q
|V |,0
V
V,V
Multisensor PHD and CPHD Filters
283
Remark 35 (General PHD filter, more than two sensors) For three sensors (“undotted,” “dotted,” and “double-dotted”), the analog of (10.22) is
˜
L
¨k+1 (x)
Zk+1 ,Z˙ k+1 ,Z
=
∑
1 − π̃D (x) +
ωP
(10.24)
˜k+1
P⊟Z
∑
¨ ¨ (x)
LW (x) · L˙ W˙ (x) · L
W


δ
·
δ
·
κ
¨ |,0
˙ |,0
W
|W
|W
˙ ⊎W
¨ ∈P
W ⊎W
 +δ|W
¨ |,0 · δ|W |,0 · κ̇W
˙ 


 +δ ˙ · δ|W |,0 · κ̈ ¨ 
|W |,0
W
+τW,W˙ ,W
¨
where
π̃D (x)
=
τW,W˙ ,W
¨
=
1 − (1 − pD (x)) · (1 − ṗD (x)) · (1 − p̈D (x))
∫
¨ ¨ (x) · Dk+1|k (x)dx
LW (x) · L˙ W˙ (x) · L
W
(10.25)
(10.26)

ωP
=

δ |W
¨ |,0 · δ|W
˙ |,0 · κW
 +δ|W
∏
¨ |,0 · δ|W |,0 · κ̇W
˙ 


˙ ⊎W
¨ ∈P 
W ⊎W
+δ|W˙ |,0 · δ|W |,0 · κ̈W
¨ 
+τW,W˙ ,W
¨

 .(10.27)
δ|V¨ |,0 · δ|V˙ |,0 · κV
 +δ|V¨ |,0 · δ|V |,0 · κ̇V˙ 
∑
∏


˜k+1
Q⊟Z
V ⊎V˙ ⊎V¨ ∈Q  +δ

|V˙ |,0 · δ|V |,0 · κ̈V¨
+τV,V˙ ,V¨
For more than three sensors, the pattern is clear.
10.4
THE MULTISENSOR CLASSICAL PHD FILTER
Assume that the undotted and dotted sensors have standard multitarget measurement
models: clutter is Poisson and targets generate at most a single measurement. Then
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from (10.21) through (10.23) we know that

if
Z=∅
 1 − pD (x)
pD (x) · Lz (x) if Z = {z}
LZ (x) =

if |Z| > 1
0

Z=∅
 −λk+1 if
κk+1 (z) if Z = {z}
κZ =

0
if |Z| > 1

if
Z˙ = ∅
 1 − ṗD (x)
˙
˙
˙
LZ˙ (x) =
ṗ (x) · Lż (x) if Z = {ż}
 D
˙ >1
0
if |Z|

Z˙ = ∅
 −λ˙ k+1 if
κ̇Z˙ =
κ̇
(ż) if Z˙ = {ż} .
 k+1
˙ >1
0
if |Z|
(10.28)
(10.29)
(10.30)
(10.31)
Then we get the following result (originally established in [166]):
˜
Dk+1|k+1 (x) = L
Zk+1 ,Z˙ k+1 (x) · Dk+1|k (x)
(10.32)
where
˜
L
Zk+1 ,Z˙ k+1 (x) = 1 − π̃D (x) +
∑
ωP
˜k+1
P⊟2 Z
∑
ρW ⊎W˙ .
(10.33)
˙ ∈P
W ⊎W
Here, the summation is taken over all partitions P of Z˜k+1 = Zk+1 ⊎ Z˙ k+1
that are “binary” in the following sense. The partition P is binary if every cell
˙ ∈ P is binary, in that it has one of the following three forms:
W ⊎W
˙ = {z},
W ⊎W
˙ = {ż},
W ⊎W
˙ = {z, ż}.
W ⊎W
Also,
ρW ⊎W˙
ωP
=







=
∑
pD (x)·ℓz (x)·(1−ṗD (x))
1+τz,∅
(1−pD (x))·ṗD (x)·ℓ˙ż (x)
1+τ∅,ż
pD (x)·ℓz (x)·ṗD (x)·ℓ˙ż (x)
τz,ż
∏
˙ ∈P dW,W
˙
W ⊎W
˜k+1
Q⊟Z
∏
V ⊎V˙ ∈Q dV,V˙
if
˙ =∅
W = {z}, W
˙ = {ż}
W = ∅, W
if
˙ = {ż}
W = {z}, W
if
(10.34)
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285
where
ρW ⊎W˙
dW ⊎W˙




=



pD (x)·ℓz (x)·(1−ṗD (x))
1+τz,∅
(1−pD (x))·ṗD (x)·ℓ˙ż (x)
1+τ∅,ż
pD (x)·ℓz (x)·ṗD (x)·ℓ˙ż (x)
τz,ż

 1 + τz,∅
1 + τ∅,ż

τz,ż
=
if
if
if
if
˙ =∅
W = {z}, W
˙ = {ż}
W = ∅, W
if
˙ = {ż}
W = {z}, W
if
˙ =∅
W = {z}, W
˙ = {ż}
W = ∅, W
˙ = {ż}
W = {z}, W
and where
ℓz (x)
=
τz,∅
=
τ∅,ż
=
τz,ż
=
Lz (x)
Lż (x)
,
ℓ˙ż (x) =
κk+1 (z)
κk+1 (ż)
∫
pD (x) · ℓz (x) · (1 − ṗD (x)) · Dk+1|k (x)dx
∫
(1 − pD (x)) · ṗD (x) · ℓ˙ż (x) · Dk+1|k (x)dx
∫
pD (x) · ℓz (x) · ṗD (x) · ℓ˙ż (x) · Dk+1|k (x)dx.
(10.35)
(10.36)
(10.37)
(10.38)
The fact that (10.21) through (10.23) reduce to these formulas is established
in Section K.20.
1
1
2
2
Example 6 (Binary partitions) Suppose that Z˜k+1 = {z1 , z2 } ∪ {z1 , z2 }. Then
˜
the binary partitions of Zk+1 are
1
P1
=
1
P2
=
=
=
=
=
=
2
2
2
2
1
2
(10.41)
2
1
1
(10.40)
2
1
1
(10.39)
2
(10.42)
2
(10.43)
2
{{z1 , z1 }, {z2 , z2 }}
1
P7
1
{{z2 , z1 }, {z1 }, {z2 }}
1
P6
2
{{z1 , z2 }, {z2 }, {z1 }}
1
P5
2
{{z2 , z2 }, {z1 }, {z1 }}
1
P4
2
{{z1 , z1 }, {z2 }, {z2 }}
1
P3
1
{{z1 }, {z2 }, {z1 }, {z2 }}
(10.44)
2
{{z1 , z2 }, {z2 , z1 }}.
(10.45)
Remark 36 (Sensor-consistency gating) One possible method for reducing computational load in the multisensor classical PHD filter is “sensor-consistency gat1 2
ing” [166]. That is, discard partitions that contain any measurement-pair {z, z}
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such that
1
1
2
2
pD Lz1 pD Lz2 ∼
= 0.
(10.46)
Such pairs are less likely to have been jointly generated by the same target. This
approach is, of course, based on the presumption that the clutter rates of both
sensors are not too large, as compared to the corresponding sensor resolutions.
Remark 37 (Classical PHD filter, more than two sensors) The three-sensor and
multisensor cases follow from (10.24) and (10.28). For according to (10.24),
∑
LZk+1 ,Z˙ k+1 ,Z¨k+1 (x) = 1 − π̃D (x) +
ωP
(10.47)
˜k+1
P⊟Z
∑
¨ ¨ (x)
LW (x) · L˙ W˙ (x) · L
W


δ
·
δ
·
κ
¨
˙
W
|
W
|,0
|
W
|,0
˙ ⊎W
¨ ∈P
W ⊎W
 +δ|W
¨ |,0 · δ|W |,0 · κ̇W
˙ 


 +δ ˙ · δ|W |,0 · κ̈ ¨ 
|W |,0
W
+τW,W˙ ,W
¨
¨ ¨ (x) vanishes unless
where, according to (10.28), the product LW (x) · L˙ W˙ (x) · L
W
˙
¨
˙
¨
the subsets W, W , W of each cell W ⊎ W ⊎ W of the partition P have no more
than a single element. Thus the only terms in the second summation that survive are
those corresponding to cells that are “ternary” in the following sense: they must
have one of the following seven forms:
˙ ⊎W
¨
W ⊎W
˙ ⊎W
¨
W ⊎W
˙ ⊎W
¨
W ⊎W
˙
¨
W ⊎W ⊎W
=
=
{z}
{ż}
(10.48)
(10.49)
=
=
{z̈}
{z, ż}
(10.50)
(10.51)
˙ ⊎W
¨
W ⊎W
˙ ⊎W
¨
W ⊎W
˙ ⊎W
¨
W ⊎W
=
=
=
{z, z̈}
{ż, z̈}
{z, ż, z̈}.
(10.52)
(10.53)
(10.54)
Consequently, the only terms in the first summation that survive are those corresponding to partitions P of Z˜k+1 = Zk+1 ⊎ Z˙ k+1 ⊎ Z¨k+1 that are “ternary” in
the following sense: every cell in P is ternary. For more than three sensors, the
pattern is clear. For s sensors, the first summation will be taken over all partitions
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287
P of the multisensor measurement set Z˜k+1 that are “s-ary” in the sense that
every cell of P is “s-ary.” And a cell is “s-ary” if it contains no more than a
single measurement from each sensor.
10.4.1
Implementations of the Exact Classical Multisensor PHD Filter
The two-sensor classical PHD filter has been implemented and compared to other
approaches in [221]. See Section 10.8 for more details.
Moratuwage, Vo, and Danwei Wang have successfully implemented this filter
in a Simultaneous Localization and Mapping (SLAM) robotics application [205].
10.5
ITERATED-CORRECTOR MULTISENSOR PHD/CPHD FILTERS
Since the exact multisensor PHD filter (and thus also the multisensor CPHD
filter) is computationally problematic, approximation techniques are necessary. The
simplest and oldest of these is the obvious heuristic approach: the iterated corrector
multisensor PHD/CPHD filters.
For the sake of conceptual clarity, assume that we have only two sensors, with
the respective sensor models
1
1
1
1
1
pD (x),
Z,
(10.55)
Lz1 (x) = f k+1 (z|x)
1
1
1
1
1
1
κk+1 (z) = λk+1 ck+1 (z),
pκk+1 (m)
2
2
(10.56)
and
2
2
2
Z,
pD (x),
(10.57)
Lz2 (x) = f k+1 (z|x)
2
2
2
2
pκk+1 (m).
κk+1 (z) = λk+1 ck+1 (z),
(10.58)
Then as the name implies, in the iterated-corrector approach one simply applies the
CPHD/PHD filter corrector step in succession, once for each of the two sensors.
Thus in the case of the PHD filter, one first applies the corrector step for the
first sensor:
first
? ?? ?
1
1
2
1
Dk+1|k+1 (x| Z (k+1) , Z (k) )
1
1
= 1 − pD (x) +
2
Dk+1|k (x|Z (k) , Z (k) )
∑
1
1
z∈Z k+1
pD (x) · Lz1 (x)
1
1
1
1
κk+1 (z) + τ k+1 (z)
(10.59)
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where
1
1
τ k+1 (z) =
∫
1
1
2
1
pD (x) · Lz1 (x) · Dk+1|k (x|Z (k) , Z (k) )dx.
(10.60)
Then one applies the corrector step for the second sensor:
first
second
? ?? ? ? ?? ?
1
2
2
2
Dk+1|k+1 (x| Z (k+1) , Z (k+1) )
2
1
= 1 − pD (x) +
2
Dk+1|k+1 (x|Z (k+1) , Z (k) )
∑
2
2
pD (x) · Lz2 (x)
2
2
2
2
κk+1 (z) + τ k+1 (z)
z∈Z k+1
(10.61)
where
first
2
2
τ k+1 (z) =
∫
? ?? ?
2
1
2
2
pD (x) · Lz2 (x) · Dk+1|k+1 (x| Z (k+1) , Z (k) )dx
(10.62)
and where the notations
first
second
? ?? ?
? ?? ?
1
Z (k+1) ,
2
, Z (k+1)
indicate that the first sensor is applied first and the second sensor is applied second.
10.5.1
Limitations of the Iterated-Corrector Approach
The iterated-corrector approach is conceptually simple but not entirely satisfactory,
from either a theoretical or a practical point of view. This is because:
• The value of the posterior PHD depends on the order in which the sensors are
applied:
first
second
? ?? ? ? ?? ?
1
second
2
Dk+1|k+1 (x| Z (k+1) , Z (k+1) )
1
̸=
first
? ?? ? ? ?? ?
2
Dk+1|k+1 (x| Z (k+1) , Z (k+1) ).
(10.63)
• As a consequence, tracking performance can be less or more effective,
depending on the sensor order.
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289
Specifically, Nagappa and Clark [219] have demonstrated that a significant
performance deterioration results when the probabilities of detection of the two
2
1
sensors are markedly different, say pD ≪ pD . In this case, the first sensor—that
is, the one with the larger pD —should be applied before the second sensor—that
is, the one with the smaller pD .
On the other hand, if the probabilities of detection are approximately equal
then tracking performance is not significantly affected by sensor order.
The ultimate source of this behavior lies in the theoretical assumptions that
are implicitly being made. In the case of the PHD filter, we begin by assuming
1
2
that the predicted distribution fk+1|k (X|Z (k) , Z (k) ) is Poisson. If the first sensor
1
2
is applied first, this results in an updated distribution fk+1|k+1 (X|Z (k+1) , Z (k) ).
When we apply the corrector step for the second sensor, we are additionally and
implicitly assuming that this distribution is Poisson. On the other hand, if the
1
2
second sensor is applied first then the distribution fk+1|k+1 (X|Z (k) , Z (k+1) ) is
implicitly assumed to be Poisson. Because the two pairs of Poisson approximations
are different, they lead to a dependence on sensor order.
Similar comments apply to the iterated-corrector CPHD filter.
10.6
PARALLEL COMBINATION MULTISENSOR PHD AND CPHD FILTERS
The exact multisensor CPHD or PHD filters are computationally problematic. The
iterated-corrector approximation is computationally tractable, but leads to potential
performance problems. So what is to be done? The following describes a theoretically principled, order-independent, and computationally tractable approximation
for multisensor PHD and CPHD filters, first proposed in 2010 [151].
The fundamental idea is as follows. Suppose that there are s sensors with
respective multitarget likelihood functions
j
j
j
(10.64)
L j (X) = f k+1 (Z|X)
Z
j
where Z denotes a measurement set collected by the jth sensor. Let
1
s
fk+1|k (X) abbr.
= fk+1|k (X|Z (k) , ..., Z (k) )
(10.65)
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j
j
j
be the prior distribution, where Z (k) : Z 1 , ..., Z k denotes the time sequence of
1
s
measurement sets collected by the jth sensor. Let Z k+1 , ..., Z k+1 be the new
measurement sets collected by the sensors. If the sensors are independent, then the
posterior distribution conditioned on the new measurements is
1
s
fk+1|k+1 (X) abbr.
= fk+1|k+1 (X|Z (k) , ..., Z (k) )
1
s
1
f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) · fk+1|k (X)
=
1..s
(10.66)
s
1
(10.67)
s
f k+1 (Z k+1 , ..., Z k+1 )
where
1..s
1
s
f k+1 (Z k+1 , ..., Z k+1 )
1
abbr.
=
=
s
1
s
f
(Z
, ..., Z k+1 |Z (k) , ..., Z (k) )
∫k+1 k+1
1
s
1
s
f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) · fk+1|k (X)δX.
(10.68)
(10.69)
Equivalently,
1
s
1
s
fk+1|k+1 (X) ∝ f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) · fk+1|k (X).
(10.70)
Now, we can rewrite fk+1|k+1 (X) as a “Bayes parallel combination” ([179], Eq.
(8.5)) of the form
1
s
fk+1|k+1 (X) ∝ fk+1|k+1 (X|Z k+1 ) · · · fk+1|k+1 (X|Z k+1 ) · fk+1|k (X)1−s
(10.71)
where
j
1
j
s
fk+1|k+1 (X|Z k+1 ) abbr.
= fk+1|k (X|Z (k) , ..., Z (k+1) , ..., Z (k) )
(10.72)
is the multitarget posterior, “singly-updated” using only the jth measurement set
j
Z k+1 . Thus, for example, if s = 2 then (10.71) becomes
1
2
fk+1|k+1 (X) ∝ fk+1|k+1 (X|Z k+1 ) · fk+1|k+1 (X|Z k+1 ) · fk+1|k (X)−1 . (10.73)
Given these preliminaries, three principled multisensor approximate filters
result from three different assumptions about the prior fk+1|k (X), the singlyj
updated posteriors fk+1|k+1 (X|Z k+1 ), and the sensor clutter processes. The
assumptions underlying these filters are as follows:
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291
1. Parallel Combination Approximate Multisensor (PCAM) CPHD Filter:
j
fk+1|k (X) and all of the fk+1|k+1 (X|Z k+1 ) are approximately i.i.d.c.;
and all of the sensor clutter processes are i.i.d.c. That is,
fk+1|k (X)
∼
=
|X|! · pk+1|k (|X|) · sX
k+1|k
∼
=
|X|! · pk+1|k+1 (|X|) · sX
k+1|k+1 (10.75)
=
|Z|! · pκk+1 (|Z|) · cZ
k+1 .
(10.74)
j
fk+1|k+1 (X|Z k+1 )
j
j
j
j
j
j
j
j
κk+1 (Z)
2. PCAM-PHD Filter: fk+1|k (X)
j
(10.76)
is approximately Poisson; and all of the
j
fk+1|k+1 (X|Z k+1 ) are approximately i.i.d.c.; and all of the sensor clutter
processes are Poisson. That is,
fk+1|k (X)
∼
=
X
e−Nk+1|k · Dk+1|k
∼
=
|X|! · pk+1|k+1 (|X|) · sX
k+1|k+1 (10.78)
=
e−λk+1 · κZ
k+1 .
(10.77)
j
fk+1|k+1 (X|Z k+1 )
j
j
j
j
j
j
j
κk+1 (Z)
3. Simplified PCAM (SPCAM) PHD Filter:
(10.79)
fk+1|k (X)
and all of the
j
fk+1|k+1 (X|Z k+1 ) are approximately Poisson; and all of the sensor clutter
processes are Poisson. That is,
fk+1|k (X)
∼
=
X
e−Nk+1|k · Dk+1|k
j
j
fk+1|k+1 (X|Z k+1 )
∼
=
j
e−N k+1|k+1 · D X
k+1|k+1
j
j
j
κk+1 (Z)
(10.80)
(10.81)
j
j
=
e−λk+1 · κZ
k+1 .
(10.82)
The basic ideas underlying the parallel-combination approximation are most
easily illustrated by looking at the simplest special case: the simplified PCAM-PHD
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filter. Substituting (10.80) through (10.82) into (10.71) we get:
1
s
fk+1|k+1 (X|Z k+1 ) · · · fk+1|k+1 (X|Z k+1 ) (10.83)
∝
fk+1|k+1 (X)
·fk+1|k (X)1−s
1
s
X
X
1−s
DX
k+1|k+1 · · · D k+1|k+1 · (Dk+1|k )
∝
1
1−s
(D k+1|k+1 · · · D k+1|k+1 · Dk+1|k
)X
=
(10.84)
s
(10.85)
and so
X
fk+1|k+1 (X) = e−Nk+1|k+1 · Dk+1|k+1
(10.86)
where
1
Dk+1|k+1 (x)
=
Nk+1|k+1
=
s
D k+1|k+1 (x) · · · D k+1|k+1 (x) · Dk+1|k (x)1−s (10.87)
∫
1
s
D k+1|k+1 (x) · · · D k+1|k+1 (x)
(10.88)
·Dk+1|k (x)1−s dx.
However, because of the measurement-update equations for the classical PHD filter,
(8.50) and (8.51), we can write
j
j
j
pD (x) · Lj (x)
∑
j
D k+1|k+1 (x)
j
z
= Lj
(x) = 1 − pD (x) +
. (10.89)
j
j
j
j
Dk+1|k (x)
Z k+1
κ
(
z)
+
τ
(
z)
k+1
k+1
j
j
z∈Z k+1
So, after substitution of these equations, (10.87) and (10.88) become the measurement update equations for the simplified PCAM-PHD filter of Section 10.6.3:
1
Dk+1|k+1 (x)
=
Nk+1|k+1
=
1
L 1 (x) · · · L 1 (x) · Dk+1|k (x)
(10.90)
Z
Z k+1
∫ k+1
1
1
L 1 (x) · · · L 1 (x) · Dk+1|k (x)dx. (10.91)
Z k+1
Z k+1
The same basic reasoning is applied to derive the measurement-update equations for the PCAM-PHD and PCAM-CPHD filters. Because the approximation
assumptions are more general, the resulting equations are more complicated than
(10.90) and (10.91).
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293
As will be pointed out shortly, the simplified PCAM-PHD filter is not entirely
satisfactory, since it does not reduce to the correct Bayesian solution in the singletarget case. Nevertheless, it has been employed with some success by a few
researchers, and therefore remains worth consideration.
The remainder of the section is organized as follows:
1. Section 10.6.1: The PCAM-CPHD filter.
2. Section 10.6.2: The PCAM-PHD filter.
3. Section 10.6.3: The simplified PCAM-PHD filter.
10.6.1
Parallel Combination Multisensor CPHD Filter
Suppose that at time tk we are given the spatial distribution sk|k (x) and the
cardinality distribution pk|k (n). Then the measurement-update of the PCAMCPHD filter consists of the following steps.
First, the usual time-update equations for the CPHD filter are used to construct the predicted spatial distribution sk+1|k (x) and the predicted cardinality
distribution pk+1|k (n).
Second, let
∑
Gk+1|k (x) =
pk+1|k (n) · xn
(10.92)
n≥0
(1)
Nk+1|k
=
(10.93)
Gk+1|k (1).
Third, assume that there are s sensors collecting measurements of the form
j
j
z ∈ Z, and that they are governed by the following models, for j = 1, ..., s:
j
j
• Probabilities of detection: pD (x) def.
= pD,k+1 (x).
j
j
j
• Single-target likelihood functions: Lj (x) abbr.
= f k+1 (z|x).
z
j
j
j
j
j
j
• Clutter intensity functions: κk+1 (z) = λk+1 · ck+1 (z) where λk+1 is the
j
j
clutter rate and ck+1 (z) is the clutter spatial distribution.
j
j
• Clutter cardinality distributions and p.g.f.’s: pκk+1 (m) and Gκk+1 (z) =
∑
j
κ
m
m≥0 pk+1 (m) · z .
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From these, define the following intermediate parameters:
∫
j
j
ϕk+1 =
(1 − pD (x)) · sk+1|k (x)dx
∫
j
j
j
j
τ̂ k+1 (z) =
pD (x) · Lj (x) · sk+1|k (x)dx.
(10.94)
(10.95)
z
Fourth and finally, suppose that at time
1
tk the s sensors collect the
j
s
j
respective measurement sets Z k+1 , ..., Z k+1 with |Z k+1 | = m. Then we are
to construct the spatial distribution sk+1|k+1 (x) and cardinality distribution
1
s
pk+1|k+1 (n), updated using all of the measurement sets Z k+1 , ..., Z k+1 .
Given these preliminaries, the measurement-updated spatial distribution and
cardinality distribution are ([151], Eqs. (9-21)):
sk+1|k+1 (x)
=
1
˜ 1
·L
(x) · sk+1|k (x)
s
Z k+1 ,...,Z k+1
Nk+1|k+1
(10.96)
pk+1|k+1 (n)
=
n
p̃(n) · θk+1
˜ k+1 )
G(θ
(10.97)
where
1
˜ 1
L
(x)
=
N k+1|k+1
=
s
Z k+1 ,...,Z k+1
1..s
s
˜ (1) (θk+1 ) LZ1 k+1 (x) · · · LZs k+1 (x)
G
· 1
(10.98)
s
˜ k+1 )
G(θ
N k+1|k+1 · · · N k+1|k+1
∫
1
s
L 1 (x) · · · L s (x) · sk+1|k (x)dx (10.99)
Z k+1
Z k+1
1..s
N k+1|k+1
θk+1
=
1
s
(10.100)
N k+1|k+1 · · · N k+1|k+1
and
Nk+1|k+1
=
p̃k+1|k+1 (n)
=
˜ k+1|k+1 (x)
G
=
˜ (1)
G
k+1|k+1 (θk+1 )
·θ
˜ k+1|k+1 (θk+1 ) k+1
G
1
s
ℓz1 (n) · · · ℓzs (n) · pk+1|k (n)
∑
p̃k+1|k+1 (n) · xn
n≥0
(10.101)
(10.102)
(10.103)
Multisensor PHD and CPHD Filters
295
and where
j
min{n,m}
∑
j
ℓj
(n)
=
Z k+1
j
j
j
(m − l)! · pκk+1 (m − l) · l!
(10.104)
l=0
j
j
j
·Cn,l · ϕn−l
k+1 · σ l (Z k+1 )
j
j
Lj
(x)
j
(10.105)
α0 · (1 − pD (x))
=
Z k+1
j
j
j
j
∑ pD (x) · Lj (x) · α(z)
z
+
j
j
j
ck+1 (z)
j
z∈Z k+1
j
j
j j
∑ τ̂ k+1 (z)
· α(z)
j
j
j
N k+1|k+1
α0 · ϕk+1 +
=
j
j
(10.106)
j
ck+1 (z)
j
z∈Z k+1
and where



j
j
j
κ
(
m
−
l)!
·
p
(
m
−
l)
k+1
l=0

j
∑m
j
j
j
(l+1)
·Gk+1|k (ϕk+1 ) · σ l (Z k+1 )

α0 = 
j
∑m
j
j
j
κ
i=0 (m − l)! · pk+1 (m − i) 

j
j
j
(i)
·Gk+1|k (ϕk+1 ) · σ i (Z k+1 )


j
∑m
j
j
j
κ
l=0 (m − l − 1)! · pk+1 (m − l − 1) 

j
j
(l+1)
j
j
j
·Gk+1|k (ϕk+1 ) · σ l (Z k+1 − {z})


α(z) =
j
∑m
j
j
j
κ
(
m
−
l)!
·
p
(
m
−
i)
k+1
i=0


j
j
j
(i)
·Gk+1|k (ϕk+1 ) · σ i (Z k+1 )


j
j
j
j
τ̂
(
z
)
j
j
k+1
j
 τ̂ k+1 (z1 )
m 
σ l (Z k+1 ) = σ j  j
, ..., j
.
j
j
m,l
ck+1 (z1 )
ck+1 (z j )
j
j
(10.107)
(10.108)
(10.109)
m
The following characteristics of the PCAM-CPHD filter should be pointed
out:
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Remark 38 (Computability of PCAM-CPHD filter) The PCAM-CPHD filter is
potentially computationally tractable, provided that we assume that, for all larger
values of n, pk+1|k (n) = 0. Given this, the computational complexity appears to
1
s
be O(m3 · · · m3 · n), where n is the current number of tracks and
current numbers of measurements collected by the jth sensor.
j
m is the
Remark 39 (PCAM-CPHD filter and CPHD filter) In the single-sensor case, the
PCAM-CPHD filter reduces to the CPHD filter.
Remark 40 (Consistency of the PCAM-CPHD filter) Suppose that there are no
missed detections, no false alarms, and that the number of targets is known a priori
to be one—that is, assume the single-target filtering scenario. Then the PCAMCPHD filters reduces to the multisensor, single-target Bayes filter.
Remark 41 (“Spooky action at a distance”) The PCAM-CPHD filter exhibits the
“spooky action at a distance” behavior described in Section 9.2 for the singlesensor CPHD filter. This means that, as with the single-sensor CPHD filter,
multitarget scenarios should be broken up into statistically noninteracting clusters,
and a separate filter assigned to each cluster.
Remark 42 (Disjoint FoVs) Because of this “spookiness,” the PCAM-CPHD filter
does not behave intuitively when the sensors have mutually disjoint fields of view.
Specifically, it does not reduce to separate, noninteracting filters, one for each field
of view (as is the case with the multisensor classical PHD filter described in Section
10.4). Since no multisensor fusion is possible in such situations, this should not pose
a difficulty in practice.
Remark 43 (Implementations of PCAM-CPHD filter) Nagappa and Clark [221]
have implemented two-sensor and three-sensor versions of the PCAM-CPHD filter.
See Section 10.8 for more details.
10.6.2
Parallel Combination Multisensor PHD Filter
Suppose that at time tk we are given the PHD Dk|k (x). Then the measurementupdate step for this filter consists of the following steps.
First, the usual time-update equation for the PHD filter is used to construct
the predicted PHD Dk+1|k (x).
Multisensor PHD and CPHD Filters
297
Second, let
Nk+1|k
=
sk+1|k (x)
=
∫
(10.110)
Dk+1|k (x)dx
Dk+1|k (x)
.
Nk+1|k
(10.111)
Third, assume that there are s sensors collecting measurements of the form
j
j
j
j
j
z ∈ Z. Let ϕk+1 and τ̂ k+1 (z) be as in (10.94) and (10.95), respectively. Then
the measurement-update for the PHD is ([151], Eqs. (22-29)):
Dk+1|k+1 (x) = L 1
s
Z k+1 ,...,Z k+1
(x) · Dk+1|k (x)
(10.112)
where
1
s
L1
˜ 1
L
(x)
=
χk+1
=
s
Z k+1 ,...,Z k+1
(x) · · · L s
(x)
Z k+1
χk+1 · 1
s
ν k+1|k+1 · · · ν k+1|k+1
Z k+1
(10.113)
(10.114)
∑
n
Nk+1|k
·θ n
1 (n + 1) · · · ℓs (n + 1) ·
n≥0 ℓz
n!
z
1
∑
s
1
s
1 (j) · · · ℓs (j) ·
j≥0 ℓz
z
j
Nk+1|k
·θ j
j!
1..s
ν k+1|k+1
θ
=
1
1..s
ν k+1|k+1
=
s
ν k+1|k+1 · · · ν k+1|k+1
∫
1
s
1
L 1 (x) · · · L s (x)
Z k+1
Z k+1
Nk+1|k
·Dk+1|k (x)dx
(10.115)
(10.116)
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and where
j
min{n,m}
∑
j
ℓj (x)
=
z
j j
j
j
j
n−l
λm−jl
k+1 · l! · Cn,l · ϕk+1 · σ l (Z k+1 )
(10.117)
l=0
j
Lj
(x)
(10.118)
=
Z k+1
j
j
∑
j
1 − pD (x) +
j
j
z∈Z k+1
pD (x) · Lj (x)
z
j
j
j
j
κk+1 (z) + Nk+1|k · τ̂ k+1 (z)
j
∑
j
j
ν k+1|k+1
=
ϕk+1 +
j
j
z∈Z k+1
j
τ̂ k+1 (z)
(10.119)
j
j
j
j
κk+1 (z) + Nk+1|k · τ̂ k+1 (z)
j
j
and where σ l (Z k+1 ) was defined in (10.109).
The following characteristics of the PCAM-PHD filter should be pointed out:
Remark 44 (Computability of PCAM-PHD filter) The PCAM-PHD filter may
be more computationally problematic than the PCAM-CPHD filter. This is because
of the infinite sums in the definition of the quantity χk+1 . For the PCAM-PHD filter
to be tractable, these series must be truncated to a suitable degree of approximation.
If the series do not converge rapidly enough, this could present a computational
issue.
Remark 45 (PCAM-PHD filter reduces to PHD filter) The PCAM-PHD filter reduces to the classical PHD filter in the single-sensor case.
Remark 46 (Consistency of the PCAM-PHD filter) Suppose that there- are no
missed detections, no false alarms, and that the number of targets is known a priori
to be one—that is, assume the single-target filtering scenario. Then the PCAMPHD filter reduces to the multisensor, single-target Bayes filter.
Remark 47 (“Spooky action at a distance”) Like the PCAM-CPHD filter, the
PCAM-PHD filter exhibits the “spooky action at a distance” behavior described
in Section 9.2. As with the PCAM-CPHD filter, scenarios should be broken up into
statistically noninteracting clusters, and a separate filter assigned to each cluster.
Remark 48 (Implementations of the PCAM-PHD filter) Nagappa and Clark [221]
have implemented two-sensor and three-sensor versions of the PCAM-PHD filter.
See Section 10.8 for more details.
Multisensor PHD and CPHD Filters
10.6.3
299
Simplified PCAM-PHD Filter
This, the simplest of the approximate multisensor PHD filters, was first proposed
without proof in [165], Eqs. (105-107). It has the following measurement-update
equation:
Dk+1|k+1 (x) = L 1
Z k+1
(x) · · · L s
Z k+1
(10.120)
(x) · Dk+1|k (x)
where
j
j
j
Lj
(x)
1 − pD (x) +
=
pD (x) · Lj (x)
∑
j
Z k+1
j
j
z
j
j
j
(10.121)
j
κk+1 (z) + τ k+1 (z)
z∈Z k+1
j
∫
j
τ k+1 (z)
=
j
j
pD (x) · Lj (x) · Dk+1|k (x)dx.
(10.122)
z
The following characteristics of this filter should be noted:
Remark 49 (Computability of simple PCAM-PHD filter) The complexity of the
j
1
s
filter is O(m · · · m · n) where n is the current number of tracks and m is the
current number of measurements collected by the jth sensor.
Remark 50 (Behavior with small target number) The simplified PCAM-PHD filter is unlikely to perform well when the number of targets is small. This is because
it does not reduce to the correct formula in the single-sensor, single-target special
case—that is, to the multisensor version of Bayes’ rule. For in this case (10.121)
reduces to
j
Lj (x)
j
z
Lj (x) = j k+1j
(10.123)
zk+1
τ k+1 (zk+1 )
where
j
j
τ k+1 (z) =
∫
j
Lj (x) · fk+1|k (x)dx.
(10.124)
z
Thus (10.120) reduces to
1
L z1
s
k+1
(x) · · · L zs k+1 (x)
· fk+1|k (x)
fk+1|k+1 (x) = 1
1
s
s
τ k+1 (zk+1 ) · · · τ k+1 (zk+1 )
(10.125)
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Advances in Statistical Multisource-Multitarget Information Fusion
whereas the correct Bayesian solution is
1
s
(x) · · · L zs k+1 (x)
L z1
fk+1|k+1 (x) = 1..sk+1 1
· fk+1|k (x)
s
τ k+1 (zk+1 , ..., zk+1 )
(10.126)
where
1..s
1
s
τ k+1 (zk+1 , ..., zk+1 ) =
∫
1
s
L z1
k+1
(x) · · · L zs k+1 (x) · fk+1|k (x)dx.
(10.127)
Remark 51 (Implementations) Despite this last fact, some researchers have applied the simplified PCAM-PHD filter with success. Lian, Han, Liu, and Chen have
applied it to the problem of joint tracking and sensor-bias estimation ([145], Eqs.
(11-13)). They tested a particle implementation of their filter in simulations involving four appearing and disappearing targets, a range-bearing sensor, a rangeonly sensor, and a bearing-only sensor. They demonstrated that their filter successfully estimated the translational biases in the three sensors while also successfully
tracking the targets. Also, Delande, Duflos, Heurguier, and Vanheeghie applied a
partitioning method to lower computational cost [63]. They tested a particle implementation of their filter in simulations involving 11 maneuvering, appearing, and
j
disappearing targets, and five Gaussian sensors with different choices of pD (x),
j
j
j
λk+1 , and f k+1 (z|x). They demonstrated that the filter performed effectively.
10.7
AN ERRONEOUS “AVERAGED” MULTISENSOR PHD FILTER
All of the multisensor PHD and CPHD filters just described have one thing in
common: they rely on multiplication of the pseudolikelihood functions of the
sensors. For example, the simplest approximate PHD filter described in Section
10.6.3 is based on the product
×
1
L1
s
Z,...,Z
s
(10.128)
(x) = L 1 (x) · · · L s (x)
Z
Z
of the pseudolikelihoods for the different sensors:
j
j
∑
j
j
L j (x) = 1 − pD (x) +
Z
j
j
z∈Z k+1
pD (x) · Lj (x)
z
j
j
j
(10.129)
j
κk+1 (z) + τ k+1 (z)
Multisensor PHD and CPHD Filters
where
j
j
τ k+1 (z) =
∫
301
j
j
pD (x) · Lj (x) · Dk+1|k (x)dx.
(10.130)
z
The use of product-based approaches is intuitively and theoretically compelling. Statistical independence typically leads to products of fundamental descriptors; and products of probability distributions typically lead to smaller targetlocalization error-covariance matrices as long as the sources are not too greatly in
conflict.
However and to the contrary, it has been proposed [280] that the proper
“Bayesian” approach for a multisensor PHD filter should be based on the average
of the pseudolikelihoods:1
+
L1
s
Z,...,Z
(x) =
)
s
1 (1
L 1 (x) + ... + L s (x) .
Z
Z
s
(10.131)
As has been pointed out in [167] and [181], such an approach is questionable for
an obvious reason: it tends to decrease (rather than increase) target-localization
accuracy. The purpose of this section is to demonstrate this fact using simple
examples. Section 10.8 reports simulation results leading to the same conclusion.
Let us consider the simplest possible special case: a single-target with
no clutter and no missed detections. The predicted and updated PHDs are then
probability density functions: Dk+1|k (x) = fk+1|k (x) and Dk+1|k+1 (x) =
fk+1|k+1 (x); and (10.131) reduces to
 1

s
L
(x)
1
s
L
(x)
1 z
Lz,...,
+ ... + s z s  .
1
s (x) =
1
z
s τ1 k+1 (z)
τ k+1 (z)
+
1
(10.132)
In [282], the author of [280] appears to retreat from this claim, stating that the filter described in
[280] was actually a “traffic filter” that “was misidentified there as a multisensor target filter” ([282],
Section I). That is, the claimed “multisensor multitarget intensity filter” in [280] is erroneous.
However, in [282] the author goes on to claim the validity of a “traffic filter” generalization of the
averaged-pseudolikelihood approach,
in which the average is replaced by a weighted average using
∑s
ℓ
functions β ℓ (x) such that
ℓ=1 β (x) = 1 identically. This claim is also not true—see Remark
10 in Section 4.2.5.2.
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Given this, the multisensor measurement update using the averaged likelihood is
+
=
=
=
j
(10.133)
f k+1|k+1 (x)
 1

s
Lzs (x)
1  Lz1 (x)
 · fk+1|k (x)
+ ... + s
s
1
s τ1 k+1 (z)
τ k+1 (z)
1

s
Lzs (x) · fk+1|k (x)
1  Lz1 (x) · fk+1|k (x)

+ ... +
s
s
1
1
s
τ k+1 (z)
τ k+1 (z)
)
1(
1
s
fk+1|k+1 (x|z) + ... + fk+1|k+1 (x|z)
s
j
(10.134)
(10.135)
j
where the τ k+1 (z) were defined in (10.130) and where fk+1|k+1 (x|z) is the
posterior distribution updated using the jth measurement alone. The following
difficulties are apparent:
+
1. Lz,...,
is not a proper likelihood function for the single-target case,
1
s (x)
z
because it is unitless and integrates to infinity rather than to unity:
∫ +
1
s
Lz,...,
(10.136)
1
s (x)dz · · · dz = ∞.
z
+
2. f k+1|k+1 (x) is not equal to the Bayes-optimal solution, namely the multisensor, single-target version of Bayes’ rule:
1
fk+1|k+1 (x) =
s
Lz1 (x) · · · Lzs (x) · fk+1|k (x)
1
s
(10.137)
τk+1 (z, ..., z)
where
1
s
τk+1 (z, ..., z) =
∫
1
s
Lz1 (x) · · · Lzs (x) · fk+1|k (x)dx.
(10.138)
+
3. Since the track distribution f k+1|k+1 (x) is a mixture distribution, it will
exhibit increased track uncertainty (compared to the single-sensor track
j
distributions fk+1|k+1 (x|z))—rather than, as should be the case, decreased
track uncertainty.
Multisensor PHD and CPHD Filters
Figure 10.1
303
The graph of (10.139).
The third point is most easily demonstrated using a simple example. Assume
that there are two bearing-only sensors in the plane, with respective likelihood
functions
(
)
1
1
(z − x)2
Lz (x, y) = Nσ2 (z − x) = √
· exp −
(10.139)
2σ 2
2πσ
(
)
2
1
(z − y)2
Lz (x, y) = Nσ2 (z − y) = √
· exp −
. (10.140)
2σ 2
2πσ
That is, the sensors are oriented so as to triangulate the position of a target located
at (x, y). The graphs of these functions are plotted in Figures 10.1 and 10.2,
respectively.
For the sake of conceptual clarity, further assume that the prior distribution
is
fk+1|k (x, y) = Nσ02 (x − x0 ) · Nσ02 (y − y0 ).
(10.141)
Then
=
∫
Lz (x, y) · fk+1|k (x, y)dxdy
=
∫
Nσ2 (z − x) · Nσ02 (x − x0 ) · Nσ02 (y − y0 )dxdy (10.143)
=
Nσ2 +σ02 (z − x0 )
1
τ k+1 (z)
1
(10.142)
(10.144)
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Advances in Statistical Multisource-Multitarget Information Fusion
Figure 10.2
The graph of (10.140).
and similarly
=
∫
=
Nσ2 +σ02 (z − y0 ).
2
τ k+1 (z)
2
(10.145)
Lz (x, y) · fk+1|k (x, y)dxdy
(10.146)
Thus with averaging, the measurement-updated track distribution is
+
1
f k+1|k+1 (x, y) =
2
=
=
(1
Lz1 (x,y)
1
τ k+1 (z1 )
2
Lz2 (x, y)
+ 2
τ k+1 (z2 )
)
·fk+1|k (x, y)
(
)
1
Nσ2 (z1 − x)
Nσ2 (z2 − y)
+
2 Nσ2 +σ02 (z1 − x0 ) Nσ2 +σ02 (z2 − y0 )
·Nσ02 (x − x0 ) · Nσ02 (y − y0 )
(
)
Nω2 (x − p0 ) · Nσ02 (y − y0 )
1
+Nσ02 (x − x0 ) · Nω2 (y − q0 )
2
(10.147)
(10.148)
(10.149)
Multisensor PHD and CPHD Filters
305
where ω 2 , p20 , and q02 are given by
1
1
1
= 2 + 2,
ω2
σ
σ0
p0
x0
z1
= 2 + 2,
ω2
σ0
σ
q0
y0
z2
= 2 + 2.
ω2
σ0
σ
(10.150)
+
It may be verified that the mean and variance of f k+1|k+1 (x, y) are
∫
+
+
µ =
(x, y) · f k+1|k+1 (x, y)dxdy
=
1
(x0 + p0 , y0 + q0 )
2
(10.151)
(10.152)
and
∫
=
1
1
− (x0 + p0 )2 − (y0 + q0 )2 +
4
4
=
·f k+1|k+1 (x, y)dxdy
) 1
1( 2
ω 2 + σ02 +
p + q02 + y02 + x20 − (x0 + p0 )2
2 0
4
1
− (y0 + q0 )2 .
4
+
σ2
(x2 + y 2 )
(10.153)
+
(10.154)
Now let σ0 → ∞, so that the prior distribution is effectively uniform. Then
ω 2 → σ 2 , p20 → z12 , and q02 → z22 , and so
+
µ
+2
σ
→
→
1
(x0 + z1 , y0 + z2 )
2
σ02 → ∞.
(10.155)
(10.156)
That is, averaging leads to an arbitrarily large uncertainty in the target localization.
By way of contrast, the track distribution computed using Bayes’ rule is
1
×
f k+1|k+1 (x, y)
=
2
Lz1 (x,y) · Lz2 (x, y)
· fk+1|k (x, y)
τ k+1 (z1 , z2 )
Nω2 (x − p0 ) · Nω2 (y − q0 ).
12
=
(10.157)
(10.158)
In this case the mean and variance can be shown to be
×
µ
×2
σ
=
=
(p0 , q0 ) −→ (z1 , z2 )
2
2ω −→ 2σ
2
(10.159)
(10.160)
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Figure 10.3
(10.158).
The graph of the track distribution for the Bayes product likelihood,
where the limit is taken as σ0 → ∞. That is, Bayes’ rule leads to a triangulated
target position with finite localization accuracy.
These analytical conclusions have been verified in simulations—see [221] and
Section 10.8.
The graph of the Bayesian solution is displayed in Figure 10.3 whereas the
graph of the averaged solution, (10.149), is displayed in Figure 10.4.
The difference between the Bayesian and the averaged-likelihood approaches
is even more pronounced when the number of sensors increases. Suppose that
additional bearing-only sensors are applied, all with orientations different than
the first two and different than each other. Then the variance of the averaged
filter increases with the number of averaged sensors—whereas the variance greatly
decreases if Bayes’ rule is used instead.
10.8
PERFORMANCE COMPARISONS
Nagappa and Clark [221] have conducted simulations comparing the multisensor
PHD and CPHD filters described in this chapter.
In a first set of simulations, they compared two-sensor versions of the following multisensor PHD filters:
Multisensor PHD and CPHD Filters
Figure 10.4
(10.149).
307
The graph of the track distribution for the averaged likelihood,
• Iterated-corrector multisensor classical PHD filter (Section 10.5).
• PCAM-PHD filter (Section 10.6.2).
• Multisensor classical PHD filter (Section 10.4).
• Averaged pseudolikelihood (APL) PHD filter (Section 10.7).
Both sensors had pD = 0.95 and clutter rate λ = 10. The evaluations
were made using the following two versions of the optimal subpattern assignment
(OSPA) metric (Section 6.2.3):
• E-OSPA—the OSPA metric, with the Euclidean metric as the base metric,
(6.15). This metric measures the error in both target number and target localization, but not the error in track uncertainty (that is, error in the covariance).
• H-OSPA—the OSPA metric, but with the Hellinger distance as the base
metric, (6.35). Unlike the E-OSPA metric, this metric does measure the error
in track uncertainty.
The results were as follows ([221], Figure 4):
• E-OSPA: The multisensor classical PHD filter performed significantly better
than the alternatives—as would be expected because it is the most accurate
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PHD filter solution. The performance of the other three was roughly comparable with each other, except that the APL-PHD filter performed somewhat
worse.
• H-OSPA: In this case the differences were much more pronounced. Once
again, the multisensor classical PHD filter performed significantly better than
the others. The PCAM-PHD filter and iterated-corrector PHD filter were
roughly comparable, with the PCAM-PHD filter performing slightly better.
The APL-PHD filter performed considerably worse than the others. This is
to be expected, given this filter’s inherent tendency to increase rather than
decrease track covariance (as described in Section 10.7).
In a second set of related simulations, Nagappa and Clark compared twosensor versions of the following multisensor CPHD filters:
• Iterated-corrector CPHD filter.
• PCAM-CPHD filter (Section 10.6.1).
The results were as follows ([221], Figure 5):
• E-OSPA and H-OSPA The PCAM-CPHD filter performed significantly better
than the iterated-corrector CPHD filter, especially towards the end of the
scenario.
In a third set of simulations, three-sensor versions of the following algorithms
were compared:
• Iterated-PHD filter.
• PCAM-PHD filter.
• Iterated CPHD filter.
• PCAM-CPHD filter.
• Averaged-pseudolikelihood (APL) PHD filter.
In this case, two sensors had pD = 0.95 and the third pD = 0.9. The results
were as follows ([221], Figure 6):
• E-OSPA: In decreasing order of performance, the filters ranked as follows:
PCAM-CPHD, PCAM-PHD, iterated CPHD, APL-PHD, iterated PHD. The
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309
poor performance of the iterated-PHD filter is attributable to its susceptibility to differing probabilities of detection (as discussed in [219] and in Section 10.5.1). Nevertheless, the iterated-CPHD filter still significantly outperformed the APL-PHD filter.
• H-OSPA: In decreasing order of performance: PCAM-CPHD, PCAM-PHD,
iterated CPHD, iterated PHD, APL-PHD. The performance of the APL-PHD
filter was particularly bad, with the iterated PHD filter having an intermediate
position.
Similar results were observed when the probability of detection of the third
sensor was decreased from pD = 0.9 to pD = 0.85 and again to pD = 0.7
([221], Table V).
Chapter 11
Jump-Markov PHD/CPHD Filters
11.1
INTRODUCTION
Rapidly maneuvering targets severely challenge the capabilities of conventional
Bayes tracking filters—including all of the RFS filters considered up to this point.
This is because the time-update step of such filters depends on a single a priori
model of target motion.
For the single-sensor, single-target Bayes filter, this model consists of a single
item: the Markov transition density fk+1|k (x|x′ )—which, in turn, is usually based
on a statistical motion model such as Xk+1|k = φk (x′ ) + Wk . According to this
model, if a target has state x′ at time tk then it must have state φk (x′ ) at time
tk+1 —except for a degree of uncertainty modeled by the plant noise Wk .
In the multitarget case, the motion model will include additional items: the
target probability of survival, pS,k+1|k (x′ ); the target appearance RFS; and,
perhaps, also a target-spawning RFS.
Such models have a common limitation: only a single model of motion can
be addressed at any given moment. The Markov density fk+1|k (x|x′ ) models a
single type of target trajectory; pS (x′ ) models a single type of target disappearance;
and so on.
In single-target tracking, the most well-known algorithms for addressing
maneuvering targets are the multiple motion model filters, such as the popular
interacting multiple-model (IMM) filter [202]. Such filters typically employ a
“library” of motion models and apply some methodology to select, on-the-fly, the
motion model (or the amalgam of motion models) that best accounts for observed
target motion.
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The conceptually simplest such algorithms run multiple filters in parallel (a
“filter bank,” that is, one filter for each motion model), gradually winnowing out
the least explanatory filters. The filter bank approach is computationally intensive.
Thus more sophisticated filters have been devised that select the motion model using
statistical methods.
Jump-Markov models provide the most well-known theoretical foundation for
this kind of filter. The basic idea is simple: append a discrete state variable o
(the “mode variable” or “jump variable”) to the kinematic state x, resulting in an
augmented state ẍ = (o, x). A jump-Markov filter is just a Bayes filter defined on
this augmented state. The ultimate goal of the filter is to continually select o so
that the observed target motion is most accurately modeled.
To address the problem of multiple and independent rapidly-maneuvering
targets, it is necessary to generalize the jump-Markov approach to the multitarget
realm. The optimal generalization, the jump-Markov multitarget Bayes filter, is
computationally intractable in general. Thus one must devise principled approximations, for example: jump-Markov generalizations of the classical PHD and CPHD
filters. That is the purpose of this chapter, which is drawn from [170]. (JumpMarkov versions of the multi-Bernoulli filter will be addressed in Section 13.5 of
Chapter 13).
Following the general philosophy outlined in Chapter 7, the following
methodology will be employed:
1. Begin with the multitarget Bayes filter as the conceptual starting point;
2. Generalize it to a multitarget jump-Markov Bayes filter.
3. Derive PHD and CPHD filter equations from this generalized filter.
This procedure ensures that the resulting PHD/CPHD filters adhere as closely
as possible to the jump-Markov theoretical framework. It is to be contrasted with
the following, more ad hoc, approach:
1. Begin with the classical PHD (or CPHD filter) as the conceptual starting
point.
2. Try to draw an analogy between this filter and the conventional single-target
jump-Markov Bayes filter.
3. On the basis of this analogy, propose filtering equations for a jump-Markov
PHD or CPHD filter.
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The major drawback of this approach is that it is excessively subjective.
What if someone else applies a different analogy, and thereby arrives at a different
filter? How does one determine if either analogy is accurate? The end result is a
potential “Tower of Babel” of jump-Markov PHD/CPHD filters, based on various
heuristically-based analogies. What reason is there to believe that any of these
filters are actually PHD or CPHD filters in the general sense, as defined in Sections
5.10.4 and 5.10.5?
11.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• The single-sensor, single-target jump-Markov Bayes filter is a conventional
single-sensor, single-target Bayes filter, but defined on an augmented state
space of the form
¨ = {1, ..., O} × X
X
(11.1)
where X is the kinematic state space and o = 1, ..., O are the indices of
the model modes. Thus the single-sensor, single-target jump-Markov filter
propagates hybrid discrete-continuous probability distributions of the form
fk|k (o, x|Z k ). See Section 11.2.
• The naı̈ve generalization of this to the multitarget realm is not correct. By
this is meant a Bayes filter on {1, ..., O} × X∞ , where X∞ is the class
of all finite subsets of X. This approach is not correct because it rests on a
hidden assumption: that all targets move in the same manner according to a
single common model. See Section 11.3.1.
• The correct multitarget jump-Markov filter is a multiobject Bayes filter on
¨ ∞ of all finite subsets of the single-target
the multiobject state space X
¨
augmented space X. Thus it propagates multiobject probability distributions
¨ (k) ) where
of the form fk|k (X|Z
¨ = {(o1 , x1 ), ..., (on , xn )}.
X
(11.2)
See Section 11.3.2.
• To achieve computationally attractive jump-Markov PHD and CPHD filters,
one must assume that the clutter multiobject probability distribution κk+1 (Z)
is independent of the mode: κk+1 (Z|o) = κk+1 (Z). See Section 11.4.1.
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• As a consequence, both the jump-Markov PHD filter and jump-Markov
CPHD filter are classical PHD and CPHD filters, but defined on the aug¨ The jump-Markov PHD filter propagates
mented single-target state space X.
(k)
PHDs of the form Dk|k (o, x|Z ); and the jump-Markov CPHD filter propagates, in addition, cardinality distributions pk|k (n|Z (k) ) where n is the
total number of all targets (that is, regardless of what mode they are in). See
Sections 11.4 and 11.5.
• The jump-Markov PHD and CPHD filters can be implemented using both
Gaussian mixture and sequential Monte Carlo (SMC) techniques. See Section 11.7.
• The usual jump-Markov approach is based on an implicit assumption: that the
target state space for each motion model is always the same. This assumption
is often not valid in practical application. However, Chen, McDonald, and
Kirubarajan have shown that the multiple-model approach can be generalized
to include model-dependent state spaces. See Section 11.6.
11.1.2
Organization of the Chapter
The chapter is organized as follows:
1. Section 11.2: A review of the single-sensor, single-target jump-Markov
Bayes filter.
2. Section 11.3: The general multitarget jump-Markov Bayes recursive filter.
3. Section 11.4: A jump-Markov version of the classical PHD filter.
4. Section 11.5: A jump-Markov version of the classical CPHD filter.
5. Section 11.6: Jump-Markov CPHD filter with mode dependent target-state
spaces.
6. Section 11.7: Implementation of jump-Markov PHD and CPHD filters using
Gaussian mixture and sequential Monte Carlo methods.
7. Section 11.8: Implemented jump-Markov PHD and CPHD filters.
Jump-Markov PHD/CPHD Filters
11.2
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JUMP-MARKOV FILTERS: A REVIEW
A jump-Markov system results when we append a discrete state variable o (the
“mode variable” or “jump variable”) to the single-target kinematic state x, with a
finite number of possible values o = 1, ..., O. This results in an augmented state
of the form ẍ = (o, x) where x ∈ X. Note that an implicit assumption is being
made:
• The target-state space for every model o is the same: X.
The case when this assumption is not true is addressed in Section 11.6.
The integral on the augmented state space is defined by
∫
O ∫
∑
f (o, x)dx,
(11.3)
which hereafter will be abbreviated as
∫
∑∫
f (ẍ)dẍ =
f (o, x)dx.
(11.4)
f (ẍ)dẍ =
o=1
o
Given this, the likelihood function and Markov transition density for the
system must have the form
Lz (o, x) = fk+1 (z|o, x),
fk+1|k (o, x|o′ , x′ ).
(11.5)
Because of Bayes’ rule, the Markov transition function factors as
fk+1|k (o, x|o′ , x′ ) = fk+1|k (o|o′ , x′ ) · fk+1|k (x|o, o′ , x′ ).
(11.6)
It is typically assumed that the new mode is independent of the previous kinematic
state, and that the new kinematic state is independent of the new mode:
fk+1|k (o|o′ , x′ )
′
′
fk+1|k (x|o, o , x )
=
fk+1|k (o|o′ ) = χo,o′
(11.7)
=
′
(11.8)
′
fk+1|k (x|o , x ).
In summary,
fk+1|k (o, x|o′ , x′ ) = χo,o′ · fk+1|k (x|o′ , x′ )
(11.9)
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where χo,o′ is the Markov transition matrix for the Markov chain defined by the
mode variable o = 1, ..., O; and where fk+1|k (x|o′ , x′ ) is the Markov transition
density (motion model) corresponding to the mode variable o′ .
Also, for the purposes of maneuvering-target tracking, the likelihood function
is usually assumed to be independent of the mode:
fk+1 (z|o, x) = fk+1 (z|x).
(11.10)
In this case, the problem of determining the best motion model is the same as the
problem of determining the value of o′ that corresponds to the most accurate
motion model fk+1|k (x|o′ , x′ ).
When fk+1 (z|o, x) and fk+1|k (x|o′ , x′ ) are linear-Gaussian for each choice
of o, x (respectively each choice of o′ , x′ ), then the jump-Markov system is called
a jump-Markov linear system (JMLS).
11.2.1
The Jump-Markov Bayes Recursive Filter
The single-sensor, single-target Bayes filter was described in Section 2.2.7. The
Time evolution of a jump-Markov system is described by the Bayes recursive filter
defined on the augmented state ẍ = (o, x):
... → fk|k (o, x|Z k ) → fk+1|k (o, x|Z k ) → fk+1|k+1 (o, x|Z k+1 ) → ...
where fk|k (o, x|Z k ) is the probability (density) that the target has kinematic state
x and is subject to the model corresponding to the mode o. This filter is defined
by the equations
fk+1|k (ẍ|Z k )
=
fk+1|k+1 (ẍ|Z k+1 )
=
fk+1 (zk+1 |Z k )
=
∫
fk+1|k (ẍ|ẍ′ ) · fk|k (ẍ′ |Z k )dẍ′
fk+1 (zk+1 |ẍ) · fk+1|k (ẍ|Z k )
fk+1 (zk+1 |Z k )
∫
fk+1 (zk+1 |ẍ) · fk+1|k (ẍ|Z k )dẍ
(11.11)
(11.12)
(11.13)
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or, equivalently, by
fk+1|k (o, x|Z k )
=
∑
χo,o′
∫
fk+1|k (x|o′ , x′ )
(11.14)
o′
·fk|k (o′ , x′ |Z k )dx′
fk+1|k+1 (o, x|Z k+1 )
=
fk+1 (zk+1 |Z k )
=
fk+1 (zk+1 |o, x) · fk+1|k (o, x|Z k )
fk+1 (zk+1 |Z k )
∫
∑
fk+1 (zk+1 |o, x)
(11.15)
(11.16)
o
·fk+1|k (o, x|Z k )dx.
11.2.2
State Estimation for Jump-Markov Filters
What is the best way of determining the state of a jump-Markov system? This
question has been addressed by Boers and Driessen [26]. Among others, they
considered the following state estimators:
• Classical MAP estimator:
(ôk|k , x̂k|k ) = arg sup fk|k (o, x|Z k ).
(11.17)
o,x
This is the most probable estimate. The target is estimated to have kinematic
state x = x̂k|k and is, simultaneously, estimated to be governed by mode
o = ôk|k .
• Marginally maximal target-state corresponding to the marginally maximal
mode:
∫
ôk|k = arg max fk|k (o, x|Z k )dx
(11.18)
o
x̂k|k
=
arg sup fk|k (ôk|k , x|Z k ).
(11.19)
x
That is, marginalize out the kinematic state and determine the marginally
most probable mode; then determine the most probable kinematic state that
has this mode.
• Marginally maximal target state:
x̂k|k = arg sup
∑
x
o
fk|k (o, x|Z k ).
(11.20)
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This estimator is well-suited for situations in which one wishes to estimate
the target state, but determination of the jump mode is not necessary. The
mode is integrated out as a nuisance variable, and the target state estimate is
the MAP estimator applied to the marginal target-state distribution.
11.3
MULTITARGET JUMP-MARKOV SYSTEMS
The purpose of this section is to extend the theory of jump-Markov systems to the
multitarget realm. The following two issues are addressed:
1. Section 11.3.1: What is the proper definition of a multitarget jump-Markov
system?
2. Section 11.3.2: The multitarget jump-Markov Bayes recursive filter.
11.3.1
What Is a Multitarget Jump-Markov System?
A certain degree of care is required for the proper definition of a multitarget jumpMarkov system. Given the definition of a jump-Markov system in Section 11.2, one
might be tempted to define it as follows:
• Naı̈ve definition of a multitarget jump-Markov system: Append the jump
variable o to the multitarget state X, resulting in an augmented state of
the form (o, X).
This approach can be expected to result in deteriorated performance. The
reason is that all targets in X = {x1 , ..., xn } are presumed to be moving in the
same way—that is, to have the motion model corresponding to the value of o.
To better understand why this is the case ([170], Section III-B), consider
the simplest multitarget motion model: independent target motions and no target
disappearance and target appearance. Let the state sets at times k and k + 1 be,
respectively,
X ′ = {x′1 , ..., x′n },
X = {x1 , ..., xn }
(11.21)
with |X ′ | = n′ and |X| = n. Also, by (7.71), the multitarget Markov density
must have the form
Jump-Markov PHD/CPHD Filters
=
319
fk+1 (o, X|o′ , X ′ )
(11.22)
∑
′
′
′
′
δn,n′ · χo,o′
fk+1|k (xπ1 |o , x1 ) · · · fk+1|k (xπn |o , xn )
π
where the summation is taken over all permutations π on 1, ..., n. Suppose now
that the optimal mode has been determined to be ôk|k . Then we are implicitly
stating that the optimally inferred motion-mode of each and every target is the
motion model corresponding to ôk|k .
As a more concrete example, suppose that ôk|k is the mode value corresponding to a right-hand turn. Then a right-hand turn is being presumed to optimally
describe the motion of all of the targets in the scene. Clearly, this is problematic. It
is also clear that the summation in the right side of (11.22) should actually have the
form
∑
δn,n′
χoπ1 ,o′1 · fk+1|k (xπ1 |o′1 , x′1 ) · · · χoπn ,o′n · fk+1|k (xπn |o′n , x′n ). (11.23)
π
This would allow each target to be governed by its own motion model.
A better modeling approach permits this. The jump variable o should be
appended not to the multitarget state X, but (as with single-target jump-Markov
systems) to the single-target state, resulting in single-target states of the form
ẍ = (o, x). Given this, a multitarget state must be a finite set of such augmented
states: not (o, X) but, rather,
¨ = {ẍ1 , ..., ẍn } = {(o1 , x1 ), ..., (on , xn )}.
X
(11.24)
That is, target x1 is subject to mode o1 , target x2 is subject to mode o2 , and so
on. In this case, (7.71) gives us
=
¨ X
¨ ′)
fk+1 (X|
∑
δn,n′
fk+1|k (ẍπ1 |ẍ′1 ) · · · fk+1|k (ẍπn |ẍ′n )
(11.25)
π
=
δn,n′
∑
χoπ1 ,o′1 · fk+1|k (xπ1 |o′1 , x′1 )
π
· · · χoπn ,o′n · fk+1|k (xπn |o′n , x′n ),
which has the right form.
The following fact is an immediate consequence of this reasoning:
(11.26)
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• The following statistics must be mode-independent: the p.g.fl. Gk|k [g], the
p.g.f. Gk|k (x), and the cardinality distribution pk|k (n).
For, suppose that they had the form Gk|k [g|o], Gk|k (x|o), pk|k (n|o). Then
it would follow that a single value of the mode o is being imposed on all targets
simultaneously.
11.3.2
The Multitarget Jump-Markov Filter
Given the discussion in the previous section, the Time evolution of the multitarget
jump-Markov system is described by the Bayes filter
¨ (k) ) → fk+1|k (X|Z
¨ (k) ) → fk+1|k+1 (X|Z
¨ (k+1) ) → ...
... → fk|k (X|Z
where, by (5.1) through (5.3),
¨ (k) )
fk+1|k (X|Z
=
¨ (k+1) )
fk+1|k+1 (X|Z
=
fk+1 (Zk+1 |Z (k) )
=
∫
¨ X
¨ ′ ) · fk|k (X
¨ ′ |Z (k) )δ X
¨′
fk+1|k (X|
(11.27)
¨ · fk+1|k (X|Z
¨ (k) )
fk+1 (Zk+1 |X)
(11.28)
(k)
fk+1 (Zk+1 |Z )
∫
¨ · fk+1|k (X|Z
¨ (k) )δ X.(11.29)
¨
fk+1 (Zk+1 |X)
Here, the integrals are set integrals, but they must be defined in terms of the integral
on the augmented single-target state space, (11.4):
∫
¨ X
¨
f (X)δ
(11.30)
∫
∑ 1
=
f ({ẍ1 , ..., ẍn })dẍ1 · · · dẍn
n!
n≥0
∑ 1 ∑ ∫
=
f ({(o1 , x1 ), ..., (on , xn )})dx1 · · · dxn . (11.31)
n! o ,...,o
n≥0
11.4
1
n
JUMP-MARKOV PHD FILTER
The purpose of this section is to apply this top-down perspective to the derivation of
a jump-Markov PHD filter. This filter was first proposed by Pasha, Vo, Tuan, and
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Ma [234], [232], [233], [297]. In their approach, the jump-Markov PHD filter is the
classical PHD filter, but defined on augmented states ẍ = (o, x). Thus its time- and
measurement-update equations can be obtained by substituting (o, x) whenever x
occurs in the classical PHD filter equations in Sections 8.4.1 and 8.4.3.
The following topics are considered:
1. Section 11.4.1: Modeling assumptions for the jump-Markov PHD filter.
2. Section 11.4.2: Time update equations for the jump-Markov PHD filter.
3. Section 11.4.3: Measurement update equations for the jump-Markov PHD
filter.
4. Section 11.4.4: Multitarget state estimation for the jump-Markov PHD filter.
11.4.1
Jump-Markov PHD Filter: Models
The jump-Markov PHD filter is based on the following motion and measurement
models:
• Jump-Markov state transition density:
fk+1|k (o, x|o′ , x′ ) = χo,o′ · fk+1|k (x|o′ , x′ )
(11.32)
where fk+1|k (x|o′ , x′ ) is the Markov transition density corresponding to
mode o′ ; and where χo,o′ is the transition matrix for the mode.
• Mode-dependent target probability of survival: pS (o′ , x′ ) abbr.
= pS,k+1|k (o′ , x′ ).
• Mode-dependent PHD for target appearances: bk+1|k (o, x).
• Mode-dependent PHD for target spawning: bk+1|k (o, x|o′ , x′ ).
• Mode-dependent likelihood function: Lz (o, x) abbr.
= fk+1 (z|o, x).
• Mode-dependent probability of detection: pD (o, x) abbr.
= pD,k+1 (o, x).
We must also consider the Poisson clutter process. It turns out that, if we are
to achieve a computationally reasonable PHD filter, it cannot be mode dependent.
For, assume to the contrary that the clutter intensity function is mode dependent:
κok+1 (z) = λok+1 · cok+1 (z).
(11.33)
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Then expressed in more complete notation, clutter will be dependent upon the
single-target state variable ẍ = (o, x):
κk+1 (z|o, x) = κok+1 (z).
State-dependent Poisson clutter models for PHD filters were considered in [179],
pp. 424-426. The PHD filter equations corresponding to this model were given
in Section 8.7. There it was shown that the measurement-update equation for this
filter involves a combinatorial sum over all partitions of the current measurement
set. Consequently, if we want a jump-Markov PHD filter that is both theoretically
rigorous and computationally practical, the Poisson clutter RFS cannot be allowed
to depend on the jump variable o. Thus we have one more model:
• Mode-independent clutter intensity function:
κk+1 (z) = λk+1 · ck+1 (z)
(11.34)
where, as usual, λk+1 is the clutter rate and ck+1 (z) is the clutter spatial
distribution.
These considerations have the following consequence: the jump Markov
PHD filter is just an ordinary PHD filter, but defined on augmented states ẍ = (o, x)
rather than purely kinematic states x. Thus its time- and measurement-update
equations can be obtained by substituting (o, x) whenever x occurs in the
conventional PHD filter equations in Sections 8.2.2 and 8.4.3.
11.4.2
Jump-Markov PHD Filter: Time Update
The time-update equations for the jump-Markov PHD filter are:
Dk+1|k (o, x) = bk+1|k (o, x)+
∑∫
Fk+1|k (o, x|o′ , x′ )·Dk|k (o′ , x′ )dx′ (11.35)
o′
where the PHD filter pseudo-Markov density is
Fk+1|k (o, x|o′ , x′ ) = bk+1|k (o, x|o′ , x′ ) + χo,o′ · pS (o′ , x′ ) · fk+1|k (x|o′ , x′ ).
(11.36)
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323
The expected number of predicted targets is, therefore,
=
(11.37)
Nk+1|k
∑∫
Dk+1|k (o, x)dx
o
=
B
Nk+1|k
+
∑∫ (
B
Nk+1|k
(x′ ) + pS (o′ , x′ )
)
(11.38)
o′
′
·Dk|k (o′ , x )dx′
where
B
Nk+1|k
=
∑∫
bk+1|k (o, x)dx
(11.39)
o
is the expected number of appearing targets and where
∑∫
B
′
′
Nk+1|k (o , x ) =
bk+1|k (o, x|o′ , x′ )dx
(11.40)
o
is the expected number of targets spawned by a target with joint state (o′ , x′ ).
11.4.3
Jump–Markov PHD Filter: Measurement Update
The measurement-update equations for the jump-Markov PHD filter are:
∑ pD (o, x) · Lz (o, x)
Dk+1|k+1 (o, x)
= 1 − pD (o, x) +
Dk+1|k (o, x)
κk+1 (z) + τk+1 (z)
(11.41)
z∈Zk+1
where
τk+1 (z) =
∑∫
pD (o, x) · Lz (o, x) · Dk+1|k (o, x)dx.
(11.42)
o
The expected number of updated target tracks is
∑
τk+1 (z)
κk+1 (z) + τk+1 (z)
(11.43)
(1 − pD (o, x)) · Dk+1|k (o, x)dx.
(11.44)
Nk+1|k+1 = Dk+1|k [1 − pD ] +
z∈Zk+1
where
Dk+1|k [1 − pD ] =
∑∫
o
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Remark 52 (Computational complexity) The measurement-update step consists
of conventional PHD filter measurement updates for each PHD Dk+1|k+1 (o, x),
for o = 1, ..., O. Thus the computational complexity is O(mnO) where m is the
current number of measurements and n is the current number of tracks.
11.4.4
Jump-Markov PHD Filter: State Estimation
State-estimation for the jump-Markov PHD filter combines single-target jumpMarkov state estimation, as described in Section 11.2.2, with PHD filter state
estimation, as considered in Section 8.4.4.
The purpose of state estimation for jump-Markov systems is, usually, to more
accurately estimate target states using multiple motion models. It is usually not
necessary to determine which model is applicable to which target at any given time.
Therefore, the jump variable o can be integrated out as a nuisance variable to get
the PHD for targets alone:
=
(11.45)
Dk+1|k+1 (x)
∑
Dk+1|k+1 (o, x)
o
=
∑
o

1 − pD (o, x) +
∑
z∈Zk+1

pD (o, x) · Lz (o, x) 
κk+1 (zj ) + τk+1 (z)
(11.46)
·Dk+1|k (o, x).
Given this, one can employ the usual PHD estimation process. That is,
determine the expected number of targets, Nk+1|k+1 as in (11.43). Then round
Nk+1|k+1 off to the nearest integer ν, and find states x1 , ..., xν corresponding to
the ν largest suprema of Dk+1|k+1 (x).
11.5
JUMP-MARKOV CPHD FILTER
In this section, the top-down perspective of Section 11.3 is applied to the derivation
of a jump-Markov CPHD filter, as proposed in [170]. The following topics are
considered:
1. Section 11.5.1: Modeling assumptions for the jump-Markov CPHD filter.
2. Section 11.5.2: Time update equations for the jump-Markov CPHD filter.
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325
3. Section 11.5.3: Measurement update equations for the jump-Markov CPHD
filter.
4. Section 11.5.4: Multitarget state estimation for the jump-Markov CPHD
filter.
11.5.1
Jump-Markov CPHD Filter: Modeling
The motion and measurement models for the jump-Markov CPHD filter are the
same as those for the jump-Markov PHD filter (Section 11.4.1), with the following
exceptions:
• As with the classical CPHD filter, there is no target-spawning model.
• Mode-independent cardinality distribution for target appearance: pB
k+1|k (n),
where it must be the case that the two different ways of computing the targetbirth rate are the same:
∑
∑∫
B
Nk+1|k
=
n · pB
(n)
=
bk+1|k (o, x)dx.
(11.47)
k+1|k
o
n≥0
• Mode-independent cardinality distribution for clutter: pκk+1 (m) and modeindependent clutter intensity function κk+1 (z), where it must be the case
that the two different ways of computing the clutter rate are the same:
λk+1 =
∑
m · pκk+1 (m) =
∫
κk+1 (z)dz.
(11.48)
m≥0
As was explained in Section 11.4.1, the clutter RFS cannot be mode dependent if we are to achieve a computationally tractable filter. Thus (11.48) must be
independent of the mode variable o.
11.5.2
Jump-Markov CPHD Filter: Time Update
As was the case with the jump-Markov PHD filter, the jump Markov CPHD filter
is just an ordinary CPHD filter, but defined on augmented states ẍ = (o, x). Thus
its time- and measurement-update equations can be obtained by substituting (o, x)
whenever x occurs in the classical CPHD filter equations in Sections 8.5.2 and
8.5.4.
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We are given a spatial distribution sk|k (o, x) and a cardinality distribution
pk|k (n). We are to determine the predicted spatial distribution sk+1|k (o, x), the
predicted expected number of targets Nk+1|k , and either the predicted cardinality
distribution pk+1|k (n) or the predicted p.g.f. Gk+1|k (x). The predicted spatial
distribution is given by
∫
(
)
∑
bk+1|k (o, x) + Nk|k o′ χo,o′ pS (o′ , x′ )
·fk+1|k (x|o′ , x′ ) · sk|k (o′ , x′ )dx′
sk+1|k (o, x) =
(11.49)
Bk+1|k + Nk|k · ψk
or, alternatively, the predicted PHD is given by
Dk+1|k (o, x)
=
bk+1|k (o, x) +
∑
χo,o′
∫
pS (o′ , x′ )
(11.50)
o′
·fk+1|k (x|o′ , x′ ) · Dk|k (o′ , x′ )dx′
where
ψk = sk|k [pS ] =
∑∫
pS (o, x) · sk|k (o, x)dx.
(11.51)
o
The predicted cardinality distribution and its corresponding p.g.f. are given by
Gk+1|k (x)
=
pk+1|k (n)
=
GB
k+1 (x) · Gk|k (1 − ψk + ψk · x)
∑
pk+1|k (n|n′ ) · pk|k (n′ )
(11.52)
(11.53)
n′ ≥0
where
pk+1|k (n|n′ ) =
n
∑
′
i
n −i
pB
.
k+1|k (n − i) · Cn′ ,i · ψk (1 − ψk )
(11.54)
i=0
The predicted expected number of targets is
B
Nk+1|k = Nk+1|k
+ Nk|k · ψk .
11.5.3
(11.55)
Jump-Markov CPHD Filter: Measurement Update
We are given a predicted spatial distribution sk+1|k (o, x) and a predicted cardinality distribution pk+1|k (n), such that the two different ways of computing the
Jump-Markov PHD/CPHD Filters
327
expected number of targets are the same:
Nk+1|k =
∑∫
sk+1|k (o, x)dx =
o
∑
(11.56)
n · pk+1|k (n).
n≥0
We are to determine the measurement-updated spatial distribution sk+1|k+1 (o, x),
the measurement-updated expected number of targets Nk+1|k+1 , and either the
measurement-updated cardinality distribution pk+1|k+1 (n) or the measurementupdated p.g.f. Gk+1|k+1 (x).
The spatial distribution is given by
(11.57)
sk+1|k+1 (o, x) = LZk+1 (o, x) · sk+1|k (o, x)
where the CPHD filter pseudolikelihood function is

1
LZk+1 (o, x) =
Nk+1|k+1


ND
∑
+
(1 − pD (o, x)) · L Zk+1
pD (o,x)·Lz (o,x) D
· LZk+1 (z)
z∈Zk+1
ck+1 (z)
 (11.58)
where
( ∑m
)
( ∑m
)
κ
j=0 (m − j)! · pk+1 (m − j)
(j+1)
·σj (Zk+1 ) · Gk+1|k (ϕk )
ND
L Zk+1
=
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
( ∑m−1
κ
i=0 (m − i − 1)! · pk+1 (m − i − 1)
(i+1)
·σi (Zk+1 − {zj }) · Gk+1|k (ϕk )
( ∑m
)
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
D
LZk+1 (zj )
=
(l)
Gk+1|k (ϕk )
=
∑
(11.59)
)
pk+1|k (n) · l! · Cn,l · ϕn−l
k
(11.60)
(11.61)
n≥l
(j+1)
Gk+1|k (ϕk )
=
∑
n≥j+1
pk+1|k (n) · (j + 1)! · Cn,j+1 · ϕn−j−1
k
(11.62)
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and where
ϕk
=
=
sk+1|k [1 − pD ]
∑∫
(1 − pD (o, x)) · sk+1|k (o, x)dx
(11.63)
(11.64)
o
(
σi (Zk+1 )
=
σm,i
σi (Zk+1 − {zj })
=
σm−1,i
)
τ̂k+1 (z1 )
τ̂k+1 (zm )
, ...,
ck+1 (z1 )
ck+1 (zm )


τ̂?
τ̂k+1 (z1 )
k+1 (zj )
 ck+1 (z1 ) , ..., ck+1 (zj ) 
(11.65)
.
(11.66)
k+1 (zm )
, ..., τ̂ck+1
(zm )
Also, the measurement-updated number of targets is
ND
Nk+1|k+1 = ϕk · L Zk+1 +
∑ τ̂k+1 (z) D
· LZk+1 (z)
ck+1 (z)
(11.67)
z∈Zk+1
where
τ̂k+1 (z) =
∑∫
pD (o, x) · Lz (o, x) · sk+1|k (o, x)dx.
(11.68)
o
The measurement-updated cardinality distribution and p.g.f. are, respectively,
pk+1|k+1 (n)
=
Gk+1|k+1 (x)
=
ℓZ (n) · pk+1|k (n)
∑ k+1
l≥0 ℓZk+1 (l) · pk+1|k (l)
( ∑m j
)
κ
j=0 x · (m − j)! · pk+1 (m − j)
·G(j) (x · ϕk ) · σj (Zk+1 )
( ∑m
)
κ
i=0 (m − i)! · pk+1 (m − i)
·G(i) (ϕk ) · σi (Zk+1 )
where the cardinality pseudolikelihood function is
( ∑
)
min{m,n}
κ
(m
−
j)!
·
p
(m
−
j)
·
j!
·
C
n,j
k+1
j=0
·ϕn−j
· σj (Zk+1 )
k
) .
ℓZk+1 (n) = ( ∑m
κ
l=0 (m − l)! · pk+1 (m − l) · σl (Zk+1 )
(l)
·Gk+1|k (ϕk )
(11.69)
(11.70)
(11.71)
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Remark 53 (Computational complexity) The measurement-update step requires
a conventional CPHD filter measurement-update for each of the spatial distributions sk+1|k+1 (o, x), for o = 1, ..., O. Thus the computational complexity is
O(m3 nO) where m is the current number of measurements and n is the current
number of tracks and O is the number of jump modes.
11.5.4
Jump-Markov CPHD Filter: State Estimation
State estimation for the jump-Markov CPHD filter is the same as for the jumpMarkov PHD filter (Section 8.4.4), except that the target-number estimate is the
MAP estimate of the cardinality distribution.
11.6
VARIABLE STATE SPACE JUMP-MARKOV CPHD FILTERS
The jump-Markov CPHD filter just described is based on an implicit assumption:
the target state space for every motion model must be the same. However, this will
not necessarily be adequate for many practical applications. The following is a
simple example of two common single-target motion models with different singletarget state spaces:
1. Constant-velocity (CV) motion model: During the time-interval ∆t =
tk+1 − tk , the target is assumed to move along a straight line path from its
current position p = (x, y)T , with its current velocity v = (vx , vy )T . The
state space consists of all vectors (x, y, vx , vy )T ∈ R4 . The state transition
function is defined as

 


x
1 0 ∆t 0
x
 y   0 1 0 ∆t   y 
 

.
φk+1|k 
(11.72)
 vx  =  0 0 1
0   vx 
vy
0 0 0
1
vy
2. Coordinated-turn (CT) motion model: The target is assumed to move from
its current position p = (x, y)T to the left or the right along a circular arc
with unknown angular turn rate ω (radians per second). The state space
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consists of all vectors (ω, x, y, vx , vy )T ∈ R5 . The state transition model is

 

ω
ω
 x   x + vx · sin ω∆t − vy · 1−cos ω∆t 
ω
ω

 

1−cos ω∆t
 
φk+1|k 
(11.73)
+ vy · sin ωω∆t 
 y  =  y + vx ·
.
ω
 vx  

vx · cos ω∆t − vy · sin ω∆t
vy
vx · sin ω∆t + vy · cos ω∆t
Chen Xin, McDonald, and Kirubarajan have devised a means of addressing
such problems [36]. The purpose of this section is to describe their approach.
The basic idea is as follows. Let o = 1, ..., O be the model modes and let
o
X be the target state space corresponding to the mode o. Chen Xin et al. define the
multimode state space1
1
O
¨ = X ⊎ ... ⊎ X,
X
(11.74)
where as usual ‘⊎’ denotes disjoint union. They then construct a conventional
¨ can have any of O
CPHD filter on this state space.2 An element ẍ of X
o
o
o
possible forms: ẍ =x for x ∈ X and o = 1, ..., O. Thus the integral of a
¨ has the form
function f¨(ẍ) on X
∫
∫
∫
∑∫
o
o
1
1
O
O
¨(x)d
¨(x)d
f¨(ẍ)dẍ =
f
x
=
f
x
+
...
+
f¨(x)dx.
(11.75)
o
1
O
o
X
X
X
Chen et al.’s approach raises the following issues, which will be addressed in
the subsections that follow:
¨ probability of
• Modeling: the following functions must be defined on X:
target survival p̈S (ẍ); Markov transition density f¨k+1|k (ẍ|ẍ′ ); target
appearance PHD ¨bk+1|k (ẍ); probability of detection p̈D (ẍ); and likelihood
¨ z (ẍ) = f¨k+1 (z|ẍ).
function L
• Multimode PHDs: There will be a PHD
o
o
o
(k)
¨ k|k (x|Z
D k|k (x|Z (k) ) def.
=D
)
(11.76)
o
defined on a different state space X for each choice o = 1, ..., O. This PHD
describes the density of the targets that are in mode o at time tk .
1
2
¨ is being used differently than earlier in the chapter, where previously
Note that here the symbol X
¨ = {1, ..., O} × X. The intended meaning in this section will be clear from context.
X
This approach is related to that used for the “clutter agnostic” CPHD filters of Chapter 18—which
also involves a disjoint union of state spaces.
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331
• Cardinality estimation: The CPHD filter cardinality distribution has the form
1
O
o
p̈k|k (n̈|Z (k) ) where n̈ = n + ... + n, where n is the number of targets that
are in mode o, for o = 1, ..., O. Thus the MAP estimate
n̈k|k = arg sup p̈k|k (n̈|Z (k) )
(11.77)
n̈
o
provides an estimate of n̈ but not of the individual n. Consequently, it
is necessary to determine a formula for the following item: the cardinality
o
o
distribution pk|k (n|Z (k) ) for the number of targets that are in mode o.
Given this, one could then take
o
o
o
nk|k = arg sup pk|k (n|Z (k) )
(11.78)
o
n
to be the MAP estimate of the number of targets in state o. This issue is
addressed in Section 11.6.4.
The section is organized as follows:
1. Section 11.6.1: Modeling for the variable state–space CPHD filter.
2. Section 11.6.2: The time-update equations for the variable state space CPHD
filter.
3. Section 11.6.3: The measurement-update equations for the variable state
space CPHD filter.
4. Section 11.6.4: Multitarget state estimation for the variable state space CPHD
filter.
11.6.1
Variable State Space CPHD Filters: Modeling
The following models are assumed:
• Probability of target survival:
o
o
o
o
o
p̈S (x) def.
= pS (x) abbr.
= pS,k+1|k (x)
o
(11.79)
o
where pS (x) is the probability of target survival for a target with mode o
o
and state x.
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Advances in Statistical Multisource-Multitarget Information Fusion
• Markov transition density:
o,o′
o o′
o o′
f¨k+1|k (x| x ′ ) def.
= χo,o′ · f k+1|k (x| x ′ )
(11.80)
where χo,o′ is the probability that a target in mode o′ will transition to
o,o′
o o′
mode o; and where f k+1|k (x| x ′ ) is the probability (density) that a target
o′
o
that had mode o′ and state x ′ at time tk will have the state x
transitions from mode o′ to mode o.
if it
• i.i.d.c. target appearance process: the PHD of the target appearance process
is
o
o
o
def.
¨bk+1|k (x)
= bk+1|k (x)
(11.81)
o
o
where bk+1|k (x) is the PHD of the target appearance process for mode o;
where
∑
1
1
OB
O
def.
p̈B
pB
(11.82)
k+1|k (n̈) =
k+1|k (n) · · · p k+1|k (n);
1
O
n+...+ n=n̈
o
o
o
and where pB
k+1|k (n)
appear.
is the probability that n targets in mode o will
• Probability of detection:
o
o
o
o
o
p̈D (x) def.
= pD (x) abbr.
= pD,k+1 (x)
(11.83)
o
o
where pD (x) is the probability of target detection for a target with mode o
o
and state x.
• Likelihood function:
o
o
o
o
o
f¨k+1 (z|x) def.
= Lz (x) abbr.
= f k+1 (z|x)
(11.84)
o
o
where f k+1 (z|x) is the probability (density) that a target with mode o
o
and state x will generate measurement z, given that the target has been
detected.
• i.i.d.c. clutter process: As with a conventional CPHD filter, the spatial
distribution of the clutter is ck+1 (z) and the probability that m clutter
measurements will be generated is pκk+1 (m).
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333
Given these preliminaries, the variable state space CPHD filter consists of
O + 1 tightly coupled filters:
... →
p̈k|k (n̈|Z (k) )
→
p̈k+1|k (n̈|Z (k) )
... →
D k|k (x|Z (k) )
→
D k+1|k (x|Z (k) )
..
.
..
.
..
.
..
.
1
1
1
→
↑↓
p̈k+1|k+1 (n̈|Z (k+1) )
→
↑↓
..
.
D k+1|k+1 (x|Z (k+1) )
1
1
→ ...
1
→ ...
..
.
↑↓
O
... →
O
O
D k|k (x|Z (k) )
→
O
O
D k+1|k (x|Z (k) )
→
O
D k+1|k+1 (x|Z (k+1) )
→ ...
Here, the top row is a filter on the cardinality distribution p̈k|k (n̈|Z (k) ) on the total
o
o
number n̈ of targets. The other rows are filters for the PHDs D k|k (x|Z (k) ) of
o
those targets that are in mode o and have state-variable x.
11.6.2
Variable State Space CPHD Filters: Time Update
From Section 8.5.2 we know that the time-update equations for the CPHD filter
¨ are (suppressing dependence on the measurementdefined on the state space X
(k)
history Z ):
p̈k+1|k (n̈)
=
∑
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
n̈′ ≥0
¨ k+1|k (ẍ)
D
= ¨bk+1|k (ẍ) +
∫
¨ k|k (ẍ′ )dẍ′
·D
p̈S (ẍ′ ) · f¨k+1|k (ẍ|ẍ′ )
(11.85)
(11.86)
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Advances in Statistical Multisource-Multitarget Information Fusion
where
ψ¨k
=
p̈k+1|k (n̈|n̈′ )
=
∫
(11.87)
p̈S (ẍ) · s̈k|k (ẍ)dẍ
n̈
∑
¨i
¨ n̈′ −i
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i · ψk (1 − ψk )
(11.88)
i=0
s̈k|k (ẍ)
=
¨k|k
N
=
¨ k|k (ẍ)
D
¨k|k
N
∫
∑
¨ k|k (ẍ)dẍ.
n̈ · p̈k|k (n̈) = D
(11.89)
(11.90)
n̈≥0
Abbreviate
o
o
o
¨ k|k (x|Z (k) )
D k|k (x) abbr.
= D
o
o
(11.91)
o
¨ k+1|k (x|Z (k) )
D k+1|k (x) abbr.
= D
(11.92)
to be the PHDs for targets that are in mode o. Then given the modeling assumptions
in Section 11.6.1, the time-update equations are:
∑
p̈k+1|k (n̈) =
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(11.93)
n̈′ ≥0
o
o
o
o
D k+1|k (x)
=
bk+1|k (x) +
∑
χ
o,o′
o
o
∫
o′
o,o′
o′
o o′
p S ( x ′ ) · f k+1|k (x| x ′ ) (11.94)
o′
o
o′ ′
o′ ′
·D k|k ( x )d x
where
ψ¨k
=
1 ∑
¨k|k
N
∫
o
o
o
pS (x) · D k|k (x)dx
(11.95)
¨i
¨ n̈′ −i
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i · ψk (1 − ψk )
(11.96)
o
p̈k+1|k (n̈|n̈′ )
=
n̈
∑
i=0
1
¨k|k
N
=
o
N k|k
=
O
N k|k + ... + N k|k
∫
o
o
o
N k|k (x)dx.
(11.97)
(11.98)
Jump-Markov PHD/CPHD Filters
11.6.3
335
Variable State Space CPHD Filters: Measurement Update
From Section 8.5.4 we know that the measurement-update equations for the CPHD
¨ are
filter defined on the state space X
p̈k+1|k+1 (n̈)
=
¨ k+1|k+1 (ẍ)
D
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
¨ Z (ẍ) · D
¨ k+1|k (ẍ)
L
(11.99)
(11.100)
k+1
where
ℓ¨Zk+1 (n̈)
=
( ∑
min{m,n̈}
(m − j)! · pκk+1 (m − j) · j! · Cn,j
j=0
·ϕ¨n−j
· σ̈j (Zk+1 )
k
( ∑m
κ
l=0 (m − l)! · pk+1 (m − l) · σ̈l (Zk+1 )
(l)
¨
·Gk+1|k (ϕ¨k )
¨ Z (ẍ)
L
k+1
1
=
¨k+1|k
N


ND
(1 − p̈D (ẍ)) · L Zk+1
+
∑m
¨ z (ẍ) D
p̈D (ẍ)·L
j
· LZk+1 (zj )
j=1
ck+1 (zj )
(11.101)
)
)

 (11.102)
and where
( ∑m
ND
L Zk+1
=
(
( ∑m−1
(m − i − 1)! · pκk+1 (m − i − 1)
¨ (i+1) (ϕ¨k )
·σ̈i (Zk+1 − {zj }) · G
k+1|k
( ∑m
)
κ
l=0 (m − l)! · pk+1 (m − l)
¨ (l) (ϕ¨k )
·σ̈l (Zk+1 ) · G
k+1|k
i=0
D
LZk+1 (zj )
=
)
κ
j=0 (m − j)! · pk+1 (m − j)
¨ (j+1) (ϕ¨k )
·σ̈j (Zk+1 ) · G
k+1|k
)
∑m
κ
(m
−
l)!
·
p
k+1 (m − l)
l=0
¨ (l) (ϕ¨k )
·σ̈l (Zk+1 ) · G
k+1|k
(11.103)
)
(11.104)
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¨ (l) (ϕ¨k )
G
k+1|k
=
∑
p̈k+1|k (n̈) · l! · Cn̈,l · ϕ¨n̈−l
k
(11.105)
n̈≥l
¨ (j+1) (ϕ¨k )
G
k+1|k
∑
=
p̈k+1|k (n̈) · (j + 1)! · Cn̈,j+1 · ϕ¨n̈−j−1
(11.106)
k
n̈≥j+1
and where
∫
1
ϕ¨k
=
σ̈i (Zk+1 )
=
σ̈i (Zk+1 − {zj })
=
τ̈k+1 (z)
=
¨ k+1|k (ẍ)dẍ (11.107)
(1 − p̈D (ẍ)) · D
¨k+|k
N
(
)
τ̈k+1 (z1 )
τ̈k+1 (zm )
σm,i
, ...,
(11.108)
ck+1 (z1 )
ck+1 (zm )


τ̈?
τ̈k+1 (z1 )
k+1 (zj )
, ..., ck+1
(zj ) 
σm−1,i  ck+1 (z1 ) τ̈ (z
(11.109)
m)
k+1
, ..., ck+1
(zm )
∫
1
¨ z (ẍ) · D
¨ k+1|k (ẍ)dẍ.(11.110)
p̈D (ẍ) · L
¨k+|k
N
Also,
ND
¨k+1|k+1 = ϕ¨k · L Z
N
+
k+1
∑ τ̈k+1 (z) D
· LZk+1 (z).
ck+1 (z)
(11.111)
z∈Zk+1
Given the modeling assumptions in Section 11.6.1, these equations take the
form
p̈k+1|k+1 (n̈)
o
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
=
LZk+1 (x) · D k+1|k (x)
o
o
D k+1|k+1 (x)
o
o
(11.112)
o
(11.113)
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337
where
ℓ¨Zk+1 (n̈)
=
( ∑
(
o
min{m,n̈}
(m − j)! · pκk+1 (m − j) · j! · Cn̈,j
j=0
·ϕ¨kn−j · σ̈j (Zk+1 )
)
∑m
κ
l=0 (m − l)! · pk+1 (m − l) · σ̈l (Zk+1 )
¨ (l) (ϕ¨k )
·G
k+1|k

1
o
LZk+1 (x)
=
¨k+1|k
N

ND
o
o
(1 − pD (x)) · L Zk+1
+
∑m
o
o
o
o
pD (x)·Lzj (x) D
· LZk+1 (zj )
j=1
ck+1 (zj )
(11.114)
)

 (11.115)
and where
1
¨k+1|k
N
=
O
(11.116)
N k+1|k + ... + N k+1|k
ND
=
(11.117)
L Zk+1
∑m
κ
¨ (j+1) ¨
j=0 (m − j)! · pk+1 (m − j) · σ̈j (Zk+1 ) · Gk+1|k (ϕk )
∑m
κ
¨
¨ (l)
l=0 (m − l)! · pk+1 (m − l) · σ̈l (Zk+1 ) · Gk+1|k (ϕk )
( ∑m−1
κ
i=0 (m − i − 1)! · pk+1 (m − i − 1)
...
¨ (i+1) (ϕ¨k )
· σ i (Zk+1 − {zj }) · G
k+1|k
( ∑m
)
κ
l=0 (m − l)! · pk+1 (m − l)
¨ (l) (ϕ¨k )
·σ̈l (Zk+1 ) · G
k+1|k
D
LZk+1 (zj )
=
¨ (l) (ϕ¨k )
G
k+1|k
=
∑
p̈k+1|k (n̈) · l! · Cn̈,l · ϕ¨n̈−l
k
)
(11.118)
(11.119)
n̈≥l
¨ (j+1) (ϕ¨k )
G
k+1|k
=
∑
n̈≥j+1
p̈k+1|k (n̈) · (j + 1)! · Cn̈,j+1 · ϕ¨n̈−j−1
(11.120)
k
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Advances in Statistical Multisource-Multitarget Information Fusion
and where
ϕ¨k
(11.121)
=
∑∫
σ̈i (Zk+1 )
=
σ̈i (Zk+1 − {zj })
=
1
o
o
o
o
(1 − pD (x)) · D k+1|k (x)dx
¨k+1|k
N
o
(
)
τ̈k+1 (zm )
τ̈k+1 (z1 )
σm,i
, ...,
(11.122)
ck+1 (z1 )
ck+1 (zm )


τ̈?
τ̈k+1 (z1 )
k+1 (zj )
,
...,
ck+1 (zj ) 
σm−1,i  ck+1 (z1 ) τ̈ (z
(11.123)
m)
k+1
, ..., ck+1
(zm )
1
τ̈k+1 (z)
=
¨k+1|k
N
o
o
∑∫ o o o o
pD (x) · Lz (x)
(11.124)
o
o
o
·D k+1|k (x)dx.
Also, for the next time-update step we need to know
ND
¨k+1|k+1 = ϕ¨k · L Z
N
+
k+1
∑ τ̈k+1 (z) D
· LZk+1 (z).
ck+1 (z)
(11.125)
z∈Zk+1
11.6.4
Variable State Space CPHD Filters: State Estimation
State estimation for the variable state space CPHD filter is the same as for the
conventional CPHD filter (see Section 8.5.5). First, estimate the total number of
targets (irrespective of the mode that they are in) using a MAP estimator:
ν = arg sup p̈k|k (n̈|Z (k) ).
(11.126)
n̈
o1
oν
Given this, the states of the targets can be estimated by finding the states x 1 , ..., x ν
¨ k|k (ẍ|Z (k) ). To accomplish this, one
corresponding to the ν largest peaks of D
must determine the peaks of the mode-specific PHDs
1
1
O
O
D k|k (x|Z (k) ), ..., D k|k (x|Z (k) )
and then find the ν largest of those. This requires that we first estimate the number
of targets that are in any given mode o.
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339
Towards this end, Chen, McDonald, and Kirubarajan have shown that it is
possible to estimate the number of targets that are in a given mode o. Let
¨ k|k (x) be
p̈k|k (n̈|Z (k) ) be the cardinality distribution at time tk and let G
its p.g.f. Let
∫
o
o
N k|k =
D k|k (x|Z (k) )dx
o
o
(11.127)
be the expected number of targets that are in mode o, and let
o
N k|k
o
(11.128)
r k|k = 1
O
N k|k + ... + N k|k
be the fraction of targets that are in mode o. Then the cardinality distribution for
those targets that are in mode o is:
oo
o
o
pk|k (n|Z
(k)
r nk|k
)=
o
n!
o
(n)
o
¨ (1 − r k|k ).
·G
k|k
(11.129)
Given this, the MAP estimate of the number of targets that have mode o is
o
o
o
nk|k = arg sup pk|k (n|Z (k) ).
(11.130)
o
n
¨ k|k be the random joint state set for
To see why these equations are true, let Ξ
¨ k|k
all targets, irrespective of mode. Then the (random) number of elements of Ξ
that are in mode o is
o
¨ k|k ∩ X|.
|Ξ
(11.131)
o
¨ k|k ∩ X is the cardinality distribution of the number
The cardinality distribution of Ξ
o
¨ k|k ∩ X is
of actual targets in mode o. According to (4.137), the p.g.f. of Ξ
G¨
o
Ξk|k ∩X
(11.132)
(x) = GΞ¨ k|k [1 − 1 o + x · 1 o ].
X
X
¨ k+1|k+1 is i.i.d.c., this becomes
Since Ξ
G¨
o
(x)
=
Ξk|k ∩X
¨ k|k (N
¨ −1 D
¨
o
o
G
k|k k|k [1 − 1 + x · 1 ])
(11.133)
¨ k|k (1 − N
¨ −1 D
¨
o
G
k|k k|k [1 ]
(11.134)
X
=
X
X
¨ −1 D
¨
o
+x · N
k|k k|k [1X ])
=
¨ k|k (1 − ro k|k + x · ro k|k ).
G
(11.135)
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The last equation follows because
∫
∫ o
o
o
¨ k|k (ẍ)dẍ
o D
D k|k (x)dx
−1
X
¨ D
¨ k|k [1 o ] =
N
=
(11.136)
1
O
1
O
k|k
X
N k|k + ... + N k|k
N k|k + ... + N k|k
o
N k|k
o
=
1
O
(11.137)
= r k|k .
N k|k + ... + N k|k
Thus the cardinality distribution for the number of targets in mode o is:
[
]
o
1 dn ¨
o
o
o
o
pk|k (n) =
(11.138)
o
o Gk|k (1 − r k|k + x · r k|k )
n! dxn
x=0
 o

o
o
r nk|k
¨ (n) (1 − ro k|k + x · ro k|k )
=  o ·G
(11.139)
k|k
n!
x=0
o
on
r k|k
=
o
n!
11.7
o
(n)
o
¨ (1 − r k|k ).
·G
k|k
(11.140)
IMPLEMENTING JUMP-MARKOV PHD/CPHD FILTERS
The purpose of this section is to briefly describe the practical implementation
of jump-Markov PHD and CPHD filters. Gaussian mixture implementation is
addressed in Section 11.7.1, and sequential Monte Carlo (SMC) implementation
is addressed in Section 11.7.2.
11.7.1
Gaussian Mixture Jump-Markov PHD/CPHD Filters
Gaussian mixture implementation of jump-Markov PHD and CPHD filters is a
direct generalization of Gaussian mixture implementation of the classical PHD and
CPHD filters, as described in Sections 9.5.4 and 9.5.5. For the sake of conceptual
clarity, only the jump-Markov PHD filter will be addressed. Gaussian-mixture
implementation of the jump-Markov CPHD filter can be accomplished similarly.
11.7.1.1
Jump-Markov GM-PHD Filter Models
The GM implementation of the PHD filter requires the following modeling assumptions:
Jump-Markov PHD/CPHD Filters
341
• Probability of target survival: does not depend on the target kinematic state
but does depend on the mode, pS,k+1 (o, x) = pS,k+1 (o) abbr.
= poS .
• Single-target Markov transition density—is linear-Gaussian and mode dependent:
fk+1|k (x|o, x′ ) = NQok (x − Fko x′ ).
(11.141)
• Target-appearance PHD—is mode dependent, and is a Gaussian mixture for
each o:
B
νk+1|k
bk+1|k (o, x) =
∑
k+1|k
k+1|k
· NB k+1|k (x − bi,o
bi,o
(11.142)
)
i,o
i=1
in which case the expected number of appearing targets in mode o is
B
νk+1|k
B,o
Nk+1|k
=
∑
k+1|k
bi,o
(11.143)
,
i=1
whereas the total expected number of appearing targets is
νB
O k+1|k
∑
∑ k+1|k
B
Nk+1|k =
bi,o .
(11.144)
o=1 i=1
• Target-spawning PHD—is mode dependent and is a Gaussian mixture for
each o, o′ :
S
νk+1|k
′
′
bk+1|k (o, x|o , x ) =
∑
k+1|k
k+1|k ′
ej,o,o′ · NGk+1|k (x − Ej,o′
x ).
(11.145)
j,o′
j=1
• Probability of detection—is independent of the target kinematic state but dependent on the mode, pD,k+1 (o, x) = pD,k+1 (o) abbr.
= poD . (This assumption
can be removed using the approximation described in Section 9.5.6.)
• Sensor likelihood function: is possibly mode dependent and linear-Gaussian:
o
o
Lz (o, x) = fk+1 (z|o, x) = NRk+1
(z − Hk+1
x).
(11.146)
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• Clutter intensity function: is mode-independent, κk+1 (z) = λk+1 · ck+1 (z)
where λk+1 is the clutter rate and ck+1 (z) is the clutter spatial distribution.
The Gaussian mixture implementation of the jump-Markov PHD filter is
based on the assumption that, for each o and all k ≥ 0, Dk|k (o, x) and
Dk+1|k (o, x) can be represented as Gaussian mixtures:
νk|k
Dk|k (o, x)
∑
=
k|k
k|k
(11.147)
wi,o · NP k|k (x − xi,o )
i,o
i=1
νk+1|k
Dk+1|k (o, x)
∑
=
k+1|k
k+1|k
· NP k+1|k (x − xi,o
wi,o
).
(11.148)
i,o
i=1
Here,
o
Nk|k
=
∫
νk|k
Dk|k (o, x)dx =
∑
k|k
wi,o
(11.149)
i=1
is the expected number of targets that are in mode o, and
Nk|k =
k|k
O ν
∑
∑
k|k
(11.150)
wi,o
o=1 i=1
is the total expected number of targets, regardless of mode.
This means that the Gaussian-mixture representation of a PHD can be equivalently replaced by a system of Gaussian components
k|k
k|k
k|k
k|k ν
,O
k|k
(ℓi,o , wi,o , Pi,o , xi,o )i=1;o=1
k|k
where ℓi,o is the track label associated with each component.
11.7.1.2
Jump-Markov GM-PHD Filter Time Update
We are given a system of Gaussian components
k|k
k|k
k|k
k|k ν
,O
k|k
(ℓi,o , wi,o , Pi,o , xi,o )i=1;o=1
We are to determine formulas for the predicted system of Gaussian components
k+1|k
(ℓi,o
k+1|k
, wi,o
k+1|k
, Pi,o
k+1|k νk+1|k ,O
)i=1;o=1 .
, xi,o
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343
These will actually have the structure
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k νk|k +ν
B
,O
(ℓi.o,o′ , wi,o,o′ , Pi,o,o′ , xi,o,o′ )i=1;o,ok+1|k
,
′ =1
k+1|k
νk|k ,ν S
,O
k+1|k
(ℓi,j,o,o′ , wi,j,o,o′ , Pi,j,o,o′ , xi,j,o,o′ )i=1;j=1;o,o
′ =1
as defined by the following equations (as demonstrated in Section K.17):
• Time updated number of GM components:
B
S
νk+1|k = νk|k · O 2 + νk+1|k
· O 2 + νk|k · vk+1|k
· O2 .
(11.151)
Here, there are νk|k · O 2 components corresponding to persisting targets,
B
νk+1|k
·O 2 components corresponding to newly appearing targets, and
S
νk|k · vk+1|k
· O 2 components corresponding to spawned targets. The timeupdated components are indexed as follows:
=
1, ..., νk|k ; o = 1, ..., O
i
=
B
νk|k + 1, ..., νk|k + νk+1|k
;
i
=
1, ..., νk|k ;
i
(11.152)
′
o, o = 1, ..., O
(11.153)
o, o′ = 1, ..., O. (11.154)
S
j = 1, ..., vk+1|k
;
• Persisting-target GM components, for i = 1, ..., νk|k , o, o′ = 1, ..., O:
k+1|k
k|k
ℓi,o,o′
=
k+1|k
(11.155)
ℓi,o′
′
k|k
wi,o,o′
=
wi,o′ · χo,o′ · poS
(11.156)
′
k+1|k
xi,o,o′
k+1|k
Pi,o,o′
=
k|k
Fko xi,o′
(11.157)
=
′
′
′
k|k
Qok + Fko Pi,o′ (Fko )T .
(11.158)
B
• Appearing-target GM components, for i = νk|k + 1, ..., νk|k + νk+1|k
,
o = 1, ..., O:
k+1|k
ℓi,o
k+1|k
wi,o
k+1|k
xi,o
=
new label
(11.159)
=
k+1|k
bi−νk+1|k,o
(11.160)
=
k+1|k
bi−νk+1|k,o
(11.161)
k+1|k
=
Bi−νk+1|k,o .
(11.162)
k+1|k
Pi,o
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S
• Spawned-target GM components, for i = 1, ..., νk|k and j = 1, ..., vk+1|k
′
and o, o = 1, ..., O:
k+1|k
ℓi,j,o,o′
=
new label
=
ej,o,o′ · wi,o′
(11.164)
=
k+1|k k|k
Ej,o′ xi,o′
(11.165)
=
k+1|k
k+1|k k|k
k+1|k
Gj,o′ + Ej,o′ Pi,o′ (Ej,o′ )T .
(11.166)
k|k
k+1|k
wi,o,o′ ,j
k+1|k
xi,o,o′ ,j
k+1|k
Pi,o,o′ ,j
11.7.1.3
(11.163)
k|k
Jump-Markov GM-PHD Filter Measurement Update
We are given the predicted system of Gaussian components:
k+1|k
(ℓi,o
k+1|k
, wi,o
k+1|k νk+1|k ,O
)i=1;o=1 .
k+1|k
, Pi,o
, xi,o
We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | =
mk+1 . We are to determine formulas for the measurement-updated system of
Gaussian components
k+1|k+1
(ℓi,o
k+1|k+1
, wi,o
k+1|k+1
, Pi,o
k+1|k+1 νk+1|k+1,O
)i=1;o=1 .
, xi,o
This will actually have the structure
k+1|k+1 νk+1|k ,O
)i=1;o=1 ,
k+1|k+1
k+1|k+1
k+1|k+1
k+1|k+1 νk+1|k ,O,mk+1
(ℓi,o,j
, wi,o,j
, Pi,o,j
, xi,o,j
)i=1;o=1;j=1
k+1|k+1
(ℓi,o
k+1|k+1
, wi,o
k+1|k+1
, Pi,o
, xi,o
as defined by the following equations (as shown in Section K.17):
• Measurement updated number of GM components for the PHD:
νk+1|k+1 = νk+1|k · O + mk+1 · νk+1|k · O
(11.167)
where there are νk+1|k · O components for undetected tracks and mk+1 ·
νk+1|k · O components for detected tracks. The measurement-update components are indexed as follows:
i
=
1, ..., νk+1|k ; o = 1, ..., O
(11.168)
i
=
1, ..., νk+1|k ; o = 1, ..., O; j = 1, ..., mk+1 .
(11.169)
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345
• Measurement updated nondetection components: for i = 1, ..., νk+1|k and
o = 1, ..., O,
k+1|k+1
k+1|k
ℓi,o
(11.170)
=
ℓi,o
=
(1 − poD ) · wi,o
(11.171)
=
k+1|k
xi,o
(11.172)
=
k+1|k
Pi,o .
(11.173)
• Measurement updated detection components:
o = 1, ..., O and j = 1, ..., mk+1 ,
for i = 1, ..., νk+1|k and
k+1|k+1
k+1|k
wi,o
k+1|k+1
xi,o
k+1|k
Pi,o
k+1|k+1
k+1|k
ℓi,o,j
=
(11.174)
ℓi
νk+1|k
τk+1 (zj )
∑ ∑
=
o
k+1|k
wi,o
· poD
(11.175)
i=1
k+1|k
·NRo
k+1|k
o
o
(Hk+1
)T
k+1 +Hk+1 Pi,o
k+1|k+1
o
(zj − Hk+1
xi,o
k+1|k
xi,o,j
=
k+1|k+1
Pi,o,j
=
k+1
Ki,o
k+1|k
k+1
o
xi,o + Ki,o
(zj − Hk+1
xi,o
(
) k+1|k
k+1 o
I − Ki,o Hk+1 Pi,o
(11.176)
)
(11.177)
k+1|k
o
Pi,o (Hk+1
)T
=
(
k+1|k
o
· Hk+1
Pi,o
)
(11.178)
o
o
(Hk+1
)T + Rk+1
)−1
and
k+1|k+1
wi,o,j
(
(11.179)
)
k+1|k
wi,o
· poD
k+1|k
·NRo
k+1|k
o
o
(Hk+1
)T
k+1 +Hk+1 Pi,o
o
(zj − Hk+1
xi,o
)
.
=
κk+1 (zj ) + τk+1 (zj )
11.7.1.4
Jump-Markov GM-PHD Filter Multitarget State Estimation
State estimation for the GM-PHD filter is accomplished as indicated in Section
11.4.4. We are given the measurement-updated system
k+1|k+1
(ℓi,o
k+1|k+1
, wi,o
k+1|k+1
, Pi,o
k+1|k+1 νk+1|k+1 ,O
)i=1;o=1
, xi,o
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with associated PHD
νk+1|k+1
Dk+1|k+1 (o, x) =
∑
k+1|k+1
k+1|k+1
· NP k+1|k+1 (x − xi,o
wi,o
).
(11.180)
i,o
i=1
The total expected number of targets is
Nk+1|k+1 =
∑ νk+1|k+1
∑
o
k+1|k+1
wi,o
.
(11.181)
i=1
Let n be the integer nearest to Nk+1|k+1 and determine those n Gaussian
k+1|k+1
k+1|k+1
components for which wi
is largest. Then the associated xi
are the
k+1|k+1
track state estimates, and the associated Pi
are their track covariances.
11.7.2
Particle Implementation of Jump-Markov PHD and CPHD Filters
SMC implementation of jump-Markov PHD and CPHD filters is essentially the
same as SMC implementation of the classical PHD and CPHD filters. The primary
k|k
k|k
k|k νk|k
difference is that a particle system has the form {(oi , xi , wi )}i=1
rather
k|k
k|k νk|k
than {(xi , wi )}i=1
.
11.8
IMPLEMENTED JUMP-MARKOV PHD/CPHD FILTERS
A handful of researchers have addressed jump-Markov PHD and CPHD filters. All
have adopted a bottom-up theoretical approach. That is, they take the PHD filter or
the CPHD filter as their starting point, and then attempt to generalize it to jumpMarkov systems. The purpose of this section is to summarize this work:
1. Section 11.8.1: The jump-Markov PHD filter of Pasha, Vo, Tuan, and Ma.
2. Section 11.8.2: The IMM-like jump-Markov PHD filter of Punithakumar,
Kirubarajan, and Sinha.
3. Section 11.8.3: The Best-Fitting-Gaussian (BFG) PHD filter of Wenling and
Yingmin.
4. Section 11.8.4: The jump-Markov CPHD filter of Georgescu and Willett.
Jump-Markov PHD/CPHD Filters
347
5. Section 11.8.5: The Current Statistical Model (CSM) PHD filter of Mengjun,
Shaohua, Zhiguo, and Kangsheng.
6. Section 11.8.6: The variable state space CPHD filter of Chen Xin, McDonald,
and Kirubarajan.
11.8.1
Jump-Markov PHD Filter of Pasha et al.
This approach [234], [232], [233], [297] is consistent with a top-down approach—
that is, this jump-Markov PHD filter is identical to the one described in Section
11.4. The material in this section is drawn from [234].
The approach differs from that in Section 11.4 only in that the time-update
equations are slightly different. In order to implement the filter using Gaussian
mixture techniques, the authors examine specific target appearance and targetspawning models:
bk+1|k (o, x)
bk+1|k (o, x|o′ , x′ )
=
=
pk+1|k (o) · bk+1|k (x)
(11.182)
′
′
′
′
pk+1|k (o|x, o , x ) · bk+1|k (x|o , x ). (11.183)
That is, modes of appearing targets are independent of the targets themselves.
Likewise, the modes of spawned targets depend on the modes and states of the
targets that generated them, but not on the targets themselves.
11.8.2
IMM-Type JM-PHD Filter of Punithakumar et al.
These authors were the first to propose a jump-Markov PHD filter, in 2004 [243],
[244]. The material in this section is drawn from [244]. Their intention was to
devise a jump-Markov PHD filter inspired by the well-known interacting multiple
motion (IMM) model approach. The single-target IMM filter is known to be suboptimal, but has achieved considerable popularity because of its attractive balancing
of the competing goals of computational tractability and tracking performance.
Since the authors were avowedly pursuing a suboptimal approach, it is not
surprising that their jump-Markov PHD filter is different from the one described in
Section 11.4. The purpose of this section is to summarize their approach.
Their time-update equation (Eq. (10) in [244]) is, when expressed in the
notation of this book:
)
∫ (
pS (x′ ) · fk+1|k (x|o, x′ )
˜ k|k (o, x′ )dx′
Dk+1|k (o, x) = bk+1|k (o, x) +
·D
+bk+1|k (x|o, x′ )
(11.184)
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˜ k|k (o, x′ ) is a mode-mixed version of Dk|k (o, x):
where (Eq. (9) in [244]) D
˜ k|k (o, x) =
D
∑
χo,o′ · Dk|k (o′ , x).
(11.185)
o′
Also, the probability of target survival pS (o, x) = pS (x) is mode-independent.
Their measurement-update equation (Eq. (11) in [244]) is as follows, once
again expressed in the notation of this book,
∑ pD (x) · Lz (o, x)
Dk+1|k+1 (o, x)
= 1 − pD (x) +
o (z)
Dk+1|k (o, x)
κk+1 (z) + τk+1
(11.186)
z∈Zk+1
where (Eq. (12) in [244]):
∫
o
τk+1
(z) = pD (x) · Lz (o, x) · Dk+1|k (o, x)dx.
(11.187)
One question that arises is this: Is the authors’ approach consistent with the
top-down statistical analysis advocated in Section 11.1? A different approach
might begin by trying to specify an IMM version of the multitarget jump-Markov
filter of Section 11.3.2. Then one would derive an IMM PHD filter from it. This
approach would require that one first have a formulation of the IMM filter defined
at the density-function level, not at the state-vector level.
The authors have implemented their approach using sequential Monte Carlo
(SMC, “particle”) techniques.
11.8.3
Best-Fitting-Gaussian PHD Filter of Wenling Li and Yingmin Jia
In this approach [143], an approximation is used to replace a jump-Markov linear
system with an approximate single-model linear dynamical system. Specifically,
the jump-Markov linear system
k
Xok+1|k
= Fkok x + Gokk Wkok
(11.188)
is replaced by a linear system
Xk+1|k = Φk x + Wk .
(11.189)
The best-fitting-Gaussian (BFG) approximation [106] permits this. The basic idea
is, at each recursive cycle, the multimodel prior density is recursively approximated
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349
by its BFG approximation. This allows the multiple-model time-update to be
replaced by a single-model time-update.
The approach was implemented using unscented Kalman filter (UKF) Gaussian mixture techniques. It was tested with a filter in two-dimensional simulations
involving a range-bearing sensor with medium, uniformly distributed Poisson clutter (λ = 24). The authors report improved performance in comparison to a filter
based on the approach of Pasha et al.
11.8.4
JM-CPHD Filter of Georgescu et al.
This was the first attempt to propose a jump-Markov CPHD filter [91]. These authors employed the modeling approach identified as problematic in Section 11.3.1.
That is, they use the multitarget jump-Markov state representation (o, {x1 , ..., xn })
rather than {(o1 , x1 ), ..., (on , xn )}.3 Their jump-Markov CPHD filter was derived
using the “bin occupancy” approach of Erdinc et al. ([80], [179], pp. 599-609; Section 8.4.6.8).
As a consequence, the authors do not address the conceptual and other issues
discussed in Section 11.3.2 and more fully in Section IV-D of [170]. For example:
• Shouldn’t performance suffer if a single mode o is imposed on all targets
simultaneously?
• What does the concept of a CPHD filter approximation even mean for
distributions of the form fk+1|k (o, X|Z (k) )? That is, the CPHD filter is
based on the assumption that the predicted multitarget distribution is i.i.d.c.
But fk+1|k (o, X|Z (k) ) cannot be i.i.d.c. because of the discrete variable o
(see [170] for more details).
11.8.5
Current Statistical Model (CSM) PHD Filter of Mengjun et al.
This approach is not, strictly speaking, a jump-Markov approach [197], but is
included here for completeness. In the CSM approach, the state is assumed to
include the acceleration variables in addition to position and velocity. Instead
of a library of motion models, the CMS approach presumes that, at any given
time-step, the distribution of the mean of the scalar acceleration is Rayleigh. The
variance of this distribution can be expressed in terms of the current estimated scalar
3
This is easily seen from the fact that their cardinality distribution is mode dependent, see [91], Eq.
(9). Thus a single value of o is being imposed on all targets simultaneously.
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Advances in Statistical Multisource-Multitarget Information Fusion
acceleration. It is appended as an additional state variable, and it is time-updated
and measurement-updated independently as a separate variable.
The authors have implemented the filter using particle methods, and tested it
in two-dimensional simulations assuming a range-bearing sensor, light, uniformly
distributed Poisson clutter (λ = 10), and unity probability of detection.
A PHD filter reported by Shaohua Hong et al. [270] employs a slight variation
on this approach. It employs a constant-velocity (CV) model for nonmaneuvering
targets and a CSM model for maneuvering targets.
11.8.6
The Variable State Space CPHD Filter of Chen et al.
Chen Xin, McDonald, and Kirubarajan have tested a Gaussian mixture implementation of the variable state space CPHD filter described in Section 11.6 [36]. Four
targets in the plane are observed by a Cartesian, position-measuring sensor in light
clutter (clutter rate λ = 3). Two targets appear at time t = 0, and another two at
around midscenario. All four targets follow typical air traffic control trajectories:
that is, straight-line segments occasionally punctuated by coordinated turns.
The variable state space filter utilized the two models described at the beginning of Section 11.6: a constant velocity (CV) model and a coordinated turn (CT)
model. At any given time tk , the number of targets in modes 1 or 2 were estimated
using both the MAP estimator
(k)
nMAP
)
k|k = arg sup pk|k (n|Z
o
o
o
(11.190)
o
n
and the EAP estimator
nEAP
k|k =
o
∑o o
o
n · pk|k (n|Z (k) ).
(11.191)
o
n≥0
The authors observed that the number of targets in the modes were accurately
estimated using both methods, with the MAP approach being somewhat more
accurate. However, the state estimates of the targets exhibited a downward bias.
The authors observed that the same phenomenon occurs with single-target IMM
filters.
Chapter 12
Joint Tracking and Sensor-Bias Estimation
12.1
INTRODUCTION
Current multitarget detection and tracking algorithms are typically based on the
presumption that all sensors are correctly spatially and temporally registered. That
is:
• Spatial sensor registration—the position, velocity, and physical orientations
of all sensors are known precisely with respect to some reference spatial
coordinate system.
• Temporal sensor registration—the exact time of the measurement collections
for all sensors are known precisely, with respect to some reference clock.
In real-world applications, neither of these assumptions are necessarily true,
as the following examples illustrate:
• The time stamps of measurements may be inaccurately aligned with each
other, because different clocks have been used by different sensors.
• In ground-target tracking, sensor position detections are typically overlaid on
a geographical map. However, this map may be inaccurate because it has an
unknown translational and/or rotational offset.
• The presumed line-of-sight of a gimbaled optical or infrared sensor may
actually be inaccurate, because the gimbal axes have become misaligned or
because they were inaccurately calibrated to begin with.
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Advances in Statistical Multisource-Multitarget Information Fusion
Such imperfections are called sensor biases, and can result in severely degraded target detection, tracking, and localization performance. For example, two
sensors observing the same moving target can, if one is translationally misregistered, produce what appear to be two targets moving along parallel trajectories.
Biases pose a particularly serious challenge if neither GPS nor accurate terrain
maps are available, since the sensor platform’s inertial navigation system (INS) will
drift as time passes.
The purpose of this chapter is to address multitarget detection and tracking
when sensor data is corrupted by unknown biases of unknown types. (Temporal bias
will not be considered.) As we shall see, the problem has a solution in principle—
Bayesian unified registration and tracking (BURT)—provided that (1) the sensors
are all within localization range of each other; and (2) the number of unknown
targets of opportunity is sufficiently large.
The remainder of this Introduction is organized as follows:
1. Section 12.1.1: A simple example of joint target tracking and sensor registration: “gridlocking” of stationary sensor platforms.
2. Section 12.1.2: Gridlocking in general.
3. Section 12.1.3: A summary of the major lessons learned in this chapter.
4. Section 12.1.4: The organization of the chapter.
12.1.1
Example: “Gridlocking” of Sensor Platforms
Let us begin with the simplest registration problem. Two or more stationary
sensor-bearing platforms are to cooperate in the detection and tracking of targets.
Consequently, their positions must be known. Suppose, however, that the platforms
lack accurate map references and, for whatever reason, do not have access to
GPS. Despite this “GPS-denied” and “map-denied” environment, it is nevertheless
possible for them to use their sensors and communications systems to determine
their positions with respect to each other in a relative coordinate system, using a
procedure sometimes called gridlocking.
In what follows this process will be illustrated using three successively more
detailed examples:
1. Section 12.1.1.1: Gridlocking without sensor biases.
2. Section 12.1.1.2: The impossibility of gridlocking when sensor biases exist.
3. Section 12.1.1.3: Simultaneous target-localization and gridlocking.
Joint Tracking and Sensor-Bias Estimation
12.1.1.1
353
Gridlocking Without Sensor Biases
Suppose that we have two motionless position-measuring sensors, with infinite
fields of view, unity probability of detection, and no clutter. Suppose further that
1
2
they are located at positions x and x in some absolute coordinate frame. In their
respective local coordinate systems, the measurement models of the two sensors are
j
j
Zk+1 = x + Vk+1
(12.1)
(j = 1, 2)
j
where Vk+1 are zero-mean random vectors. Assume that the sensors measure the
following positions of the locations of the other sensor:
• Position of Sensor 2 as measured by Sensor 1:
1,2
2
1
1
1
Z k+1 = x − x + Vk+1 = ∆x + Vk+1
2
(12.2)
1
where ∆x = x − x is the separation vector between the first and second
platforms.
• Position of Sensor 1 as measured by Sensor 2:
2,1
1
2
2
2
Z k+1 = x − x + Vk+1 = −∆x + Vk+1 .
(12.3)
By averaging enough measurements over time, the first sensor can, in its own
coordinate frame, estimate the location of the second sensor as
1,2
Z k+1 = ∆x,
(12.4)
and the second sensor can, in its coordinate frame, estimate the location of the first
sensor as
2,1
Z k+1 = −∆x.
(12.5)
In other words, the two sensor platforms have deduced each other’s locations
relative to each other. If we arbitrarily choose the position of one of the sensors
as the origin—the first sensor, say—then we have determined the positions of
both sensors in a relative coordinate system. The position of the first sensor is
1
x = x = 0 and the position of the second sensor is x = ∆x.
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12.1.1.2
Gridlocking with Biases
Now suppose that both sensors have unknown translational biases. In this case
their measurement models become
j
∗j
j
Zk+1 = x + t + Vk+1
(j = 1, 2)
(12.6)
∗j
where the vectors t are the translational biases. Given this, the sensors collect the
following random measurements:
• Position of Sensor 2 as measured by Sensor 1:
1,2
2
Z k+1
=
(12.7)
1
∗1
=
1
∗1
1
x − x + t + Vk+1
∆x + t + Vk+1 .
(12.8)
• Position of Sensor 1 as measured by Sensor 2:
2,1
1
Z k+1
=
2
∗2
∗2
=
2
x − x + t + Vk+1
(12.9)
2
−∆x + t + Vk+1 .
(12.10)
The first sensor’s estimate of the second sensor’s position is
1,2
∗1
(12.11)
Z k+1 = ∆x + t
and the second sensor’s estimate of the first sensor’s location is
2,1
∗2
(12.12)
Z k+1 = −∆x + t .
∗1 ∗2
∗1 ∗2
We now have two equations in the three unknowns ∆x, t , t . The biases t , t
cannot be estimated unless ∆x is known. But such knowledge requires access to
additional information of some kind.
12.1.1.3
Simultaneous Gridlocking and Target Localization
In this section, assume that we have the following additional information:
• A motionless “target of opportunity” is located at an unknown position x0 .
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Given this we have, in addition to (12.8) and (12.10), access to the following
two additional measurement collections:
• Position of unknown target as measured by Sensor 1:
1,0
1
∗1
1
(12.13)
Z k+1 = x0 − x + t + Vk+1 .
• Position of unknown target as measured by Sensor 2:
2,0
2
∗2
2
(12.14)
Z k+1 = x0 − x + t + Vk+1 .
1,2
1,0
2
Suppose that Z k+1 and Z k+1 are transmitted to the sensor located at x.
2
Then the fusion system at x can deduce the following:
1,2
Z k+1
∗1
=
2,1
Z k+1
(12.15)
∆x + t
∗2
(12.16)
=
−∆x + t
=
x0 + ∆x − x + t
=
x0 − x + t .
1,0
∗1
2
Z k+1
2,0
2
Z k+1
(12.17)
∗2
(12.18)
∗1 ∗2
2
We now have four linear equations in the following five unknowns: ∆x, t , t , x0 , x.
2
Choose x = 0 to be the origin of a relative coordinate system, which leaves four
∗1 ∗2
equations in four unknowns ∆x, t , t , x0 . We can then difference (12.15) through
(12.18) to get:
1,0
x0
=
2,0
∆x
=
t
=
t
=
(12.19)
1,2
2,1
1,0
(12.20)
1,2
(12.21)
Z k+1 − Z k+1 + Z k+1
1,2
∗1
1,0
Z k+1 − Z k+1 + Z k+1 − Z k+1
2,0
∗2
1,2
Z k+1 − Z k+1
2,1
2,0
1,0
1,2
Z k+1 + Z k+1 − Z k+1 + Z k+1 − Z k+1 .
(12.22)
That is, we have simultaneously determined:
1. The positions of the platforms in a relative coordinate system (gridlocking).
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2. The sensor biases (sensor registration).
3. The location of the unknown target (target localization).
This process can be described as joint tracking and sensor registration.
12.1.2
Gridlocking in General
Now consider the problem of joint tracking and bias estimation in general. The
simple examples just presented rely on unrealistic assumptions: the sensors and the
targets of opportunity are motionless and well separated; the sensors are positionmeasuring; the sensor biases are purely translational; and so on. Nevertheless they
illustrate the following points:
1. Joint sensor registration and gridlocking are possible in principle, even when
all sensors are biased, and even if we do not have access to GPS or other
inertial information.
2. However, additional “ground truth” information must be available, in the form
of an unknown number of unknown “targets of opportunity.”
3. The number of targets of opportunity must be sufficiently large.
4. Bias estimation (registration), gridlocking, and target tracking must be accomplished jointly and simultaneously using a single, fully unified algorithmic procedure.
5. In general, this procedure will be highly nonlinear.
The challenge, then, is to devise a formal probabilistic framework that generalizes our simple example to scenarios of arbitrary complexity. This is the purpose
of this chapter.
12.1.3
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• If there are a sufficient number of unknown “targets of opportunity” in a
scenario, then—at least in principle—it is possible to devise an optimal
procedure for estimating the sensor biases while simultaneously detecting and
tracking those targets. See Section 12.3.
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357
• This procedure is a generalization of gridlocking—which, in turn, is a type of
SLAM (simultaneous localization and mapping). Thus the “Bayesian unified
registration and tracking” (BURT) approach in this chapter can be regarded
as a generalization of SLAM.
• The optimal BURT approach is computationally intractable in general. However, it can be approximated as a two-filter version of a PHD filter. See
Section 12.4. (A CPHD filter approximation is also possible but will not
be considered in this chapter.)
• BURT-type PHD filters are apparently computationally tractable only for
translational biases in the target-state variable.
• In particular, a BURT-PHD filter for constant translational biases in the
target-state variable, due to Ristic and Clark, appears to offer promising
performance. See Section 12.5.1.
• A heuristic BURT-PHD filter for constant translational biases in the targetstate variable appears to offer surprisingly effective performance. See Section
12.5.2.
• These two BURT-PHD filters have been implemented and evaluated in simulation, for simple translational sensor biases. See Section 12.6.
12.1.4
Organization of the Chapter
The chapter is organized as follows:
1. Section 12.2: The modeling of general sensor biases.
2. Section 12.3: Optimal joint multitarget tracking and sensor registration: the
single-filter and two-filter BURT filters.
3. Section 12.4: An approximation of the optimal BURT filter—the two-filter
version the BURT-PHD filter.
4. Section 12.5: Single-filter versions of the BURT-PHD filter.
5. Section 12.6: Implemented PHD filters for joint sensor registration and
tracking.
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12.2
Advances in Statistical Multisource-Multitarget Information Fusion
MODELING SENSOR BIASES
The purpose of this section is to introduce a general model for sensor biases. It
depends on the notion of the perceived sensor state versus the actual sensor state.
∗
As was noted in Section 10.2 of Chapter 10, a sensor has a state-vector x—for
example, a state of the form
∗
˙ η̇, φ̇, µ, χ)
x = (x, y, z, ẋ, ẏ, ż, ℓ, θ, η, φ, θ,
(12.23)
where x, y, z and ẋ, ẏ, ż are the position and velocity coordinates of the sensor˙ η̇, φ̇ are the sensor’s bodycarrying platform, ℓ is its fuel level, θ, η, ϕ and θ,
frame coordinates and their rates, µ is the sensor mode, and χ is the current
communications transmission path employed by the sensor. Thus the sensor’s
measurement model actually has the form
∗
Zk+1 = ηk+1 (x, xk+1 ) + Vk+1
(12.24)
∗
where xk+1 is the actual sensor state at time tk+1 .
∗
Complicating matters still further, any of the variables xk|k , x or Zk+1 can
be contaminated by a spatial bias—for example, translational biases
∗
Zk+1
=
ηk+1 (x + xb , xk+1 ) + Vk+1
Zk+1
Zk+1
=
=
ηk+1 (x, xk+1 + xb ) + Vk+1
∗
ηk+1 (x, xk+1 ) + zb + Vk+1
∗
∗
(12.25)
(12.26)
(12.27)
or, more generally, affine biases
Zk+1
=
Zk+1
=
Zk+1
=
∗
ηk+1 (Tb x + xb , xk+1 ) + Vk+1
∗
∗
∗
ηk+1 (x, xk+1 + T b xb ) + Vk+1
∗
T˜b ηk+1 (x, xk+1 ) + zb + Vk+1
(12.28)
(12.29)
(12.30)
∗
where Tb , T b , T˜b are rotation matrices.
In this case (12.24) actually has the form
Zk+1
=
=
∗
ηk+1 (b, x, xk+1 ) + Vk+1
∗
ηk+1 (x̊, xk+1 ) + Vk+1
(12.31)
(12.32)
Joint Tracking and Sensor-Bias Estimation
359
where the vector b denotes the concatenation of all bias variables; and where the
augmented state-vector
x̊ = (xT , bT )T
(12.33)
consolidates all unknown variables—the unknown state-vectors x and b—into a
single unknown state-vector x̊. This model is general enough to represent the most
common kinds of bias.
∗
Now, notationally suppress the sensor state xk+1 and assume that the sensor
model is linear-Gaussian. Then consider the following examples:
• Affine bias in the measurement:
Zk+1
=
T Hk+1 x + b + Vk+1
(12.34)
ηk+1 (x̊)
=
=
ηk+1 (T, b, x)
T Hk+1 x + b.
(12.35)
(12.36)
Zk+1
=
Hk+1 (T x + b) + Vk+1
(12.37)
ηk+1 (x̊)
=
=
ηk+1 (T, b, x)
Hk+1 T x + Hk+1 b.
(12.38)
(12.39)
• Affine bias in the target state:
12.3
OPTIMAL JOINT TRACKING AND REGISTRATION
The multisensor-multitarget Bayes recursive filter was introduced in Section 10.2.
The purpose of this section is to describe its extension to the joint tracking and
registration problem.
As in Section 10.2, suppose that there are s sensors and let:
j
Z
:
measurement space for jth sensor
:
measurements for jth sensor
:
measurement sets for jth sensor
j
z
j
Z
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Advances in Statistical Multisource-Multitarget Information Fusion
j
j
j
Z (k)
:
Z 1 , ..., Z k : time sequence of measurement sets for jth sensor
Z
=
Z ⊎ ... ⊎ Z: multisensor measurement space
Z
=
Z ⊎ ... ⊎ Z: multisensor measurement set
Z (k)
:
Z (k) , ..., Z (k) : time sequence of multisensor measurement sets.
1
s
1
s
1
s
When the jth sensor has a bias, the multitarget likelihood function for this sensor
has the form
j
j
j
j
j
j
j
∗j
L j (b, X) = f k+1 (Z|b, X) abbr.
= fk+1 (Z|b, X, x).
(12.40)
Z
Assume that the sensors are independent and let
1
s
b = (bT , ..., bT )T
(12.41)
be the joint bias vector. Then according to the discussion in Section 3.5.3, the
multisensor-multitarget likelihood function has the form
1
fk+1 (Z|b, X)
=
s
1
=
s
(12.43)
fk+1 (Z, ..., Z|b, X)
1
=
(12.42)
fk+1 (Z ⊎ ... ⊎ Z|b, X)
1
1
s
s
s
f k+1 (Z|b, X) · · · f k+1 (Z|b, X).
(12.44)
Thus the total unknown state of the system is the pair (b, X). We are to estimate
b and X simultaneously.
Given this, two versions of the optimal BURT filter will be discussed: a
single-filter version and a two-filter version.
12.3.1
Optimal BURT Filter: Single-Filter Version
The Bayes-optimal filter for the BURT problem—the optimal BURT filter—has the
form
... → fk|k (b, X|Z (k) ) → fk+1|k (b, X|Z (k) ) → fk+1|k+1 (b, X|Z (k+1) ) → ...
Joint Tracking and Sensor-Bias Estimation
361
where
fk+1|k (b, X|Z
(k)
)
∫
=
fk+1|k (b, X|b′ , X ′ )
(12.45)
·fk|k (b′ , X ′ |Z (k) )db′ δX ′
fk+1|k+1 (b, X|Z (k+1) )
(12.46)
fk+1 (Zk+1 |b, X) · fk+1|k (b, X|Z
fk+1 (Zk+1 |Z (k) )
∫
fk+1 (Zk+1 |b, X)
=
fk+1 (Zk+1 |Z (k) )
=
(k)
)
(12.47)
·fk+1|k (b, X|Z (k) )dbδX
or, in more detailed notation,
1
∫
s
fk+1|k (b, ..., b, X|Z (k) )
=
1
s
1′
s′
fk+1|k (b, ..., b, X|b , ..., b , X ′ )(12.48)
1′
s′
·fk|k (b , ..., b , X ′ |Z (k) )
1′
s′
·db · · · db δX ′
and
1
s
1
s
fk+1|k+1 (b, ..., b, X|Z (k+1) , ..., Z (k+1) )
( 1
)
1
s
s
1
s
f k+1 (Z k+1 |b, X) · · · f k+1 (Z k+1 |b, X)
1
s
1
(12.49)
s
·fk+1|k (b, ..., b, X|Z (k) , ..., Z (k) )
=
1
s
1
s
fk+1 (Z k+1 , ..., Z k+1 |Z (k) , ..., Z (k) )
and
1
=
s
1
s
fk+1 (Z, ..., Z|Z (k) , ..., Z (k) )
∫
1
s
1 1
s s
f k+1 (Z|b, X) · · · fk+1 (Z|b, X)
1
s
1
s
(12.50)
1
s
·fk+1|k (b, ..., b, X|Z (k) , ..., Z (k) ) · db · · · dbδX.
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12.3.2
Optimal BURT Filter: Two-Filter Version
The single-filter optimal BURT filter is inconvenient for the purpose of computational approximation. Therefore, it is necessary to consider an alternative form of
this filter. Since the system state has the hybrid form (b, X), the mixed-state
“factored filter” analysis of Section 5.9 applies. We therefore know that the optimal
BURT filter can be written in a two-filter form. From Bayes’ rule,
fk|k (b, X|Z (k) ) = fk|k (b|Z (k) ) · fk|k (X|b, Z (k) )
(12.51)
where fk|k (b|Z (k) ) is a probability distribution on b; and where fk|k (X|b, Z (k) )
is a multitarget probability distribution on X, given that the multisensor bias is b.
Also because of Bayes rule, the Markov transition density can be written as
fk+1|k (b, X|b′ , X ′ ) = fk+1|k (b|b′ , X ′ ) · fk+1|k (X|b, b′ , X ′ ).
(12.52)
Assume that
fk+1|k (b|b′ , X ′ )
fk+1|k (X|b, b′ , X ′ )
=
=
fk+1|k (b|b′ )
fk+1|k (X|X ′ ).
(12.53)
(12.54)
That is, the transition of the sensor biases does not depend on the past states of
the targets; and the transition of the target states does not depend on the current or
previous sensor biases. Then from Section 5.9 we know that the two-filter version
of the optimal BURT filter has the form
... → fk|k (b)
... → fk|k (X|b)
where:
→
↑↓
→
fk+1|k (b)
fk+1|k (X|b)
→
↑↓
→
fk+1|k+1 (b) → ...
fk+1|k+1 (X|b) → ...
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363
• Optimal BURT filter time-update:
fk+1|k (b|Z (k) )
=
∫
fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )db′ (12.55)
fk+1|k (X|b, Z (k) )
( ∫
=
f˜k+1|k (X|b′ , Z (k) )
=
∫
′
′
(k)
fk+1|k (b|b ) · fk|k (b |Z
·f˜k+1|k (X|b′ , Z (k) )db′
fk+1|k (b|Z (k) )
fk+1|k (X|X ′ )
)
(12.56)
)
(12.57)
·fk|k (X ′ |b′ , Z (k) )δX ′ .
• Optimal BURT filter measurement-update:
fk+1|k+1 (b|Z (k+1) )
(12.58)
(k)
=
fk+1|k (b|Z ) · fk+1 (Zk+1 |b, Z
fk+1 (Zk+1 |Z (k) )
(k)
)
and
fk+1|k+1 (X|b, Z (k+1) )
=
(12.59)
fk+1 (Zk+1 |b, X) · fk+1|k (X|b, Z (k) )
fk+1 (Zk+1 |b, Z (k) )
where
fk+1 (Zk+1 |Z (k) )
fk+1 (Zk+1 |b, Z (k) )
=
∫
fk+1|k (b|Z (k) )
(12.60)
=
·f
(Z
|b, Z (k) )db
∫ k+1 k+1
fk+1 (Zk+1 |b, X)
(12.61)
·fk+1|k (X|b, Z (k) )δX
and where, for fixed b, the last equation is a conventional multitarget Bayes
factor.
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Advances in Statistical Multisource-Multitarget Information Fusion
The following special case is of interest, because it is often true that a sensor’s
bias does not appreciably change over time. That is, at least approximately we can
sometimes assume that
fk+1|k (b|b′ ) = δb′ (b).
(12.62)
Given this assumption, (12.55) through (12.61) reduce to:
fk+1|k (b|Z (k) )
=
fk+1|k (X|b, Z (k) )
=
fk|k (b|Z (k) )
(12.63)
∫
fk+1|k (X|X ′ ) · fk|k (X ′ |b, Z (k) )δX ′ (12.64)
and
fk+1|k+1 (b|Z (k+1) )
=
fk|k (b|Z (k) ) · fk+1 (Zk+1 |b, Z (k) )
fk+1 (Zk+1 |Z (k) )
fk+1|k+1 (X|b, Z (k+1) )
(12.65)
(12.66)
=
fk+1 (Zk+1 |b, X) · fk+1|k (X|b, Z
fk+1 (Zk+1 |b, Z (k) )
(k)
)
where
fk+1 (Zk+1 |Z (k) )
fk+1 (Zk+1 |b, Z (k) )
=
∫
fk+1|k (b|Z (k) )
(12.67)
=
·f
(Z
|b, Z (k) )db
∫ k+1 k+1
fk+1 (Zk+1 |b, X)
(12.68)
·fk+1|k (X|b, Z (k) )δX.
12.3.3
Optimal BURT Procedure
Assume that every sensor is within the field of view of every other sensor. Then the
optimal BURT procedure consists of the following steps.1
1. Step 1: Arbitrarily select a sensor and adopt its coordinate system as the
reference coordinate system for all sensors.
2. Step 2: Use the BURT filter to process a sequence Z (k) : Z1 , ..., Zk
of multisensor measurement sets, collected from all targets in the scene
(including the sensor-carrying platforms).
1
The procedure described here is slightly different than the one presented in [192].
Joint Tracking and Sensor-Bias Estimation
365
3. Step 3: At each time-step, integrate out X as a nuisance variable to get the
marginal distribution
∫
fk+1|k+1 (b|Z (k+1) ) = fk+1|k+1 (X, b|Z (k+1) )δX.
(12.69)
4. Step 4: Estimate the sensor biases using a MAP estimator:
bk+1|k+1 = arg sup fk+1|k+1 (b|Z (k+1) ).
(12.70)
b
5. Step 5 (Optimal Registration): When the time sequence b1|1 , ..., bk+1|k+1
converges to a stable (that is, small-variance) value
1
s
bk+1|k+1 = (bTk+1|k+1 , ..., bTk+1|k+1 )T ,
(12.71)
we have achieved an optimal solution to the registration part of the problem.
Note that his procedure will not necessarily converge. This will occur if, for
example, there are too few targets of opportunity in the scenario.
6. Step 6: Employ this information to correctly register whatever multitarget
tracking algorithms are being used on the platforms. If these algorithms are
j
j
MHTs, for example, use (12.70) with b = bk+1|k+1 (j = 1, ..., s) in the
extended Kalman filters (EKFs) that are employed in the MHTs.
7. Step 7 (Optimal Gridlocking): Use the JoM estimator, (5.9), or the MaM
estimator, (5.10), to determine the number and states of the targets:
JoM
Xk+1|k+1
= arg sup
X
c|X| · fk+1|k+1 (X|bk+1|k+1 , Z (k+1) )
.
|X|!
(12.72)
Some of these targets will be those of the platforms that carry the sensors
(localized relative to the coordinate system chosen in Step 1).
This completes the joint gridlocking-registration-tracking process.
12.4
THE BURT-PHD FILTER
The optimal BURT filters of Sections 12.3.1 and 12.3.2 will be computationally intractable in general. Therefore, principled approximations are necessary. The purpose of this section is to introduce a PHD filter approximation of the optimal-BURT
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Advances in Statistical Multisource-Multitarget Information Fusion
filter. This must be accomplished using the two-filter approach of Section 12.3.2.
This is because it is not possible for the joint distribution fk+1|k (b, X|Z (k) )
to be Poisson (because of the variable b); whereas the conditional distribution
fk+1|k (X|b, Z (k) ) can be assumed to be Poisson. Thus from (12.51), since
fk|k (b, X|Z (k) ) = fk|k (b|Z (k) ) · fk|k (X|b, Z (k) )
(12.73)
it follows from (4.74) that the joint PHD will have the form
Dk|k (b, X|Z (k) ) = fk|k (b|Z (k) ) · Dk|k (x|b, Z (k) ).
(12.74)
The section is organized as follows:
1. Section 12.4.1: The single-sensor BURT-PHD filter.
2. Section 12.4.2: The multisensor BURT-PHD filter, using the iteratedcorrector approach.
3. Section 12.4.3: The multisensor BURT-PHD filter, using the parallelcombination approach.
12.4.1
BURT-PHD Filter: Single-Sensor Case
For the purposes of this section, assume that only a single sensor is present. Then b
is the bias vector for only that sensor. The PHD filter approximation of the optimal
BURT filter has the two-filter form
... → fk|k (b|Z (k) )
... → Dk|k (x|b, Z (k) )
12.4.1.1
→
↑↓
→
fk+1|k (b|Z (k) )
Dk+1|k (x|b, Z (k) )
→
↑↓
→
fk+1|k+1 (b|Z (k+1) ) → ...
Dk+1|k+1 (x|b, Z (k+1) ) → ...
Single-Sensor BURT-PHD Filter: Time Update
For the sake of conceptual clarity, ignore the target-spawning model. Then the
time-update equations are
fk+1|k (b|Z
(k)
)
Dk+1|k (x|b, Z (k) )
=
∫
fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )db′
=
∫
˜ k+1|k (x|b′ , Z (k) ) · fk|k+1 (b′ |b)db′ (12.76)
D
(12.75)
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367
where
˜ k+1|k (x|b, Z (k) )
D
=
fk|k+1 (b′ |b)
=
bk+1|k (x)
(12.77)
∫
+ pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ |b, Z (k) )dx′
fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )
fk+1|k (b|Z (k) )
(12.78)
and where, recall, fk|k+1 (b′ |b) is a retrodictive or “reverse” Markov transition
density, as in (5.77).
To see why these equations are true, note that from the integral formula for
PHDs, (4.74), the PHDs of fk+1|k (X|b, Z (k) ) and f˜k+1|k (X|b′ , Z (k) ) in (12.57)
and (12.57) are
∫
Dk+1|k (x|b, Z (k) ) =
fk+1|k ({x} ∪ X|b, Z (k) )δX
(12.79)
∫
˜ k+1|k (x|b, Z (k) ) =
D
f˜k+1|k ({x} ∪ X|b′ , Z (k) )δX.
(12.80)
Therefore, from (12.55) and (12.57) we immediately get
∫
fk+1|k (b|Z (k) ) =
fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )db′
(12.81)
( ∫
)
fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )
˜ k+1|k (x|b′ , Z (k) )db′
·D
Dk+1|k (x|b, Z (k) ) =
(12.82)
fk+1|k (b|Z (k) )
∫
˜ k+1|k (x|b′ , Z (k) ) · fk|k+1 (b′ |b)db′ .(12.83)
=
D
Since (12.57) is a conventional multitarget time-update for fixed b′ , it follows that
the time-update for its PHD must be a conventional PHD filter time-update:
˜ k+1|k (x|b, Z (k) )
D
=
=
bk+1|k (x|b) +
(12.84)
∫
pS (x′ |b) · fk+1|k (x|b, x′ ) · Dk|k (x′ |b, Z (k) )dx′
∫
bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ )
(12.85)
·Dk|k (x′ |b, Z (k) )dx′
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where the last equation follows from the fact that target appearances and disappearances cannot depend on the sensor bias.
Now consider a special case: the sensor bias is approximately constant over
time. Then fk+1|k (b|b′ ) = δb′ (b) and these equations reduce to
fk+1|k (b|Z (k) )
=
Dk+1|k (x|b, Z (k) )
=
fk|k (b|Z (k) )
(12.86)
∫
bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) (12.87)
·Dk|k (x′ |b, Z (k) )dx′ .
Remark 54 As an example of the retrodictive density in (12.78), let
fk+1|k (b|b′ )
′
fk|k (b |Z
(k)
)
=
NQ (b − F b′ )
(12.88)
=
′
(12.89)
NP (b − bk|k ).
Then
fk|k+1 (b′ |b) = NC (b′ − CP −1 bk|k − CF T Q−1 b)
(12.90)
C −1 = P −1 + F T Q−1 F.
(12.91)
where
12.4.1.2
Single-Sensor BURT-PHD Filter: Measurement Update
The measurement-update equations for this filter are
fk+1|k+1 (b|Z (k+1) )
Dk+1|k+1 (x|b, Z (k+1) )
Dk+1|k (x|b, Z (k) )
(12.92)
=
fk+1 (Zk+1 |b, Z (k) ) · fk+1|k (b|Z (k) )
fk+1 (Zk+1 |Z (k) )
=
1 − pD (b, x)
+
(12.93)
∑ pD (b, x) · fk+1 (z|b, x)
κk+1 (z) + τk+1 (z|b)
z∈Zk+1
where the likelihood function fk+1 (Zk+1 |b, Z (k) ) for the bias is
∏
fk+1 (Zk+1 |b, Z (k) ) = e−λk+1 −Dk+1|k [pD |b]
(κk+1 (z) + τk+1 (z|b))
z∈Zk+1
(12.94)
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369
and where
)
=
∫
τk+1 (z|b)
=
∫
=
·Dk+1|k (x|b, Z (k) )dx
∫
pD (b, x) · Dk+1|k (x|b, Z (k) )dx.
fk+1 (Zk+1 |Z
(k)
Dk+1|k [pD |b]
fk+1|k (b|Z (k) ) · fk+1 (Zk+1 |b, Z (k) )db(12.95)
pD (b, x) · fk+1 (z|b, x)
(12.96)
(12.97)
Remark 55 (Limitations) Notice that (12.93) is valid only for biases of the target
state x, and not for biases of the sensor state or the measurements. The reason for
this restriction is as follows. A translational bias on measurements z will cause
clutter measurements to be translationally shifted, whereas this will not be the case
for a translational bias on target states. In this case, κk+1 (z) would actually
have the form κk+1 (z|b). Similar remarks apply to a translational bias of the
∗
∗
sensor state x, since κk+1 (z) is actually an abbreviation for κk+1 (z|x). Thus
the clutter is dependent on b in the joint state (b, X). Because of this and as
was discussed in Section 8.7, the measurement-update for the PHD would involve
a computationally problematic combinatorial sum over all partitions of the current
measurement set.
Remark 56 The bias likelihood function for the bias, (12.94), is highly nonlinear.
Thus practical implementation of these equations will require sequential Monte
Carlo (SMC) techniques.
To see why (12.93) through (12.97) are true, note that, for fixed b, (12.59) is
a conventional multitarget measurement-update. Therefore, the PHD of
fk+1|k+1 (X|b, Z (k+1) )
is a conventional PHD filter measurement-update. Therefore, the equations corresponding to (12.58) through (12.59) are
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Advances in Statistical Multisource-Multitarget Information Fusion
fk+1|k+1 (b|Z (k+1) )
(12.98)
(k)
Dk+1|k+1 (x|b, Z (k+1) )
Dk+1|k (x|b, Z (k) )
=
fk+1 (Zk+1 |b, Z ) · fk+1|k (b|Z
fk+1 (Zk+1 |Z (k) )
=
1 − pD (b, x)
+
(k)
)
(12.99)
∑ pD (b, x) · fk+1 (z|b, x)
κk+1 (z) + τk+1 (z|b)
z∈Zk+1
where
τk+1 (z|b)
=
∫
pD (b, x) · fk+1 (z|b, x)
(12.100)
·Dk+1|k (x|b, Z (k) )dx
fk+1 (Zk+1 |Z (k) )
(12.101)
=
∫
fk+1 (Zk+1 |b, Z (k) ) · fk+1|k (b|Z (k) )db.
For fixed b, (12.58) is a conventional multitarget Bayes normalization factor.
Given our models and (8.56), it is
fk+1 (Zk+1 |b, Z (k) ) = e−λk+1 −Dk+1|k [pD |b]
∏
(κk+1 (z) + τk+1 (z|b))
z∈Zk+1
(12.102)
where
Dk+1|k [pD |b] =
12.4.1.3
∫
pD (b, x) · Dk+1|k (x|b, Z (k) )dx.
(12.103)
Single-Sensor BURT-PHD Filter: State Estimation
State estimation is accomplished in the obvious fashion. First, estimate the current
sensor bias:
bk+1|k+1 = arg sup fk+1|k+1 (b|Z (k+1) ).
(12.104)
b
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Given this, apply the usual PHD filter state-estimation approach in Section 8.4.4 to
Dk+1|k+1 (x|bk+1|k+1 , Z (k+1) ). That is, round off
Nk+1|k+1 =
∫
Dk+1|k+1 (x|bk+1|k+1 , Z (k+1) )dx
(12.105)
to the nearest integer ν, and then find the states corresponding to the ν largest
peaks of Dk+1|k+1 (x|bk+1|k+1 , Z (k+1) ).
12.4.2
BURT-PHD Filter: Multisensor Case Using Iterated Corrector
1
Suppose that there are
1
sensors with respective biases
s
s
b, ..., b
and let
s
T
T T
b = (b , ..., b ) be the joint multisensor bias-vector. Then apply the usual
iterated-corrector approach of Section 10.5 by repeating (12.93) and (12.93) once
for each sensor in turn.
Thus for the first sensor we get
1
s
1
2
s
fk+1|k+1 (b, ..., b|Z (k+1) , Z (k) , ..., Z (k) )
1
=
1
s
(12.106)
1
s
1
s
fk+1 (Z k+1 |b, ..., b, Z (k) ) · fk+1|k (b, ..., b|Z (k) , ..., Z (k) )
1
1
s
fk+1 (Z k+1 |Z (k) , ..., Z (k) )
and
1
s
1
2
s
Dk+1|k+1 (x|b, ..., b, Z (k+1) , Z (k) , ..., Z (k) )
1
s
(12.107)
1
s
Dk+1|k (x|b, ..., b, Z (k) , ..., Z (k) )
1
=
1 − pD (x) +
1
1
1
∑
pD (x) · f k+1 (z|b, x)
1
κk+1 (z) + τk+1 (z|b)
.
1
1
1
z∈Z k+1
Similarly, for the second sensor we get
1
s
1
2
s
fk+1|k+1 (b, ..., b|Z (k+1) , Z (k+1) , ..., Z (k) )
(
)
1
s
2
fk+1 (Z k+1 |b, ..., b, Z (k) )
1
=
s
1
2
s
·fk+1|k (b, ..., b|Z (k+1) , Z (k) , ..., Z (k) )
2
1
2
s
fk+1 (Z k+1 |Z (k+1) , Z (k) , ..., Z (k) )
(12.108)
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and
1
s
1
2
s
Dk+1|k+1 (x|b, ..., b, Z (k+1) , Z (k+1) , ..., Z (k) )
1
s
(12.109)
1
2
s
Dk+1|k (x|b, ..., b, Z (k+1) , Z (k) , ..., Z (k) )
2
=
1 − pD (x) +
2
2
2
∑
pD (x) · f k+1 (z|b, x)
2
κk+1 (z) + τk+1 (z|b)
2
2
2
z∈Z k+1
and so on.
12.4.3
BURT-PHD Filter: Multisensor Case Using Parallel Combination
In this case, the iterated-corrector procedure in the previous section is replaced by
the procedure described in Section 10.6.
12.5
SINGLE-FILTER BURT-PHD FILTERS
The purpose of this section is to describe simpler versions of the BURT-PHD filter,
in which—in effect—only the joint PHD Dk|k (x|b, Z (k) ) is propagated. The
section is organized as follows:
1. Section 12.5.1: A single-filter BURT-PHD filter for the case of static sensor
biases.
2. Section 12.5.2: Heuristic single-filter BURT-PHD filters.
12.5.1
Single-Filter BURT-PHD Filter for Static Biases
Ristic and Clark have shown that it is possible to derive a single-filter version of the
BURT-PHD filter, given that the sensor biases are static. It has the following form
[254], [253], [255]:
... → fk|k (b|Z (k) ) →
... → Dk|k (x|b, Z (k) ) → Dk+1|k (x|b, Z (k) )
fk+1|k+1 (b|Z (k+1) ) → ...
↗↑
→ Dk+1|k+1 (x|b, Z (k+1) ) → ...
Note that the two filters are not actually coupled, since the bias distributions in the
top filter are derived entirely in terms of the conditional PHDs in the bottom filter.
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Thus this filter has, in essence, a single-filter form. It is defined by the following
equations:
• Time update (target spawning neglected):
Dk+1|k (x|b, Z (k) )
=
bk+1|k (x)
(12.110)
∫
+ pS (x′ ) · fk+1 (x|x′ ) · Dk|k (x′ |b, Z (k) )dx′ .
• Measurement update:
Dk+1|k+1 (x|b, Z (k+1) )
Dk+1|k (x|b, Z (k) )
1 − pD (b, x)
=
(12.111)
∑ pD (b, x) · fk+1 (z|b, x)
κk+1 (z) + τk+1 (z|b)
+
z∈Zk+1
and
fk+1|k+1 (b|Z (k+1) ) =
fk+1 (Zk+1 |b, Z (k) ) · fk|k (b|Z (k) )
fk+1 (Zk+1 |Z (k) )
(12.112)
where
τk+1 (z|b)
=
∫
pD (b, x) · fk+1 (z|b, x)
(12.113)
·Dk+1|k (x|b, Z (k) )dx
fk+1 (Zk+1 |b, Z (k) )
=
e−λk+1 −Dk+1|k [pD |b]
(12.114)
∏
·
(κk+1 (z) + τk+1 (z|b))
z∈Zk+1
and where
fk+1 (Zk+1 |Z
(k)
)
Dk+1|k [pD |b]
=
∫
=
·fk|k (b|Z (k) )db
∫
pD (b, x)
fk+1 (Zk+1 |b, Z (k) )
·Dk+1|k (x|b, Z (k) )dx.
(12.115)
(12.116)
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To see why these equations are true, recall that the two-filter version of the
BURT-PHD filter is based on (12.51),
fk+1|k+1 (b, X|Z (k+1) ) = fk+1|k+1 (b|Z (k+1) ) · fk+1|k+1 (X|b, Z (k+1) ),
(12.117)
from which follows, because of (4.74),
Dk+1|k+1 (b, x|Z (k+1) ) = fk+1|k+1 (b|Z (k+1) ) · Dk+1|k+1 (x|b, Z (k+1) )
(12.118)
and where, by (12.93),
Dk+1|k+1 (x|b, Z (k+1) )
Dk+1|k (x|b, Z (k) )
=
1 − pD (b, x)
+
(12.119)
∑ pD (b, x) · fk+1 (z|b, x)
.
κk+1 (z) + τk+1 (z|b)
z∈Zk+1
Ristic and Clark noted that, if b is static, then fk+1|k+1 (b|Z (k+1) ) can be
derived directly rather than recursively. An application of Bayes’ rule yields
fk+1|k+1 (b|Z (k+1) ) = ∫
fk+1 (Z (k+1) |b) · f0|0 (b)
fk+1 (Z (k+1) |b′ ) · f0|0 (b′ )db′
(12.120)
where f0|0 (b) is the prior distribution on b and fk+1 (Z (k+1) |b) is the likelihood
function for the variable b. A second application of Bayes’ rule shows that
fk+1 (Z (k+1) |b)
=
=
fk+1 (Zk+1 |b, Z (k) ) · fk (Zk |b, Z (k−1) ) (12.121)
· · · f2 (Z2 |b, Z (1) ) · f1 (Z1 |b)
fk+1 (Zk+1 |b, Z (k) ) · fk (Z (k) |b).
(12.122)
However, from (12.94) we know that—given that the predicted multitarget distributions are Poisson for every k ≥ 1—that, for all l = 1, ..., k + 1,
fl (Zl |b, Z (l−1) ) = e−λl −Dl|l−1 [pD |b]
∏
(κl (z) + τl (z|b)) .
(12.123)
z∈Zl
Consequently,
fk+1 (Z (k+1) |b) can be computed directly from the PHDs
(l)
Dl|l−1 (x|b, Z ) for l ≥ 1. Thus the measurement-updated posterior on b
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can be computed recursively as
fk+1|k+1 (b|Z (k+1) )
∝
fk+1 (Z (k+1) |b) · f0|0 (b)
(k)
(12.124)
(12.125)
=
fk+1 (Zk+1 |b, Z )
·fk (Z (k) |b) · f0|0 (b)
∝
fk+1 (Zk+1 |b, Z (k) ) · fk|k (b|Z (k) ). (12.126)
As time progresses, the BURT-PHD filter will implicitly estimate the value
of b. If it is desired that this value be explicitly estimated, then this can be
accomplished using the MAP estimator:
bk+1|k+1 = arg sup fk+1|k+1 (b|Z (k+1) ).
(12.127)
b
12.5.2
A Heuristic Single-Filter BURT-PHD Filter
It is possible to approach the problem of devising a BURT-PHD filter in a naı̈ve
fashion. Consider the single-sensor case first. We naı̈vely apply the usual PHD
filtering equations to the joint PHD Dk|k (b, x|Z (k) ):
∫
Dk+1|k (b, x) = bk+1|k (x) + pS (x′ )
(12.128)
·fk+1|k (b, x|b′ , x′ )
·Dk|k (b′ , x′ )db′ dx′
and
fk+1|k (b, x|b′ , x′ )
=
fk+1|k (b|b′ ) · fk+1|k (x|x′ )
(12.129)
Dk+1|k+1 (b, x)
Dk+1|k (b, x)
=
1 − pD (b, x)
(12.130)
+
∑ pD (b, x) · fk+1 (z|b, x)
κk+1 (z) + τk+1 (z)
z∈Zk+1
τk+1 (z)
=
∫
pD (b, x) · fk+1 (z|b, x)
(12.131)
·Dk+1|k+1 (b, x)dbdx.
For the multisensor case, in place of (12.130) we apply one of the multisensor PHD
measurement-update equations described in Chapter 10.
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Strictly speaking, such an approach is not theoretically justifiable. This is
because, according to (12.74), the joint Dk+1|k+1 (b, x|Z (k+1) ) actually has the
form
Dk+1|k+1 (b, x|Z (k+1) ) = fk+1|k+1 (b|Z (k+1) ) · Dk+1|k+1 (x|b, Z (k+1) ).
(12.132)
Here, and as the analysis leading up to that equation demonstrates, the PHD
measurement-update equation can be applied only to Dk+1|k+1 (x|b, Z (k+1) ), not
to Dk+1|k+1 (b, x|Z (k+1) ).
The naı̈ve BURT-PHD filter was proposed independently in 2011 by Lian,
Han, Liu, and Chen [145] and by Mahler [192]. The two approaches differ in
two minor respects. First, to address multisensor biases, Lian et al. employed the
product-pseudolikelihood approach of Section 10.6.3; whereas Mahler employed
the iterated-corrector approach of Section 10.5 of the same chapter. Second, to
estimate the multisensor bias, Lian et al. proposed the EAP estimator
bEAP
k+1|k+1 =
∫
b · Dk+1|k+1 (b, x|Z (k+1) )dbdx
∫
,
Dk+1|k+1 (b′ , x′ |Z (k+1) )db′ dx′
(12.133)
whereas Mahler proposed the MAP estimator,
AP
bM
k+1|k+1 = arg sup
b
∫
Dk+1|k+1 (b, x|Z (k+1) )dx.
(12.134)
Lian et al. also implemented and tested their approach in the multisensor
case. Surprisingly—given the heuristic nature of the naı̈ve approach and given
its substantial deviation from the rigorous, two-filter approach—it exhibited good
performance (see Section 12.6.2). It is unclear why this should be the case.
12.6
IMPLEMENTED BURT-PHD FILTERS
The purpose of this section is to report two implemented BURT-PHD filters: the
filter of Ristic and Clark as described in Section 12.5.1, and the filter of Lian, Han,
Liu, and Chen, as described in Section 12.5.2.
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12.6.1
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The BURT-PHD Filter of Ristic and Clark
This filter was discussed in Section 12.5.1. The authors implemented and tested it
using sequential Monte Carlo (SMC) techniques. The discussion in this section is
drawn from [255]. The following two asynchronous sensors were considered:
• Range-bearing sensor, collects at odd-numbered time-steps, is located at
(0, 0)T , with covariance diag((50km)2 , (0.5o )2 ), probability of detection
pD = 0.95, uniformly distributed Poisson clutter with clutter rate λ = 10,
and static measurement translational bias (6.8km, −3.50o )T .
• Range-bearing sensor, collects at even-numbered time-steps, is located at
(120km, 35km)T , with variance diag((50km)2 , (0.5o )2 ), probability of
detection pD = 0.95, uniformly distributed Poisson clutter with clutter rate
λ = 10, and static measurement translational bias (−5km, 2o )T .
Because the sensors collect measurements asynchronously, the single-sensor
PHD filter can be applied (that is, no multisensor PHD filter is necessary). These
sensors were used to observe three to five moving, appearing and disappearing
targets. Because the primary purpose of the experiments was to determine the
accuracy of the bias estimates, the track trajectories were very short.
The authors report that their filter’s estimates of the biases were quite close to
the actual values: (6.863km, −3.578o )T and (−5.104km, 1.806o )T , respectively.
12.6.2
The BURT-PHD Filter of Lian et al.
This filter was discussed in Section 12.5.2. In [145], the authors described the
performance of a sequential Monte Carlo (SMC) implementation of it, in simple
two-dimensional scenarios. The following three sensors were considered:
• Range-bearing sensor, located at (600, 400)T , with covariance
diag(2.5m)2 , (2.5mrad)2 ),
probability of detection pD = 0.8, uniformly distributed Poisson clutter with clutter rate λ = 60, and static measurement translational bias
(50m, −50mrad)T .
• Range-only sensor, located at (0, 0)T , with variance (2.5m)2 , probability of
detection pD = 0.9, uniformly distributed Poisson clutter with clutter rate
λ = 50, and static measurement translational bias 30m.
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• Bearing-only sensor, located at (−600, −400)T , with variance (2.5mrad)2 ,
probability of detection pD = 0.7, uniformly distributed Poisson clutter with
clutter rate λ = 40, and static measurement translational bias −40mrad.
These sensors were used to observe up to four moving, appearing and disappearing targets moving along curvilinear trajectories.
The authors reported that, after a settling-in period of about 15 time steps,
their SMC-BURT-PHD filter converged to the correct estimates of the biases of
the three sensors. It also essentially correctly estimated the number of targets at
any given time—whereas a conventional SMC-PHD filter exhibited a significant
downward bias. Finally, their filter accurately localized the target locations, with a
significantly smaller error than the standard SMC-PHD filter, as measured using the
OSPA metric (Section 6.2.2).
Lian et al. also compared their filter with the multisensor JPDA filter, suitably
generalized to address joint tracking and registration. The authors reported that the
multisensor JPDA filter outperformed the SMC-BURT-PHD filter when the clutter
rate was relatively low, but the reverse was true when the clutter rate was large.
Chapter 13
Multi-Bernoulli Filters
13.1
INTRODUCTION
Beyond the underlying modeling assumptions, the measurement-update equations
for the classical PHD filter require the following simplifying assumption:
• The predicted multitarget distribution fk+1|k (X|Z (k) ) is approximately
Poisson, for every k ≥ 0.
Likewise, the classical CPHD filter requires the following two simplifying
assumptions:
• The multitarget distributions fk|k (X|Z (k) )
approximately i.i.d.c. for every k ≥ 0.
and fk+1|k (X|Z (k) ) are
The main subject of this chapter, the cardinality-balanced multitarget multiBernoulli (CBMeMBer) filter, follows this pattern. It requires the following simplifying assumption:
• The multitarget distributions fk|k (X|Z (k) ) and fk+1|k (X|Z (k) ) are
approximately multi-Bernoulli for every k ≥ 0, where multi-Bernoulli RFSs
were defined in Section 4.3.4.
The CBMeMBer filter is conceptually different than the PHD and CPHD
filters, however, in that the information in fk|k (X|Z (k) ) or fk+1|k (X|Z (k) ) is not
compressed into summary statistical moments. Rather, an attempt is being made
to:
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• Directly approximate fk|k (X|Z (k) ) and fk+1|k (X|Z (k) ) as full multitarget
probability distributions.
In certain circumstances, this approximation will not be a good one. Let
pk|k (n|Z (k) ) be the cardinality distribution of fk|k (X|Z (k) ). If fk|k (X|Z (k) ) is
multi-Bernoulli, then the variance of pk|k (n|Z (k) ) can never exceed the mean of
pk|k (n|Z (k) )—see (4.133). Thus:
• The CBMeMBer filter cannot be expected to perform well when the variance
of pk|k (n|Z (k) ) exceeds its mean—that is, when target number is being
poorly estimated.
As was noted in Section 5.10.6, the derivation of the measurement-update
equations for the original MeMBer filter made use of an ill-considered first-order
Taylor’s linearization ([179], p. 681, Eq. (17.176)). This linearization caused a
serious upward bias in the target number estimate [310]. To correct for this bias, Vo,
Vo, and Cantoni [310] devised the CBMeMBer filter, which is the primary subject
of this chapter.
Vo, Vo, and their associates have extended the CBMeMBer filter to address
unknown clutter and detection-profile backgrounds [312], [313]. This additional
work will be described in Section 18.7.1
A secondary purpose of this chapter is to describe the Bernoulli filter, which
was briefly introduced in Section 5.10.7. The CBMeMBer filter is in certain
respects more general than the Bernoulli filter, in that the Bernoulli filter can track at
most a single target. In other respects, however, the Bernoulli filter is more general
than the CBMeMBer filter. Specifically, its clutter model can be arbitrary, whereas
the clutter model for the CBMeMBer filter is presumed to be Poisson.
The discussions of both the CBMeMBer filter and the Bernoulli filter in the
following sections will be at a fairly high level. For details—especially in regard
to implementation issues—see the book Particle Filters for Random Set Models by
Ristic [250].
13.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
1
Ouyang, Ji, and Li have pointed out a limitation of the CBMeMBer filter. They proposed a heuristic
remedy for this problem, but it does not seem to be generally applicable [228].
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• The Bernoulli filter is—given its modeling assumptions—the Bayes-optimal
approach for detecting and tracking at most a single target in an arbitrary
clutter and detection background. See Section 13.2.
• The CBMeMBer filter is conceptually different than the PHD and CPHD
filters. Whereas the latter assume that the multitarget probability distribution
fk|k (X|Z (k) ) can be approximated by statistical moments, the CBMeMBer
filter is based on a direct approximation of fk|k (X|Z (k) ) itself.
• This multi-Bernoulli approximation is not accurate when the CBMeMBer
filter is not accurately estimating target number—specifically, when the variance of the cardinality distribution exceeds the mean of the cardinality distribution.
• The instantaneous computational complexity of the CBMeMBer filter is
roughly the same as that of the PHD filter: O(mn), where m is the current
number of measurements and n is the current number of target tracks.
However, n tends to increase with time, so that pruning and merging of
tracks is required.
• The performance of the Gaussian mixture (GM) implementation of the CBMeMBer filter is not appreciably better than that of the GM-CPHD filter.
However, it is significantly more computationally efficient.
• Both the computational efficiency and tracking performance of the sequential
Monte Carlo (SMC) implementation of the CBMeMBer filter are appreciably
better than that of the SMC-CPHD filter.
• Consequently, the SMC-CBMeMBer filter is probably most appropriate for
problems with larger pD , but with significant motion and/or measurement
nonlinearities.
• The CBMeMBer filter can be extended to incorporate multiple motion models, using jump-Markov techniques. See Section 13.5.
13.1.2
Organization of the Chapter
The chapter is organized as follows:
1. Section 13.2: The Bernoulli filter.
2. Section 13.3: The multisensor Bernoulli filter.
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3. Section 13.4: The cardinality-balanced multitarget multi-Bernoulli (CBMeMBer) filter.
4. Section 13.5: A jump-Markov version of the CBMeMBer filter.
13.2
THE BERNOULLI FILTER
Given its modeling assumptions, the Bernoulli filter is:
• The Bayes-optimal approach for detection and tracking in a single-sensor
scenario that is known to contain at most a single target, in an arbitrary clutter
background and detection profile.
Under certain modeling assumptions, the Bernoulli filter is just the multitarget
Bayes filter, given that target number is known a priori to be 0 or 1—that is, when
the initial multitarget distribution has the form f0|0 (X) = 0 if |X| ≥ 2. When
clutter is i.i.d.c., the Bernoulli filter is identical to the single-target CPHD filter (see
Section 8.5.6.5).
The Bernoulli filter was independently proposed by B.-T. Vo in his doctoral dissertation [298], and by Mahler in [179], pp. 514-528. Vo’s terminology,
“Bernoulli filter,” is more technically accurate and descriptive, and has also become
the accepted usage. It is therefore adopted in place of Mahler’s usage in [179],
“joint target-detection and tracking” (JoTT) filter.
As was noted in [179], the Bernoulli filter is a generalization of the integrated probabilistic data association (IPDA) filter, derived by Musicki, Evans and
Stankovic using a bottom-up methodology [215].2 Challa, Vo, and Wang subsequently demonstrated that, given the same modeling assumptions, the IPDA filter
could be derived using the FISST methodology [32].
The Bernoulli filter consists of two coupled filters of the form
... →
pk|k (Z (k) )
... →
sk|k (x|Z (k) )
→
↑↓
→
pk+1|k (Z (k) )
sk+1|k (x|Z (k) )
→
↑↓
→
pk+1|k+1 (Z (k+1) )
→ ...
sk+1|k+1 (x|Z (k+1) )
→ ...
where pk|k is the probability that the target exists at time tk ; and where, if it does
exist, sk|k (x) is its track distribution—that is, the probability (density) that it has
2
The Bernoulli filter generalizes the IPDA filter in that (1) target appearance is modeled; (2)
probability of detection is state-dependent; and (3) the state-independent clutter process is arbitrary
rather than Poisson.
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state x. At any time-step, the Bernoulli filter is related to the multitarget Bayes
filter as follows:

1 − pk|k
if
X=∅

pk|k · sk|k (x) if X = {x} .
fk|k (X|Z (k) ) =
(13.1)

0
if |X| ≥ 2
The purpose of this section is to summarize this filter. A detailed tutorial, by
Ristic, Vo, Vo, and Farina, can be found in [262]. See also Ristic’s book, Particle
Filters for Random Set Models [250].
The section is organized as follows:
1. Section 13.2.1: Modeling assumptions for the Bernoulli filter.
2. Section 13.2.2: Bernoulli filter time-update equations.
3. Section 13.2.3: Bernoulli filter measurement-update equations.
4. Section 13.2.4: State estimation for the Bernoulli filter.
5. Section 13.2.5: Error estimation for the Bernoulli filter.
6. Section 13.2.6: The Bernoulli filter is equivalent to an exact PHD filter.
7. Section 13.2.7: Implementing the Bernoulli filter.
8. Section 13.2.8: Algorithmic implementations of the Bernoulli filter.
13.2.1
Bernoulli Filter: Modeling
The following models are required for the Bernoulli filter:
• Probability that, if it is in the scene, the target will not disappear if it has state
x—pS (x) abbr.
= pS,k+1|k (x).
• Probability that, if it is not in the scene, the target will appear or reappear—
pB abbr.
= pB,k+1|k .
• Spatial distribution of the target if it appears—ˆbk+1|k (x). Thus the PHD of
the target-birth RFS is
bk+1|k (x) = pB · ˆbk+1|k (x).
• Single-target Markov density—Mx (x′ ) abbr.
= fk+1|k (x|x′ ).
(13.2)
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• Single-sensor, single-target likelihood function—Lz (x) abbr.
= fk+1 (z|x).
• Multiobject probability distribution for an arbitrary clutter RFS—κk+1 (Z).
13.2.2
Bernoulli Filter: Time-Update
Suppose that we are given the prior probability of existence pk|k and the prior track
distribution sk|k (x) at time tk . Then the time-update equations for the Bernoulli
filter are ([179], p. 519):
pk+1|k
=
sk+1|k (x)
=
pB · (1 − pk|k ) + pk|k · sk|k [pS ]
pB · (1 − pk|k ) · ˆbk+1|k (x) + sk|k [pS Mx ]
(13.3)
(13.4)
pk+1|k
where
sk|k [pS ]
sk|k [pS Mx ]
13.2.3
=
∫
pS (x′ ) · sk|k (x′ )dx′
(13.5)
=
∫
pS (x′ ) · Mx (x′ ) · sk|k (x′ )dx′ .
(13.6)
Bernoulli Filter: Measurement Update
Suppose that we are given the predicted probability of existence pk+1|k and
the predicted track distribution sk+1|k (x) at time tk+1 . Let Zk+1 be the
newly collected measurement set. Then the measurement-update equations for the
Bernoulli filter are ([179], p. 520):
(13.7)
pk+1|k+1
∑
(Zk+1 −{z})
1 − sk+1|k [pD ] + z∈Zk+1 sk+1|k [pD Lz ] · κk+1
κk+1 (Zk+1 )
∑
κk+1 (Zk+1 −{z})
p−1
z∈Zk+1 sk+1|k [pD Lz ] ·
k+1|k − sk+1|k [pD ] +
κk+1 (Zk+1 )
=
and
sk+1|k+1 (x)
sk+1|k (x)
=
∑
(Zk+1 −{z})
1 − pD (x) + pD (x) z∈Zk+1 Lz (x) · κk+1
κk+1 (Zk+1 )
∑
(Zk+1 −{z})
1 − sk+1|k [pD ] + z∈Zk+1 sk+1|k [pD Lz ] · κk+1
κk+1 (Zk+1 )
(13.8)
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where
sk+1|k [pD ]
sk+1|k [pD Lz ]
=
∫
pD (x) · sk+1|k (x)dx
(13.9)
=
∫
pD (x) · Lz (x) · sk+1|k (x)dx.
(13.10)
When Zk+1 = ∅, the summations vanish by convention. That is,
pk+1|k+1
=
1 − sk+1|k [pD ]
−1
pk+1|k − sk+1|k [pD ]
(13.11)
sk+1|k+1 (x)
=
1 − pD (x)
· sk+1|k (x).
1 − sk+1|k [pD ]
(13.12)
Remark 57 In 2007, Vo, Vo, and Cantoni proposed a version of the Bernoulli filter
that addresses state-dependent Poisson clutter [309]. The clutter process in this
case has the specific form
κk+1 (Z|x) = e−λk+1 (x)
∏
κk+1 (z|x)
(13.13)
zseZ
where κk+1∫(z|x) is the state-dependent clutter intensity function and where
λk+1 (x) = κk+1 (z|x)dz is the state-dependent clutter rate. This filter was
shown to significantly outperform conventional approaches such as the probabilistic
data association (PDA) filter.
13.2.4
Bernoulli Filter: State Estimation
State estimation for the Bernoulli filter requires that the following two questions
be answered: Is a target present? If so, what is its state? The answers to these
questions involve the JoM or MaM multitarget state estimators of (5.9) and (5.10).
In the case of the MaM estimator, a target can be declared to exist at time
tk+1 if pk+1|k+1 > 1/2—or, equivalently, if ([179], Eq. (14.212))
1
pk+1|k >
2 − sk+1|k [pD ] +
∑
. (13.14)
z∈Zk+1 sk+1|k [pD Lz ] ·
κk+1 (Zk+1 −{z})
κk+1 (Zk+1 )
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Given this, the estimated state is3
x̂MaM
k+1|k+1 = arg sup sk+1|k+1 (x).
(13.15)
x
In the case of the JoM estimator, let c > 0 be a constant with the same units
as x, which is equal in magnitude to the desired target-localization accuracy. Then
a target is declared to exist if4
pk+1|k+1 < c · sup sk+1|k+1 (x).
(13.16)
x
The target state estimate is the same as for the MaM estimator.
13.2.5
Bernoulli Filter: Error Estimation
Error estimation requires two things: (1) an estimate of the error in the target
number estimate, and (2) an estimate of the error in the target state estimate (if
such exists). The former is given by the variance ([179], Eq. (14.229)):
2
σk+1|k+1
= pk+1|k+1 · (1 − pk+1|k+1 ).
(13.17)
The latter is given by the covariance of sk+1|k+1 (x) ([179], Eq. (14.232)):
∫
Pk+1|k+1 = (x − x̄k+1|k+1 )(x − x̄k+1|k+1 )T · sk+1|k+1 (x)dx
(13.18)
where x̄k+1|k+1 is the JoM or MaM estimate of the target state.
13.2.6
The Bernoulli Filter as an Exact PHD Filter
From (4.75) we know that the PHD of the multitarget distribution fk|k (X|Z (k) ) in
(13.1) is
Dk|k (x) = pk|k · sk|k (x)
(13.19)
with expected number of targets
Nk|k =
3
4
∫
Dk|k (x)dx = pk|k .
(13.20)
Erratum: There is a typo in the corresponding equation in [179]. Specifically, the factor fk+1|k (x)
in Eq. (14.213) should be included within the arg supx in Eq. (14.214).
Note that Eq. (14.215) in [179] is valid only if pD is constant.
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Thus when there is at most a single target, Dk|k (x) contains exactly the same
information as the two Bernoulli-filter items pk|k and sk|k (x). Consequently, the
Bernoulli filter is equivalent to a PHD filter
... →
Dk|k (x)
→
Dk+1|k (x)
→
Dk+1|k+1 (x)
→ ...
where, now, the measurement-update step is exact in the sense that it no longer
requires the assumption that fk+1|k (X|Z (k) ) is Poisson.
The time-update equation for this PHD filter is, because of the particular
motion model assumed for the Bernoulli filter:
∫
Dk+1|k (x) = bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
(13.21)
where
bk+1|k (x) = pB · (1 − pk|k ) · ˆbk+1|k (x).
(13.22)
The measurement-update equation is (see [171] and [179], pp. 631-632):
Dk+1|k+1 (x)
Dk+1|k (x)
=
(13.23)
∑
(Zk+1 −{z})
1 − pD (x) + pD (x) z∈Zk+1 Lz (x) · κk+1
κk+1 (Zk+1 )
∑
(Zk+1 −{z})
1 − Dk+1|k [pD ] + z∈Zk+1 Dk+1|k [pD Lz ] · κk+1
κk+1 (Zk+1 )
where
13.2.7
Dk+1|k [pD ]
=
∫
pD (x) · Dk+1|k (x)dx
(13.24)
Dk+1|k [pD Lz ]
=
∫
pD (x) · Lz (x) · Dk+1|k (x)dx.
(13.25)
Bernoulli Filter: Practical Implementation
The Bernoulli filter can be implemented in exact closed form using Gaussian
mixture techniques. Alternatively, it can be implemented using sequential Monte
Carlo techniques. For GM implementation, the track distribution is approximated
as a Gaussian mixture:
νk|k
sk|k (x) ∼
=
∑
i=1
k|k
wk|k · NP k|k (x − xi )
i
(13.26)
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∑νk|k
where
i=1 wk|k = 1. For SMC implementation, it is approximated as a Dirac
mixture:
νk|k
∑
∼
sk|k (x) =
wk|k · δxk|k (x).
(13.27)
i
i=1
For greater detail, see [303] or the book Particle Filters for Random Set
Models by Ristic [250].
13.2.8
Bernoulli Filter: Implementations
The tutorial [262] by Ristic, Vo, Vo, and Farina discusses several applications of the
Bernoulli filter. The reader is referred there for details.
13.3
THE MULTISENSOR BERNOULLI FILTER
Since the Bernoulli filter is just a special case of the general multitarget Bayes
filter, multiple independent sensors can be addressed using the iterated-corrector
approach.
For conceptual clarity, consider the two-sensor case. Suppose that measure1
2
ment sets Z k+1 and Z k+1 are collected by the two sensors at the same time
tk+1 . Then (13.7) and (13.8) are applied to the first sensor:
(13.28)
p̃k+1|k+1
1
1 − sk+1|k [pD ] +
∑
1
1
1
1
(Z k+1 −{z})
sk+1|k [pD Lz1 ] · κk+1
1
1
1
1
1
z∈Z k+1
κk+1 (Z k+1 )
=
∑
1
p−1
1 1
k+1|k − sk+1|k [pD ] +
z∈Z
1
1
1
1
(Z k+1 −{z})
sk+1|k [pD Lz1 ] · κk+1
1
1
1
κk+1 (Z k+1 )
k+1
and
s̃k+1|k+1 (x)
sk+1|k (x)
1
(13.29)
1
1 − pD (x) + pD (x)
∑
1
z∈Z k+1
1
1
1
1
1
(Z k+1 −{z})
Lz1 (x) · κk+1
1
1
κk+1 (Z k+1 )
=
.
1
1 − sk+1|k [pD ] +
∑
1
1
1
1
z∈Z k+1
sk+1|k [pD Lz1 ] ·
1
1
1
κk+1 (Z k+1 −{z})
1
1
κk+1 (Z k+1 )
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Then (13.7) and (13.8) are applied for the second sensor, resulting in the two-sensor
measurement-update:
(13.30)
pk+1|k+1
2
1 − s̃k+1|k+1 [pD ] +
∑
2
2
2
2
(Z k+1 −{z})
s̃k+1|k+1 [pD Lz2 ] · κk+1
2
2
2
2
2
z∈Z k+1
κk+1 (Z k+1 )
=
2
p̃−1
k+1|k+1 − s̃k+1|k+1 [pD ] +
∑
2
2
2
2
(Z k+1 −{z})
s̃k+1|k+1 [pD Lz2 ] · κk+1
2
2
2
2
2
z∈Z k+1
κk+1 (Z k+1 )
and
sk+1|k+1 (x)
s̃k+1|k+1 (x)
(13.31)
2
2
1 − pD (x) + pD (x)
∑
2
z∈Z k+1
2
2
2
2
2
(Z k+1 −{z})
Lz2 (x) · κk+1
2
2
κk+1 (Z k+1 )
=
.
2
1 − s̃k+1|k+1 [pD ] +
∑
2
2
2
2
z∈Z k+1
s̃k+1|k+1 [pD Lz2 ] ·
2
2
2
κk+1 (Z k+1 −{z})
2
2
κk+1 (Z k+1 )
An application of the multisensor Bernoulli filter, to detection and tracking of
road-constrained targets using TDOA/FDOA (time difference of arrival/frequency
difference of arrival) measurements, has been reported by B.-T. Vo, Chong Meng
See, and Wee Teck Ng [303]. The authors implemented the multisensor Bernoulli
filter in exact closed form, using the UKF variant of the Gaussian mixture (GM)
approach. They also used the approach described in Section 9.5.6 to account for
state-dependent probability of detection. Road segments were modeled as ellipses.
The primary challenge of filtering using TDOA/FDOA measurements is the
fact that the actual target is heavily obscured by a large number of “ghost targets.”
These are due to the large number of bearing-only triangulations created by clutter.
Especially when the clutter rate is large or the probability of detection is small, it
can be essentially impossible to initialize the target.
The authors employed two range-dependent TDOA/FDOA sensors, with the
measurements of both corrupted by uniformly distributed Poisson clutter. The
characteristics of these sensors were as follows:
• First sensor: maximum probability of detection pD,max = 0.95 and clutter
rate λ = 100.
• Second sensor: maximum probability of detection pD,max = 0.75 and
clutter rate λ = 10.
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The multisensor Bernoulli filter was tested against an appearing and disappearing target under two conditions: without road-constraint information, and with
this information. As expected, the filter’s performance in the first case was very
poor, largely because of the large clutter rate for the first sensor and the small
probability of detection for the second sensor. With road information, however,
performance was very good. The target’s appearances and disappearances were
detected (with a small delay), and it was tracked accurately when it was present.
The authors also found that, as expected, performance was better when both sensors
were used rather than one.
13.4
THE CBMEMBER FILTER
As previously noted in Section 5.10.6, the CBMeMBer filter is based on the
approximation of multitarget posterior distributions as multi-Bernoulli distributions
in the sense of Section 4.3.4:
νk|k
Gk|k [h|Z (k) ] ∼
=
∏
i
i
(1 − qk|k
+ qk|k
· sik|k [h]).
(13.32)
i=1
Also as previously noted, this approximation is not fully general: if pk|k (n) is the
cardinality distribution of a multi-Bernoulli RFS, then its variance is always smaller
than its mean.
The CBMeMBer filter time-update step is exact in the following sense. Suppose that both the target appearance process and the multitarget Markov density are
both multi-Bernoulli. Then if fk|k (X|Z (k) ) is multi-Bernoulli, so is the predicted
distribution fk+1|k (X|Z (k) ). The same is not true for the measurement-update
step, however. If fk+1|k (X|Z (k) ) is multi-Bernoulli then fk+1|k+1 (X|Z (k+1) ) is
usually not multi-Bernoulli. Thus one must determine a multi-Bernoulli distribution
that approximates it. In this case one gets a multitarget filter that has the form
... →
fk|k (X|Z (k) )
→
fk+1|k (X|Z (k) )
→
fk+1|k+1 (X|Z (k+1) ) → ...
where fk+1|k (X|Z (k) ) is multi-Bernoulli if fk|k (X|Z (k) ) is multi-Bernoulli,
and fk+1|k+1 (X|Z (k+1) ) is approximately multi-Bernoulli if fk+1|k (X|Z (k) )
is multi-Bernoulli. The CBMeMBer filter results from a particular choice of
approximation for fk+1|k+1 (X|Z (k+1) ).
The CBMeMBer filter propagates the multi-Bernoulli parameters rather than
the multi-Bernoulli distributions. That is, it propagates a track table
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T0|0 → T1|0 → T1|1 → · · · → Tk|k → Tk+1|k → Tk+1|k+1 → · · ·
where, for a given value of k, Tk|k consists of a list of νk|k target tracks:
ν
ν
ν
k|k
k|k
k|k
1
Tk|k = {(ℓ1k|k , qk|k
, s1k|k (x)) , ..., (ℓk|k
, qk|k
, sk|k
(x))}.
(13.33)
Here,
• ℓik|k is the identifying label of the ith track at time tk .
i
• 0 < qk|k
< 1 is the probability that the ith track is an actual target (that is,
its probability of existence) at time tk .
• sik|k (x) is the probability distribution (track distribution) of the ith track at
time tk .
Thus the CBMeMBer filter has the following form:
ν
...
→
k|k
i
{(ℓik|k , qk|k
, sik|k (x))}i=1
→
k+1|k
i
{(ℓik+1|k , qk+1|k
, sik+1|k (x))}i=1
→
k+1|k+1
i
{(ℓik+1|k+1 , qk+1|k+1
, sik+1|k+1 (x))}i=1
→ ...
ν
ν
Its computational complexity is O(mn), where m is the current number of
measurements and n is the current number of tracks ([310], p. 414). However,
n increases without bound over time, thus requiring the use of track-pruning and
track-merging techniques.
Remark 58 (“Spooky action at a distance”) The “spookiness” phenomenon in
PHD and CPHD filters was noted in Section 9.2. Vo and Ma have noted that the
CBMeMBer filter also exhibits this phenomenon, though to a significantly lesser
extent [304]. Spookiness is probably a consequence of the approximations used to
derive the measurement-update equations for the CBMeMBer filter.
The purpose of this section is to describe the CBMeMBer filter. It is organized
as follows:
1. Section 13.4.1: Modeling assumptions for the CBMeMBer filter.
2. Section 13.4.2: Time update equations for the CBMeMBer filter.
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3. Section 13.4.3: Measurement update equations for the CBMeMBer filter.
4. Section 13.4.4: Merging and pruning for the CBMeMBer filter.
5. Section 13.4.5: Multitarget state and error estimation for the CBMeMBer
filter.
6. Section 13.4.6: More efficient track management for the CBMeMBer filter.
7. Section 13.4.7: Gaussian-mixture and particle implementation of the CBMeMBer filter.
8. Section 13.4.8: Practical implementations of the CBMeMBer filter.
13.4.1
CBMeMBer Filter: Modeling
The CBMeMBer filter requires the following models:
• Target probability of survival: pS (x′ ) abbr.
= pS,k+1|k (x′ ).
• Single-target Markov density: Mx (x′ ) abbr.
= fk+1|k (x|x′ ).
• Target probability of detection: pD (x) abbr.
= pD,k+1|k (x), assumed to be large.
• Single-sensor, single-target likelihood function, Lz (x) abbr.
= fk+1 (z|x).
• Poisson clutter with clutter rate λk+1 and spatial distribution ck+1 (z), with
λk+1 assumed to be not too large, where the clutter intensity function is
(13.34)
κk+1 (z) = λk+1 · ck+1 (z).
13.4.2
CBMeMBer Filter: Predictor
The time-update for the CBMeMBer filter is the same as that for the original
MeMBer filter ([179], pp. 661-662). Suppose that we are given the prior track
table
νk|k
i
Tk|k = {(ℓik|k , qk|k
, sik|k (x))}i=1
.
(13.35)
We are to determine the time-updated track table
ν
k+1|k
i
Tk+1|k = {(ℓik+1|k , qk+1|k
, sik+1|k (x))}i=1
.
(13.36)
It consists of persisting tracks and birth (appearing) tracks
persist
birth
Tk+1|k = Tk+1|k
∪ Tk+1|k
(13.37)
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where
persist
Tk+1|k
birth
Tk+1|k
ν
=
k|k
{(ℓi , qi , si (x))}i=1
(13.38)
=
bk
B B
{(ℓB
i , qi , si (x))}i=1
(13.39)
and where there are νk|k persisting tracks and bk appearing tracks.
The appearing tracks are specified on the basis of whatever a priori knowledge
we might have about the appearances of targets. The persisting tracks have the form,
for i = 1, ..., νk|k ,
ℓi
=
ℓik|k
(13.40)
qi
=
i
qk|k
· sik|k [pS ]
(13.41)
=
sik|k [pS Mx ]
sik|k [pS ]
(13.42)
=
∫
pS (x′ ) · sik|k (x′ )dx′
(13.43)
=
∫
pS (x′ ) · Mx (x′ ) · sik|k (x′ )dx′ .
(13.44)
si (x)
where
sik|k [pS ]
sik|k [pS Mx ]
Because of the birth tracks, the number of tracks will tend to increase with
time:
(13.45)
νk+1|k = νk|k + bk .
13.4.3
CBMeMBer Filter: Corrector
Suppose that we are given the predicted track table
ν
k+1|k
i
Tk+1|k = {(ℓik+1|k , qk+1|k
, sik+1|k (x))}i=1
.
(13.46)
and suppose that a new measurement set Zk+1 = {z1 , ..., zmk+1 } is collected with
|Zk+1 | = mk+1 . We are to determine the form of the time-updated track table
ν
k+1|k+1
i
Tk+1|k+1 = {(ℓik+1|k+1 , qk+1|k+1
, sik+1|k+1 (x)) }i=1
.
(13.47)
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It consists of legacy tracks and measurement-updated tracks
legacy
meas
Tk+1|k+1 = Tk+1|k+1
∪ Tk+1|k+1
(13.48)
where
ν
legacy
Tk+1|k+1
meas
Tk+1|k+1
=
k+1|k
L L
{(ℓL
i , qi , si (x))}i=1
(13.49)
=
mk+1
U U
{(ℓU
j , qj , sj (x))}j=1
(13.50)
and where there are νk+1|k legacy tracks and mk+1 measurement-updated tracks.
Thus the total number of tracks is νk+1|k+1 = νk+1|k + mk+1 . Because of the
measurement-updated tracks, the number of tracks will tend to increase with time.
Given this, the measurement-update equations for the CBMeMBer filter are
as follows:
• Corrector equations for legacy tracks for i = 1, ..., νk+1|k ([310], Eqs.
(14,15)):
ℓL
i
=
ℓik+1|k
qiL
=
i
qk+1|k
·
sL
i (x)
=
sik+1|k (x) ·
(13.51)
1 − sik+1|k [pD ]
i
1 − qk+1|k
· sik+1|k [pD ]
1 − pD (x)
.
1 − sik+1|k [pD ]
(13.52)
(13.53)
• Corrector equations for measurement-updated tracks for i = 1, ..., νk+1|k
and j = 1, ..., mk+1 ([310], Eqs. (27,38)):
ℓU
j
=
qjU
=
ℓ∗k+1|k
i
i
∑νk+1|k qk+1|k
(1−qk+1|k
)·sik+1|k [pD Lzj ]
i
(1−qk+1|k
·sik+1|k [pD ])2
i=1
κk+1 (zj ) +
sU
j (x)
∑
=
(13.54)
i
∑νk+1|k qk+1|k
·sik+1|k [pD Lzj ]
i=1
i
νk+1|k qk+1|k
i
i=1
1−qk+1|k
∑νk+1|k
i=1
(13.55)
i
1−qk+1|k
·sik+1|k [pD ]
· sik+1|k (x) · pD (x) · Lzj (x)
i
qk+1|k
i
1−qk+1|k
(13.56)
· sik+1|k [pD Lzj ]
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where
sik+1|k [pD ]
sik+1|k [pD Lzj ]
=
∫
pD (x) · sik+1|k (x)dx
(13.57)
=
∫
pD (x) · Lzj (x) · sik+1|k (x)dx
(13.58)
and where ℓ∗k+1|k is the label of the predicted track that has the largest
contribution to the current measurement-updated probability of existence qjU
in (13.55) ([310], p. 414).
13.4.4
CBMeMBer Filter: Merging and Pruning
As time progresses, the number of tracks will increase without bound, and so
merging and pruning will be necessary. Techniques similar to those for Gaussian
mixture implementation can be employed to reduce the number of tracks ([179], pp.
665-666).
Suppose that two tracks ℓi , qi , si (x) and ℓj , qj , sj (x) are such that qi +qj <
1. Then they are eligible for merging if the association density
∫
pi,j = si (x) · sj (x)dx
(13.59)
exceeds some threshold. In this case the merged track is ℓ, q, s(x) where
ℓ
=
ℓ∗
(13.60)
q
=
(13.61)
s(x)
=
qi + qj
si (x) · sj (x)
pi,j
(13.62)
where ℓ∗ is the label of the track that has largest probability of existence: ℓ∗ = ℓi
if qi > qj .
Once tracks with small probabilities of existence have been discarded, the
remaining tracks are merged in order to keep within memory and computational
limits.
13.4.5
CBMeMBer Filter: State and Error Estimation
Following merging and pruning, Vo, Vo, and Cantoni proposed two general approaches for estimating the number and states of the targets ([310], p.414).
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• Method 1: Choose a target-detection threshold τ , and then select those tracks
i
that have the largest existence probabilities: qk+1|k+1
> τ . Then select the
means or modes of the corresponding distributions sk+1|k+1 (x).
• Method 2: The cardinality distribution is given by (4.130):
=
(13.63)
pk+1|k+1 (n)
(νk+1|k+1
)
∏
i
(1 − qk+1|k+1 )
i=1
·σνk+1|k+1 ,n
( 1
qk+1|k+1
1
qk+1|k+1
ν
k+1|k+1
qk+1|k+1
, ...,
ν
)
.
k+1|k+1
1 − qk+1|k+1
Estimate the number of targets by determining the MAP estimate
n̂k+1|k+1 = arg sup pk+1|k+1 (n).
(13.64)
n≥0
Then select those n̂k+1|k+1 tracks that have the largest probabilities of
existence. Finally, select the means or modes of the corresponding track
distributions sk+1|k+1 (x).
13.4.6
CBMeMBer Filter: Track Management
The track management scheme described in (13.40), (13.51), (13.54), and (13.60)
has the advantage of being simple to implement. However, it has been noted by
Wong, Vo, and Vo that this approach is not effective when targets intersect or are
otherwise closely-spaced [325]. This is because tracks in close proximity will
inevitably fall within the track-merging threshold.
To address this problem, these authors proposed a more sophisticated track
management approach based on the method of Shafique and Shah [269].5 In brief,
the approach has three stages:
1. Stage 1: Search for possible associations between the current estimate and
the estimates in previous time-steps.
5
Strictly speaking, in [325] Wong et al. did not propose the method of Shafique and Shah for use with
the CBMeMBer filter. Rather, this method was applied to the IO-MeMBer filter of Section 20.5.
However, it applies equally well to the CBMeMBer filter.
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397
2. Stage 2: Previously missed estimates are considered for association with
those current estimates which have not yet been associated.
3. Stage 3: Current estimates that have not yet been associated are considered
to be possible newly appearing targets, and are assigned new labels.
4. Stage 4: If an old estimate is not associated after a sufficiently large number
of times, then it is considered to be a disappearing target and is eliminated.
13.4.7
CBMeMBer Filter: Gaussian-Mixture and Particle Implementation
Gaussian mixture and sequential Monte Carlo implementation of the CBMeMBer
filter are described in [310], pp. 414-417. For more details, the reader is directed
there.
As usual, for GM implementation one must assume that the probability of
detection and probability of target survival are constant: pD (x) = pD and
pS (x′ ) = pS .
13.4.8
CBMeMBer Filter: Performance
Vo, Vo, and their associates have implemented and tested the CBMeMBer filter
in a number of applications. In this section, discussion is limited to two such
implementations: the baseline implementations described by Vo, Vo, and Cantoni
in their original paper; and an application for tracking using audio and visual data.
The section also includes a summary of a series of papers by Zhang et al., in which
the CBMeMBer filter is applied to multitarget detection and tracking using managed
sensor networks.
13.4.8.1
CBMeMBer Filter: Baseline Simulations
Vo, Vo, and Cantoni implemented the CBMeMBer filter using both Gaussian
mixture and particle techniques, with the following results.
Gaussian mixture implementation ([310], pp. 420-421). The GM-CBMeMBer
filter was tested in a scenario involving a single linear-Gaussian sensor, observing
10 targets that appear and disappear along linear trajectories. In a uniform clutter
background with clutter rate 10, the filter was able to correctly detect and track the
targets, including a simultaneous crossing of three targets at roughly mid-scenario.
The CBMeMBer filter’s average localization accuracy was the same as that of the
GM-PHD filter (23m), but worse than that of the GM-CPHD filter (17m). When
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the clutter rate was increased to 50, the GM-CBMeMBer filter evidenced a slight
upward bias in the target-number estimate (whereas the GM-PHD and GM-CPHD
filters remained unbiased). The authors noted that the GM-CBMeMBer filter performed well up to a clutter rate of 20, and down to a probability of detection of
0.90.
Particle implementation ([310], pp. 417-419): In this case, the SMCCBMeMBer filter was tested in a scenario involving a single range-bearing sensor, with 10 targets that appear and disappear along curvilinear trajectories. A
coordinated-turn motion model was assumed. In a high-SNR scenario with clutter
rate 10, the SMC-CBMeMBer filter was able to successfully detect and track the
targets, including at simultaneous target crossings. The filter had a better average
localization performance (50m) than either the SMC-CPHD filter (60m) or SMCPHD filter (70m).
This is probably due to the difficulty of multitarget state estimation for SMCPHD and SMC-CPHD filters, compared to the simplicity of estimation for the
SMC-CBMeMBer filter. This relationship held true for larger clutter rates—though,
once again, the SMC-CBMeMBer filter exhibited a slight upward bias in the targetnumber estimate for clutter rates exceeding 20.
Overall Evaluation: On the basis of their experiments, Vo et al. concluded
that the CBMeMBer filter is best suited for problems with the following characteristics:
• State-dependent probability of detection.
• Nonlinearities extreme enough to require particle implementation.
13.4.8.2
CBMeMBer Filter: Audio-Visual Tracking
In [114], Hoseinnezhad, Vo, Vo, and Suter applied the CBMeMBer filter to the
problem of tracking people using a video camera equipped with two side-mounted
microphones. In this problem, the targets are not always audible, and—because
of occlusions or crossings—not always visible. Targets were modeled as moving
rectangular templates with unknown widths and heights. Time difference of arrival
(TDOA) techniques were used to process the microphone measurements. Kernelbased background-subtraction techniques and morphological techniques were used
to process the video data. Also, an “active speaker” model was employed, in which
probability of detection is set high for video data (0.95, because targets are almost
always visible) but low for audio (0.40, because targets are usually nonspeaking).
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399
The approach was successfully tested against real audio-video data involving
two people. In part, good performance was due to the complementary nature of
the two data sources. Nonspeaking targets could be tracked visually and obscured
targets could be tracked using audio.
13.4.8.3
CBMeMBer Filter: Tracking Using Managed Sensor Networks
In a series of papers, Zhang and his associates have applied the CBMeMBer filter to
the problem of detecting and tracking multiple targets using a network of managed
sensors [327], [328], [122], [120], [123], [121], [124]. The authors assume that
the sensors are organized into sensor-clusters, each managed by its own “cluster
head” (CH). A CH is activated if its sensors are capable of efficiently sensing
at least some targets. within each cluster, those sensors with better information
about the targets—as determined using an RFS-based sensor management objective
function—transmit their measurements to the CH. The CH then sequentially
processes its local information using a CBMeMBer filter.
Since this approach employs the sensor management approach described in
Part V, further discussion is deferred until Section 26.6.3.1.
13.5
JUMP-MARKOV CBMEMBER FILTER
Dunne and Kirubarajan have extended the CBMeMBer filter to more effectively
track rapidly maneuvering targets [65], using the jump-Markov techniques described in Chapter 11 [66].6 The filtering equations for the JM-CBMeMBer filter
are essentially identical to (13.40) through (13.58), with a few minor alterations.
The main differences are that the track distributions sik|k (x) now have the form
sik|k (o, x), and that the integral on the augmented target state (o, x) has the form
∑ ∫
f (o, x)dx.
o
13.5.1
Jump-Markov CBMeMBer Filter: Modeling
The jump-Markov CBMeMBer filter requires the following models:
• Target probability of survival: pS (o′ , x′ ) abbr.
= pS,k+1|k (o′ , x′ ).
6
Jin-Long Yang, Hong-Bing Ji, and Hong-Wei Ge proposed a jump-Markov CBMeMBer filter in
2012 [119], based on generalization of the interacting multiple model (IMM) technique. This work
came to my attention too late for consideration in this book.
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Advances in Statistical Multisource-Multitarget Information Fusion
• Single-target Markov density:
Mo,x (o′ , x′ )
=
=
fk+1|k (o, x|o′ , x′ )
χo,o′ · fk+1|k (x|o′ , x′ ),
(13.65)
(13.66)
where χo,o′ is the mode transition matrix.
• Target probability of detection: pD (o, x) abbr.
= pD,k+1|k (o, x), assumed to be
large.
• Single-sensor, single-target likelihood function, Lz (o, x) abbr.
= fk+1 (z|o, x).
• Poisson clutter with clutter rate λk+1 and spatial distribution ck+1 (z), with
λk+1 assumed to be not too large, where the clutter intensity function is
(13.67)
κk+1 (z) = λk+1 · ck+1 (z).
Remark 59 (Gaussian mixture implementation) Gaussian mixture (GM) implementation of the jump-Markov CBMeMBer filter is similar to GM implementation of the jump-Markov PHD and CPHD filters, as described in Section 11.7.1.
The probability of detection and probability of target survival must be assumed
to have functional dependence only on the jump variable: pD (o, x) = poD and
′
pS (o′ , x′ ) = poS .
13.5.2
Jump-Markov CBMeMBer Filter: Predictor
We are given the prior track table
ν
k|k
i
Tk|k = {(ℓik|k , qk|k
, sik|k (o, x))}i=1
.
(13.68)
We are to determine the form of the time-updated track table
ν
k+1|k
i
Tk+1|k = {(ℓik+1|k , qk+1|k
, sik+1|k (o, x))}i=1
(13.69)
consisting of persisting tracks and appearing tracks
persist
birth
Tk+1|k = Tk+1|k
∪ Tk+1|k
(13.70)
where
ν
persist
Tk+1|k
=
k|k
{(ℓi , qi , si (o, x))}i=1
(13.71)
birth
Tk+1|k
=
bk
B B
{(ℓB
i , qi , si (o, x))}i=1
(13.72)
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401
and where there are νk|k persisting tracks and bk appearing tracks. The persisting
tracks have the form, for i = 1, ..., νk|k ,
=
ℓik|k
(13.73)
qi
=
i
qk|k
· sik|k [pS ]
sik|k [pS Mo,x ]
(13.74)
si (o, x)
=
ℓi
(13.75)
sik|k [pS ]
where
sik|k [pS ]
=
∑∫
pS (o′ , x′ ) · sik|k (o′ , x′ )dx′
(13.76)
o′
sik|k [pS Mo,x ]
=
∑∫
pS (o′ , x′ ) · Mo,x (o′ , x′ ) · sik|k (o′ , x′ )dx′ .(13.77)
o′
13.5.3
Jump-Markov CBMeMBer Filter: Corrector
Suppose that we are given the predicted track table
ν
k+1|k
i
Tk+1|k = {(ℓik+1|k , qk+1|k
, sik+1|k (o, x))}i=1
.
(13.78)
Suppose that a new measurement set Zk+1 = {z1 , ..., zmk+1 } is collected with
|Zk+1 | = mk+1 . We are to determine the form of the time-updated track table
ν
k+1|k+1
i
Tk+1|k+1 = {(ℓik+1|k+1 , qk+1|k+1
, sik+1|k+1 (o, x)) }i=1
.
(13.79)
It consists of legacy tracks and measurement-updated tracks
legacy
meas
Tk+1|k+1 = Tk+1|k+1
∪ Tk+1|k+1
(13.80)
where
legacy
Tk+1|k+1
meas
Tk+1|k+1
ν
=
k+1|k
L L
{(ℓL
i , qi , si (o, x))}i=1
(13.81)
=
mk+1
U U
{(ℓU
j , qj , sj (o, x))}j=1
(13.82)
and where there are νk+1|k legacy tracks and mk+1 measurement-updated tracks.
Thus the total number of tracks is νk+1|k+1 = νk+1|k + mk+1 .
Given this, the measurement-update equations for the CBMeMBer filter are
as follows ([310], Eqs. (14,15,27,38)).
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• Corrector equations for legacy tracks for i = 1, ..., νk+1|k ([310], Eqs.
(14,15)):
ℓL
i
=
ℓik+1|k
(13.83)
qiL
=
1 − sik+1|k [pD ]
i
qk+1|k
·
i
1 − qk+1|k
· sik+1|k [pD ]
(13.84)
sL
i (o, x)
=
sik+1|k (o, x) ·
1 − pD (o, x)
.
1 − sik+1|k [pD ]
(13.85)
• Corrector equations for measurement-updated tracks for i = 1, ..., νk+1|k
and j = 1, ..., mk+1 ([310], Eqs. (27,38)):
ℓU
j
=
qjU
=
ℓ∗k+1|k
i
i
∑νk+1|k qk+1|k
(1−qk+1|k
)·sik+1|k [pD Lzj ]
(13.86)
i
(1−qk+1|k
·sik+1|k [pD ])2
i=1
κk+1 (zj ) +
i
∑νk+1|k qk+1|k
·sik+1|k [pD Lzj ]
i=1
(13.87)
i
1−qk+1|k
·sik+1|k [pD ]
and
sU
j (o, x)
∑νk+1|k
i=1
=
(13.88)
i
qk+1|k
· sik+1|k (o, x) · pD (o, x) · Lzj (o, x)
i
1−qk+1|k
∑νk+1|k
i=1
i
qk+1|k
i
1−qk+1|k
· sik+1|k [pD Lzj ]
where
sik+1|k [pD ]
=
∑∫
pD (o, x) · sik+1|k (o, x)dx
(13.89)
o
sik+1|k [pD Lzj ]
=
∑∫
pD (o, x) · Lzj (o, x)
o
·sik+1|k (o, x)dx
(13.90)
and where ℓ∗k+1|k is the label of the predicted track that has the largest
contribution to the current measurement-updated probability of existence in
(13.55) ([310], p. 414).
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13.5.4
403
Jump-Markov CBMeMBer Filter: Performance
In [65], [66], Dunne et al. reported performance results for both particle (SMC)
and Gaussian mixture (GM) implementations of the JM-CBMeMBer filter. In a
two-dimensional scenario, three motion models were assumed: a single constantvelocity (CV) model and two constant turn rate (CT) models, one for right-hand
turns and one for left-hand turns. The scenario contained four targets with different
trajectories: straight-line, sinusoidal, elliptical, and “∝”-shaped. They were
observed by a linear-Gaussian sensor with uniform Poisson clutter (clutter rate
λ = 10) and probability of detection pD = 0.95. The performances of four
filters—SMC-CBMeMBer, GM-CBMeMBer, SMC-JM-CBMeMBer, and GM-JMCBMeMBer—were compared, using the OSPA distance (see Section 6.2.2) and the
estimates of target number.
The authors report that all four filters estimated target number reasonably
well with the GM-JM-CBMemBer filter performing best, followed by the GMCBMemBer filter, the SMC-JM-CBMeMBer filter, and the SMC-CBMeMBer filter.
Similar results were reported for overall performance, as measured using the OSPA
metric (Section 6.2.2).
The authors also assessed the ability of the two jump-Markov CBMeMBer
filters to estimate the currently correct target motion model. Both filters proved to
be quite capable in this respect.
Chapter 14
RFS Multitarget Smoothers
14.1
INTRODUCTION
Let us be given a single sensor that observes a single target with state x, with
no missed detections or clutter. Given a time sequence Z k : z1 , ..., zk of
measurements, recall that the single-target recursive Bayes filter propagates the
measurement-updated probability density
fk+1|k+1 (x|Z k+1 ) =
fk+1 (zk+1 |x, Z k ) · fk+1|k (x|Z k )
fk+1 (zk+1 |Z k )
(14.1)
where
fk+1|k (x|Z k )
fk+1 (zk+1 |Z k )
=
∫
fk+1|k (x|x′ , Z k ) · fk|k (x′ |Z k )dx′
(14.2)
=
∫
fk+1 (zk+1 |x, Z k ) · fk+1|k (x|Z k )dx
(14.3)
and where it is usually assumed that
fk+1|k (x|x′ , Z k )
=
fk+1|k (x|x′ )
(14.4)
k
=
fk+1 (zk+1 |x).
(14.5)
fk+1 (zk+1 |x, Z )
At each step k of the recursion, a Bayes-optimal state estimator is used to construct
an estimate of x from fk|k (x|Z k ).
However, the Bayes filter is not the only way to exploit the information in
Z k to arrive at an estimate of the target’s trajectory. It is also possible to exploit
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Advances in Statistical Multisource-Multitarget Information Fusion
the entire time history Z k to arrive at more accurate track estimates at all of the
preceding time-steps ℓ = 0, 1, ..., k.
A Bayes smoother is an algorithm that computes the probability distributions
fℓ|k (x|Z k )
(14.6)
for ℓ = 0, ..., k, and then uses a Bayes-optimal state estimator to compute a
smoothed estimate of x at time tℓ from fℓ|k (x|Z k ). That is, it employs the
entire measurement-stream Z k to determine the best estimate of x at each of the
intermediate times tℓ . While this requires off-line batch processing rather than
real-time processing, it is useful for track reconstruction.
Various Bayes smoothers have been proposed. The two most common are the
forward-backward smoother and the two-filter smoother.
The obvious multitarget generalization of a Bayes smoother would be an
algorithm that computes the multitarget distributions
fℓ|k (X|Z (k) )
(14.7)
for ℓ = 0, ..., k, and which then uses a Bayes-optimal multitarget state estimator
to compute a smoothed estimate of X at time tℓ from the distribution
fℓ|k (X|Z (k) ).
The purpose of this chapter is to consider the following multitarget generalizations of the forward-backward smoother:
• The general multitarget forward-backward smoother.
• A special case of the general smoother, in which target number is assumed a
priori to be at most 1: the Bernoulli forward-backward smoother.
• A Poisson approximation of the general multitarget smoother: the PHD
forward-backward smoother.
• An i.i.d.c. approximation of the general multitarget smoother, assuming that
there are no target appearances: the zero target appearances (ZTA) CPHD
smoother. (This is the smoother analog of the ZFA-CPHD filter of Section
8.6.)
14.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
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407
• The single-target forward-backward smoother can be solved in exact closed
form using Gaussian mixture methods (Section 14.2.3).
• Finite-set statistics methods lead to principled multitarget smoothers.
• In particular, the general multitarget forward-backward smoother provides a
Bayes-optimal solution to the multitarget smoothing problem (Section 14.3).
This smoother is given by (see (14.58))
fℓ|k (X|Z
(k)
) = fℓ|ℓ (X|Z
(ℓ)
)
∫
fℓ+1|k (Y |Z (k) ) · fℓ+1|ℓ (Y |X)
δY. (14.8)
fℓ+1|ℓ (Y |Z (ℓ) )
• A special case of (14.8)—the Bernoulli forward-backward smoother—optimally
and tractably addresses the single-target detection and smoothing problem in
clutter with missed detections, and performs successfully when implemented
using either particle or Gaussian mixture methods (Section 14.4).
• The forward-backward smoothed PHDs Dℓ|k (x|Z (k) ) are the PHDs of the
smoothed multitarget distributions fℓ|k (X|Z (k) ). Consequently, the rigorous
approach for determining the Dℓ|k (x|Z (k) ) is to determine the formulas for
the PHDs of the right side of (14.8). These are (see (14.96)):
Dℓ|k (x|Z (k) )
Dℓ|ℓ (x|Z (ℓ) )
ℓ+1|ℓ
=
1 − pS
(x)
ℓ+1|ℓ
+pS
(x)
∫
(14.9)
fℓ+1|ℓ (y|x) · Dℓ+1|k (y|Z (k) )
dy.
Dℓ+1|ℓ (y|Z (ℓ) )
• PHD forward-backward smoothers have somewhat (about 30%) better target
localization accuracy than PHD filters, but tend to be adversely affected by
missed detections or target disappearances (Section 14.5).
• A fast particle implementation of the PHD forward-backward smoother,
based on target labeling, can address significantly large numbers of targets
in significantly dense clutter (Section 14.5.3).
• A generalization of the PHD forward-backward smoother to the CPHD filter
case does not appear to be possible for computational reasons. However,
such a smoother is possible under the simplifying assumption that target
appearances are negligible (Section 14.6).
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14.1.2
Organization of the Chapter
The chapter is organized as follows:
1. Section 14.2: The single-sensor, single-target forward-backward smoother,
including closed-form Gaussian mixture implementation.
2. Section 14.3:
smoother.
The general single-sensor, multitarget forward-backward
3. Section 14.4: The Bernoulli forward-backward smoother, for the case when
target number is known a priori to be no larger than 1.
4. Section 14.5: The PHD forward-backward smoother.
5. Section 14.6: The zero target appearance (ZTA) CPHD forward-backward
smoother—a CPHD smoother, assuming that target appearances are negligible.
14.2
SINGLE-TARGET FORWARD-BACKWARD SMOOTHER
This smoother is defined by the equation [2]:
fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ )
∫
fℓ+1|ℓ (y|x) · fℓ+1|k (y|Z k )
dy.
fℓ+1|ℓ (y|Z ℓ )
(14.10)
It is employed using the following three steps:
1. Forward recursion: Use the recursive Bayes filter and the initial distribution
f0|0 (x) to compute the distributions fℓ+1|ℓ (y|Z l ) for ℓ = 0, ..., k − 1 and
fℓ|ℓ (y|Z ℓ ) for ℓ = 1, ..., k.
2. Backward recursion: Starting with ℓ = k − 1,
computed as
k
fk−1|k (x|Z ) = fk−1|k−1 (x|Z
k−1
)
∫
fk−1|k (x|Z k ) can be
fk|k (y|Z k ) · fk|k−1 (y|x)
dy.
fk|k−1 (y|Z k−1 )
(14.11)
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409
Given fk−1|k (x|Z k ), fk−2|k (x|Z k ) can be computed as
∫
fk−1|k (y|Z k ) · fk−1|k−2 (y|x)
dy.
fk−1|k−2 (y|Z k−2 )
(14.12)
Continuing in this fashion we eventually get to a formula for f1|k (y|Z k ),
from which we compute f0|k (y|Z k ) as
fk−2|k (x|Z k ) = fk−2|k−2 (x|Z k−2 )
k
f0|k (x|Z ) = f0|0 (x)
∫
f1|k (y|Z k ) · f1|0 (y|x)
dy.
f1|0 (y)
(14.13)
3. State estimation: At each stage of the backward recursion, apply a Bayesoptimal state estimator to fℓ|k (x|Z k ) to get smoothed estimates of x at
times tℓ = tk , tk−1 , ..., t0 .
The section is organized as follows:
• Section 14.2.1: Derivation of the single-target forward-backward smoother.
• Section 14.2.2: The Vo-Vo alternative formulation of the forward-backward
smoother.
• Section 14.2.3: The Vo-Vo exact closed-form Gaussian mixture solution of
the forward-backward filter.
14.2.1
Derivation of Forward-Backward Smoother
Let y be the target state at time tℓ+1 , in which case the total probability theorem
and Bayes’ rule gives us
∫
k
fℓ|k (x|Z ) =
fℓ,ℓ+1|k (x, y|Z k )dy
(14.14)
∫
=
fℓ+1|k (y|Z k ) · fℓ|ℓ+1,k (x|y, Z k )dy.
(14.15)
The density fℓ|ℓ+1,k (x|y, Z k ) defines the backward state transition from y at
time tℓ+1 to x at time tℓ .1 Assume that in this transition, x is independent of
1
The notation fℓ,ℓ+1|k (x, y|Z k ) indicates that x is the target state at time tℓ , y is the target
state at time tℓ+1 , and that x, y are conditioned on the measurements through time tk ; and
similarly for fℓ|ℓ+1,k (x|y, Z k ).
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those measurements zℓ+1 , ..., zk that are in its future:
fℓ|ℓ+1,k (x|y, Z k ) = fℓ|ℓ+1,ℓ (x|y, Z ℓ ).
(14.16)
Then:
k
fℓ|k (x|Z )
=
=
∫
fℓ+1|k (y|Z k ) · fℓ|ℓ+1,ℓ (x|y, Z ℓ )dy
(14.17)
∫
fℓ|ℓ+1,ℓ (x|y, Z ℓ )
fℓ|ℓ (x|Z ℓ ) fℓ+1|k (y|Z k ) ·
dy.(14.18)
fℓ|ℓ (x|Z ℓ )
Bayes’ rule then yields
fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ )
∫
fℓ+1|k (y|Z k ) · fℓ+1|ℓ,ℓ (y|x, Z ℓ )
dy
fℓ+1|ℓ (y|Z ℓ )
(14.19)
where fℓ+1|ℓ,ℓ (y|x, Z ℓ ) defines the usual forward state transition from x to y.
Further assume that y does not depend on Z ℓ —that is, fℓ+1|ℓ,ℓ (y|x, Z ℓ ) =
fℓ+1|ℓ (y|x). Then, as claimed,
fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ )
14.2.2
∫
fℓ+1|k (y|Z k ) · fℓ+1|ℓ (y|x)
dy.
fℓ+1|ℓ (y|Z ℓ )
(14.20)
Vo-Vo Alternative Form of the Forward-Backward Smoother
Because of the denominator of the quotient in the integral on the right side of
(14.20), it would appear to be impossible to implement (14.10) exactly using
Gaussian mixture techniques. However, Vo and Vo have shown [302] that the
forward-backward smoother can be reformulated in such a manner as to permit
exact closed-form GM implementation. In fact, their approach appears to be:
• The first general exact closed-form Gaussian mixture solution of the singletarget forward-backward smoother.
Specifically, for ℓ = 0, ..., k − 1, define the unitless function (the backward
corrector)
∫
fℓ+1|k (y|Z k )
Bℓ|k (x) =
· fℓ+1|ℓ (y|x)dy
(14.21)
fℓ+1|ℓ (y|Z ℓ )
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411
and note that Bℓ|ℓ (x) = 1. Also, for ℓ = 1, ..., k define the unitless function
Lℓ (zℓ |x) =
fℓ (zℓ |x)
.
fℓ (zℓ |Z ℓ−1 )
(14.22)
Then (14.10) can be equivalently replaced by the following equations, for ℓ =
0, ..., k − 1 ([302], Eqs. (15-18)):
fℓ|k (x|Z k )
=
Bℓ|k (x)
=
fℓ|ℓ (x|Z ℓ ) · Bℓ|k (x)
(14.23)
∫
Bℓ+1|k (y) · Lℓ+1 (zℓ+1 |y) · fℓ+1|ℓ (y|x)dy. (14.24)
To see why, note that (14.10) becomes
fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ ) · Bℓ|k (x).
(14.25)
From this follows
fℓ|k (x|Z k )
fℓ|ℓ−1 (x|Z ℓ−1 )
=
=
=
fℓ|ℓ (x|Z ℓ )
· Bℓ|k (x)
fℓ|ℓ−1 (x|Z ℓ−1 )
fℓ (zℓ |x)
· Bℓ|k (x)
fℓ (zℓ |Z ℓ−1 )
Lℓ (zℓ |x) · Bℓ|k (x)
(14.26)
(14.27)
(14.28)
and thus, as claimed,
Bℓ−1|k (x)
=
∫
=
∫
fℓ|k (y|Z k )
· fℓ|ℓ−1 (y|x)dy
fℓ|ℓ−1 (y|Z ℓ−1 )
(14.29)
Lℓ (zℓ |y) · Bℓ|k (y) · fℓ|ℓ−1 (y|x)dy.
(14.30)
The Vo-Vo alternative forward-backward smoother is employed using the following
three steps:
1. Forward recursion: Use the recursive Bayes filter and the initial distribution
f0|0 (x) to compute the distributions fℓ+1|l (y|Z ℓ ) for ℓ = 0, ..., k − 1 and
fℓ|ℓ (y|Z ℓ ) for ℓ = 1, ..., k.
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2. Backward recursion: Starting with ℓ = k−1, fk−1|k (x|Z k ) and Bk−1|k (x)
can be computed as
Bk−1|k (x)
=
∫
Lk (zk |y) · fk|k−1 (y|x)dy
(14.31)
fk−1|k (x|Z k )
=
fk−1|k−1 (x|Z k−1 ) · Bk−1|k (x).
(14.32)
=
∫
(14.33)
Then for ℓ = k − 2,
Bk−2|k (x)
Bk−1|k (y) · Lk−1 (zk−1 |y)
·fk−1|k−2 (y|x)dy
fk−2|k (x|Z k )
fk−2|k−2 (x|Z k−2 ) · Bk−2|k (x).
=
(14.34)
Continuing in this fashion we eventually get, with ℓ = 0,
B0|k (x)
=
∫
f0|k (x|Z k )
=
f0|0 (x) · B0|k (x).
B1|k (y) · L1 (z1 |y) · f1|0 (y|x)dy
(14.35)
(14.36)
3. State estimation: At each stage of the backward recursion, apply a Bayesoptimal state estimator to fℓ|k (x|Z k ).
14.2.3
Vo-Vo Exact Closed-Form GM Forward-Backward Smoother
Because (14.23) and (14.24) no longer involve a quotient, Gaussian mixture implementation becomes possible. Suppose that for all ℓ = 1, ..., k,
fℓ (z|x)
fℓ|ℓ−1 (y|x)
=
=
fℓ|ℓ−1 (y|Z ℓ−1 )
=
(14.37)
(14.38)
NRℓ (z − Hℓ x)
NQℓ−1 (y − Fℓ−1 x)
νℓ|ℓ−1
∑
ℓ|ℓ−1
wiℓ|ℓ−1 · NP ℓ|ℓ−1 (y − xi
)
(14.39)
i
i=1
nℓ|k
Bℓ|k (y)
=
∑ ℓ|k
ℓ|k
ci · NC ℓ|k (y − ci ).
i
i=1
(14.40)
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413
From (14.23) and (14.24) it follows that the smoothed distribution fℓ|k (x|Z k )
is also a Gaussian mixture. Specifically, in Section K.24 it is shown that
nℓ|k
Bℓ−1|k (x) =
∑ ℓ−1|k
ci
· NQ
ℓ|k
ℓ|k T
ℓ−1 +Fℓ−1 Di Fℓ−1
(di
(14.41)
− Fℓ−1 x)
i=1
where
ℓ|k
ci
ℓ|k
· NR +H C ℓ|k H T (zℓ − Hℓ ci ) (14.42)
ℓ
ℓ i
ℓ
fℓ (zℓ |Z ℓ−1 )
ℓ−1|k
ci
=
νℓ|ℓ−1
=
∑
(Di )−1
=
ℓ|k
ℓ|k
(Di )−1 di
=
i=1
ℓ|k
(Ci )−1 + HℓT Rℓ−1 Hℓ
ℓ|k
ℓ|k
(Ci )−1 ci + HℓT Rℓ−1 zℓ
fℓ (zℓ |Z
ℓ−1
)
ℓ|k
wiℓ|ℓ−1 · NR +H P ℓ|ℓ−1 H T (z − Hℓ x)
ℓ
ℓ
i
(14.43)
ℓ
(14.44)
(14.45)
or, equivalently,
ℓ|k
ℓ|k
di
=
ℓ|k
Di
=
Kℓ,k
=
ℓ|k
+ Kℓ,k (zℓ − Hℓ ci )
(14.46)
ℓ|k
(I − Kℓ,k Hℓ )Ci
ℓ|k
ℓ|k
Ci HℓT (Hℓ Ci HℓT + Rℓ )−1 .
(14.47)
ci
(14.48)
Then, the smoothed distribution is
k
fℓ−1|k (x|Z )
=
νℓ|ℓ nℓ−1|k
∑
∑
l=1
wlℓ|ℓ ciℓ−1|k
(14.49)
l=1
ℓ|k
·NQ
ℓ|k
ℓ|ℓ T
ℓ−1 +Di +Fℓ−1 Pl Fℓ−1
(di
ℓ|ℓ
− Fℓ−1 xl )
ℓ|ℓ
·NE ℓ|ℓ (y − ei,l )
i,l
where
ℓ|ℓ
(Ei,l )−1
ℓ|ℓ
ℓ|ℓ
(Ei,l )−1 ei,l
ℓ|ℓ
ℓ|k
=
T
(Pl )−1 + Fℓ−1
(Qℓ−1 + Di )−1 Fℓ−1
(14.50)
=
ℓ|ℓ
ℓ|ℓ
ℓ|k
ℓ|k
T
(Pl )−1 xl + Fℓ−1
(Qℓ−1 + Di )−1 di
(14.51)
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Advances in Statistical Multisource-Multitarget Information Fusion
or, equivalently,
ℓ|ℓ
ei,l
ℓ|ℓ
ℓ
xl + Ki,l
(di
=
=
ℓ|ℓ
Ei,l
ℓ
Ki,l
ℓ|k
=
ℓ|ℓ
− Fℓ−1 xl )
(14.52)
ℓ
(I − Ki,l
Fℓ−1 )Pl
ℓ|ℓ
(14.53)
ℓ|ℓ T
ℓ|ℓ
ℓ|k
Pl Fℓ−1
(Pl + Qℓ−1 + Di )−1 .
(14.54)
For a somewhat different formulation and more complete implementation
details, see [302].
Remark 60 (Two-filter smoother) Next to the forward-backward smoother, this
smoother is probably the next most familiar one. It has the form [136]
fℓ|k (x|Z k ) = ∫
fℓ|ℓ (x|Z ℓ ) · fℓ+1 (Z ℓ+1 |x)
fℓ|ℓ (y|Z ℓ ) · fℓ+1 (Z ℓ+1 |y)dy
(14.55)
where fℓ+1 (Z ℓ+1 |x) can be determined recursively using the following “backwardforward information filter”:
fℓ (Z ℓ |x) = fℓ (zℓ |x)
∫
fℓ+1 (Z ℓ+1 |y) · fℓ+1|ℓ (y|x)dy.
(14.56)
The two-filter smoother is not well-suited for particle implementation, which has led
Klass, Briers, de Freitas, Doucet, Maskell, and Lang to propose a more appropriate
alternative form [136], [28].
14.3
GENERAL MULTITARGET FORWARD-BACKWARD SMOOTHER
This is the obvious multitarget analog of the forward-backward smoother. Only the
single-sensor case will be considered here, although the discussion applies equally
well to the multisensor-multitarget case.
Suppose that a single sensor observes multiple targets with state set X. Given
a time sequence Z (k) : Z1 , ..., Zk of measurements, assume—in addition to
the usual assumptions underlying the multitarget Bayes filter—that the backward
multitarget state transition obeys the following conditions, for ℓ = 0, ..., k − 1:
fℓ|ℓ+1,k (X|Y, Z (k) ) = fℓ|ℓ+1,ℓ (X|Y, Z (ℓ) ).
(14.57)
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415
Then the multitarget forward-backward Bayes smoother is defined by the following
direct generalization of (14.10):
∫
fℓ+1|k (Y |Z (k) ) · fℓ+1|ℓ (Y |X)
fℓ|k (X|Z (k) ) = fℓ|ℓ (X|Z (ℓ) )
δY
(14.58)
fℓ+1|ℓ (Y |Z (ℓ) )
where fℓ+1|ℓ (Y |X) is the multitarget Markov density for the standard multitarget
motion model. Equation (14.58) is applied using the same three-step procedure
described at the end of Section 14.2.
The p.g.fl. form of the forward-backward smoother is easily shown to be:
Gℓ|k [h] =
∫
where
F˜ℓ+1|ℓ [r, h] =
δ F˜ℓ+1|ℓ
fℓ+1|k (X ′ |Z (k) )
[0,
h]
·
δX ′
δX ′
fℓ+1|ℓ (X ′ |Z (ℓ) )
∫
hX · Gℓ+1|ℓ [r|X] · fℓ|ℓ (X|Z (ℓ) )δX
and where
Gℓ+1|ℓ [r|X] =
∫
(14.59)
(14.60)
′
r X · fℓ+1|ℓ (X ′ |X)δX ′
(14.61)
is the p.g.fl. of the multitarget Markov density fℓ+1|ℓ (X ′ |X). Equation (14.59) is
the smoother analog of the p.g.fl. form of Bayes’ rule, (5.58).
Equation (14.58) can be equivalently replaced by the multitarget analogs of
(14.23) and (14.24):
fℓ|k (X|Z (k) )
=
Bℓ|k (X)
=
fℓ|ℓ (X|Z (ℓ) ) · Bℓ|k (X)
(14.62)
∫
Bℓ+1|k (Y ) · Lℓ+1 (Zℓ+1 |Y ) · fℓ+1|ℓ (Y |X)δY(14.63)
where
Bℓ|k (X)
=
Lℓ (Zℓ |X)
=
∫
fℓ+1|k (Y |Z (k) )
· fℓ+1|ℓ (Y |X)δY
fℓ+1|ℓ (Y |Z (ℓ) )
fℓ (Zℓ |X)
fℓ (Zℓ |Z (ℓ−1) )
(14.64)
(14.65)
and where, as usual, the integrals are set integrals.
In general, (14.58) or (14.62) and (14.63) will usually not be computationally
tractable. Principled approximations are required, and these are the subjects of the
following subsections.
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14.4
Advances in Statistical Multisource-Multitarget Information Fusion
BERNOULLI FORWARD-BACKWARD SMOOTHER
The Bernoulli filter, which was described in Section 13.2, is a general and Bayesoptimal approach to single-target joint detection and tracking. It is a special case
of the multitarget Bayes filter, in which target number is known, a priori, to be no
greater than 1. Thus the multitarget distribution, (13.1), has the form
fℓ|ℓ (X|Z (ℓ) ) =


1 − pℓ|ℓ
pℓ|ℓ · sℓ|ℓ (x)

0
if
if
if
X=∅
X = {x}
otherwise
(14.66)
where sℓ|ℓ (x) abbr.
= sℓ|ℓ (x|Z (ℓ) ) is the distribution of the target track at time tℓ and
abbr.
pℓ|ℓ = pℓ|ℓ (Z (ℓ) ) is its probability of existence.
The Bernoulli forward-backward smoother is a similarly general and Bayesoptimal approach for single-target detection and smoothing. It was first proposed
by Clark in 2009 [43]; first implemented using particle methods by Clark, Vo, and
Vo [50]; and subsequently implemented by Vo, Clark,Vo, and Ristic [301]. Nagappa
and Clark described a fast particle implementation in [220]. Subsequently, Vo
and Vo discovered an exact closed-form Gaussian mixture implementation, using
the alternative forward-backward formulation described earlier in Section 14.3 (see
[302]).
Clark has proposed a Bernoulli two-filter smoother [50], although this will
not be described here.
The purpose of this section is to describe the alternative formulation of the
Bernoulli forward-backward smoother due to Vo and Vo. The development here is
more direct than theirs, which employed a very general formalism.
The section is organized as follows:
1. Section 14.4.1: Modeling assumptions for the Bernoulli forward-backward
smoother.
2. Section 14.4.2: The Bernoulli forward-backward smoother equations.
3. Section 14.4.3: Exact Gaussian mixture implementation of the Bernoulli
forward-backward smoother.
4. Section 14.4.4: Implementations of the Bernoulli forward-backward smoother.
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14.4.1
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Bernoulli Forward-Backward Smoother: Modeling
As in Section 13.2.1, assume the following models (using notation that is slightly
different than in that section):
• Probability that, if the target has state x′ at time tℓ , then it will not
ℓ+1|ℓ
disappear: pS (x′ ).
ℓ+1|ℓ
• Probability that, if the target is not present at time tℓ , it will appear: pB
.
• Spatial distribution of the target if it appears at time tℓ+1 : ˆbℓ+1|ℓ (x).
• Single-target Markov density: fℓ+1|ℓ (x|x′ ).
• Single-sensor, single-target likelihood function at time tℓ+1 : fℓ+1 (z|x).
• Multiobject probability distribution for an arbitrary clutter RFS: κℓ+1 (Z).
14.4.2
Bernoulli Forward-Backward Smoother: Equations
Define the backward correctors
0
θℓ|k
=
1
θℓ|k
(x)
=
1 − pℓ+1|k
k+1|k
· (1 − pB
)
(14.67)
1 − pℓ+1|ℓ
∫
k+1|k
p
· pℓ+1|k
sℓ+1|k (y) ˆ
+ B
· bℓ+1|ℓ (y)dy
pℓ+1|ℓ
sℓ+1|ℓ (y)
1 − pℓ+1|k
ℓ+1|ℓ
· (1 − pS (x))
(14.68)
1 − pℓ+1|ℓ
∫
ℓ+1|ℓ
p
(x) · pℓ+1|k
sℓ+1|k (y)
+ S
· fℓ+1|ℓ (y|x)dy
pℓ+1|ℓ
sℓ+1|ℓ (y)
and the forward correctors
L0ℓ+1 (Zℓ+1 ) =
1
Lℓ
(14.69)
and
=
L1ℓ+1 (Zℓ+1 |x)
(
)
1 − pℓ+1
(x)
1
D
∑
fℓ+1 (z|x)·κℓ+1 (Zℓ+1 −{z})
+pℓ+1
Lℓ
D (x)
z∈Zℓ+1
κℓ+1 (Zℓ+1 )
(14.70)
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Advances in Statistical Multisource-Multitarget Information Fusion
where
Lℓ = 1 − pℓ+1|ℓ + pℓ+1|ℓ
(
sℓ+1|ℓ [1 − pℓ+1
D ]
+
∑
z∈Zℓ+1
ℓ+1
sℓ+1|ℓ [pℓ+1
]·κℓ+1 (Zℓ+1 −{z})
D Lz
κℓ+1 (Zℓ+1 )
)
(14.71)
and where
sℓ+1|ℓ [1 − pℓ+1
D ]
=
∫
(1 − pℓ+1
D (x)) · sℓ+1|ℓ (x)dx
(14.72)
ℓ+1
sℓ+1|ℓ [pℓ+1
D Lz ]
=
∫
pℓ+1
D (x) · fℓ+1 (z|x) · sℓ+1|ℓ (x)dx.
(14.73)
0
1
Note that θℓ|ℓ
= 1 and θℓ|ℓ
(x) = 1.
Then as will be shown in Section K.25, the equations for the Bernoulli
forward-backward smoother are as follows:
pℓ|k
=
sℓ|k (x)
=
0
1 − (1 − pℓ|ℓ ) · θℓ|k
(14.74)
1
pℓ|ℓ · sℓ|ℓ (x) · θℓ|k
(x)
(14.75)
pℓ|k
0
θℓ|k
=
0
θℓ+1|k
· L0ℓ+1 (Zℓ+1 )
∫
ℓ+1|ℓ
1
+pB
θℓ+1|k
(x) · L1ℓ+1 (Zℓ+1 |x) · ˆbℓ+1|ℓ (x)dx
1
θℓ|k
(x)
=
0
θℓ+1|k
· L0ℓ+1 (Zℓ+1 ) · (1 − pS (x))
(14.77)
∫
ℓ+1|ℓ
1
+pS (x) θℓ+1|k
(y) · L1ℓ+1 (Zℓ+1 |y) · fℓ+1|ℓ (y|x)dy.
(14.76)
ℓ+1|ℓ
These equations are employed as follows:
1. Forward recursion: Use the Bernoulli filter and the initial items p0|0 , s0|0 (x)
to compute the items pℓ+1|ℓ , sℓ+1|ℓ (x) for ℓ = 0, ..., k−1 and pℓ|ℓ , sℓ|ℓ (x)
for l = 1, ..., k.
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0
1
2. Backward recursion: Starting with ℓ = k − 1, the items θk−1|k
, θk−1|k
(x),
pk−1|k (x), and sk−1|k (x) can be computed as
0
θk−1|k
=
L0k (Zk )
k|k−1
+pB
(14.78)
∫
L1k (Zk |x) · ˆbk|k−1 (yx)dx
k|k−1
1
θk−1|k
(x)
=
L0k (Zk ) · (1 − pS
(x))
(14.79)
∫
k|k−1
+pS
(x) L1k (Zk |y) · fk|k−1 (y|x)dy
pk−1|k
=
0
1 − (1 − pk−1|k−1 ) · θk−1|k
(14.80)
1
pk−1|ℓ · sk−1|k−1 (x) · θk−1|k
(x)
sk−1|k (x)
=
(14.81)
.
pk−1|k
Given this, if ℓ = k − 2 then we get
0
θk−2|k
=
0
θk−1|k
· L0k−1 (Zk−1 )
∫
k−1|k−2
1
+pB
θk−1|k
(y) · L1k−1 (Zk−1 |y)
(14.82)
·ˆbk−1|k−2 (y)dy
1
θk−2|k
(x)
k−1|k−2
=
0
θk−1|k
· L0k−1 (Zk−1 ) · (1 − pS
(x)) (14.83)
∫
k−1|k−2
1
+pS
(x) θk−1|k
(y) · L1k−1 (Zk−1 |y)
·fk−1|k−2 (y|x)dy
pk−2|k
=
0
1 − (1 − pk−2|k−2 ) · θk−2|k
(14.84)
1
pk−2|k−2 · sk−2|k−2 (x) · θk−2|k
(x)
sk−2|k (x)
.
=
pk−2|k
(14.85)
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Continuing in this fashion we eventually get, for ℓ = 0,
0
θ0|k
=
0
θ1|k
· L01 (Z1 )
∫
1|0
1
+pB
θ1|k
(y) · L11 (Z1 |y) · ˆb1|0 (y)dy
1
θ0|k
(x)
=
0
θ1|k
· L01 (Z1 ) · (1 − pS (x))
(14.87)
∫
1|0
1
+pS (x) θ1|k
(y) · L11 (Z1 |y) · f1|0 (y|x)dy
p0|k
=
0
1 − (1 − p0|0 ) · θ0|k
1|0
(14.88)
1
p0|0 · s0|0 (x) · θ0|k
(x)
s0|k (x)
(14.86)
=
(14.89)
.
p0|k
3. State estimation: At each stage of the backward recursion, apply a Bayesoptimal state estimator as in Section 13.2.4 to pℓ|k , sℓ|k (x) to get smoothed
estimates of x at times tℓ = tk , tk−1 , ..., t0 .
14.4.3
Bernoulli Forward-Backward Smoother: Exact GM Implementation
This smoother is based on the following assumptions. Suppose that for all ℓ =
1, ..., k,
ℓ|ℓ−1
ℓ|ℓ−1
=
pS
(14.90)
fℓ (z|x)
=
=
pℓD
(14.91)
(14.92)
fℓ|ℓ−1 (y|x)
=
NQℓ−1 (y − Fℓ−1 x)
νℓ|ℓ−1
sℓ|ℓ−1 (x)
=
pS
(x)
pℓD (x)
NRℓ (z − Hℓ x)
∑
(14.93)
ℓ|ℓ−1
wiℓ|ℓ−1 · NP ℓ|ℓ−1 (x − xi
)
(14.94)
i
i=1
nℓ|k
1
θℓ|k
(x)
=
∑ ℓ|k
ℓ|k
ci · NC ℓ|k (x − ci ).
(14.95)
i
i=1
Then the Bernoulli forward-backward smoother equations, (14.74) through (14.77),
can be solved in exact closed form. The specific formulas for this implementation
will not be described here.
For more complete implementation details, see [302].
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14.4.4
421
Bernoulli Forward-Backward Smoother: Results
Two implementations are described in this section: an SMC implementation and a
GM implementation.
SMC implementation by Clark, Vo, Vo, and Ristic [50], [301]. The target
appears at k = 11 and disappears at k = 94, while following a curvilinear
trajectory between these two times. It is observed by a range-bearing sensor with
probability of detection pD = 0.88 and in uniformly distributed Poisson clutter
with clutter rate λ = 30. The target is declared to be present if pℓ|k > 0.5, in
which case its state estimate is the expected value of sℓ|k (x). The smoother was
not run in full batch mode (that is, backwards smoothing from ℓ = k, ..., 0) but,
rather, recursively with a two-step lag: ℓ = k−1, k−2. Performance was measured
using the OSPA metric (Section 6.2.2).
The authors report that the smoother performed better than a corresponding
SMC-PHD filter. It initialized and terminated the track two time-steps earlier than
the PHD filter, and the state estimates were slightly improved.
Exact GM implementation by Vo, Vo, and Mahler [302]. The target appears at
k = 10 and disappears at k = 80, while following a slightly curvilinear trajectory.
It is observed by a linear-Gaussian sensor with probability of detection pD = 0.98
and in uniformly distributed Poisson clutter with clutter rate λ = 7. The target is
declared to be present if pℓ|k > 0.5, in which case its state estimate is the expected
value of sℓ|k (x). The smoother was run using one-step, two-step, and three-step
lags. Performance was measured using the OSPA metric.
The authors reported the expected results: with all three lags, the smoother
initialized and terminated the tracks correctly. State estimation was successively
improved as the lag increased from 1 to 2 to 3.
14.5
PHD FORWARD-BACKWARD SMOOTHER
This smoother was independently and simultaneously proposed by:
• Nadarajah and Kirubarajan [217], [218], using the “physical-space” representation of PHD filters (mentioned in Section 8.4.6.8; see also [179], pp.
599-609).
• Mahler, Vo, and Vo [199], [196], using finite-set statistics p.g.fl. techniques.
The purpose of this section is to describe the PHD forward-backward
smoother and its implementations. The section is organized as follows:
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1. Section 14.5.1: The defining equation for the initial form of the PHD
forward-backward smoother.
2. Section 14.5.2: A sketch of the p.g.fl. derivation of the forward-backward
smoother.
3. Section 14.5.3: A fast sequential Monte Carlo implementation of the PHD
forward-backward smoother, due to Nagappa and Clark.
4. Section 14.5.4: An alternative formulation of the PHD forward-backward
smoother, due to Vo and Vo.
5. Section 14.5.5: The exact closed-form Gaussian mixture solution of the PHD
forward-backward smoother, due to Vo and Vo.
6. Section 14.5.6: Implementations of the PHD forward-backward smoother.
14.5.1
PHD Forward-Backward Smoother Equation
For ℓ = 0, ..., k − 1 let us be given:
• bℓ+1|ℓ (x), the PHD of the target appearance process at time tℓ+1 .
• fℓ+1|ℓ (y|x), the Markov transition density from time tℓ to time tℓ+1 .
• pS,ℓ+1|ℓ (x), the probability of target survival at time tℓ+1 .
Then the PHD forward-backward smoother equation is, for ℓ = 0, ..., k − 1,
given by ([196], p. 5, Proposition 1):
Dℓ|k (x|Z (k) )
Dℓ|ℓ (x|Z (ℓ) )
ℓ+1|ℓ
=
1 − pS
(x)
ℓ+1|ℓ
+pS
(x)
∫
(14.96)
fℓ+1|ℓ (y|x) · Dℓ+1|k (y|Z (k) )
dy
Dℓ+1|ℓ (y|Z (ℓ) )
ℓ+1|ℓ
=
(x)
(14.97)
∫
(k)
fℓ+1|ℓ (y|x) · Dℓ+1|k (y|Z )
ℓ+1|ℓ
+pS (x)
dy
bℓ+1|ℓ (y) + ρℓ+1|ℓ (y)
1 − pS
where the second equation follows from the PHD filter time-update equation, (8.15),
where
∫
ℓ+1|ℓ
ρℓ+1|ℓ (y) = pS (x) · fℓ+1|ℓ (y|x) · Dℓ|ℓ (x|Z (ℓ) )dx.
(14.98)
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423
If there are no target appearances or disappearances and if exactly one target
is known to exist, then (14.96) reduces to the single-target forward-backward
smoother equation, (14.10).
The derivation of (14.96) requires the following assumptions:
• The multitarget distributions fℓ|ℓ (X|Z (ℓ) ) are Poisson for ℓ = 0, 1, ..., k.
• The multitarget distributions fℓ+1|ℓ (X|Z (ℓ) ) are Poisson for ℓ = 0, ..., k−1.
If in addition the smoothed multitarget distributions fℓ+1|ℓ (X|Z (ℓ) ) are assumed to be Poisson, then it is additionally possible to determine the formula for
the smoothed cardinality distribution ([196], p. 7, Proposition 2):
∫
pℓ|k (n) = e E(y)dy
ℓ|ℓ−1
n
∑
Dℓ|ℓ [1 − pS ]n−i
i=0
(n − i)!
[
]
bℓ+1|ℓ i
· Dℓ+1|k 1 −
Dℓ+1|ℓ
(14.99)
where
E(y)
=
bℓ+1|ℓ (y) · Dℓ+1|k (y|Z (k) )
Dℓ+1|ℓ (y|Z (ℓ) )
(14.100)
−Dℓ+1|k (y|Z (k) ) + Dℓ+1|ℓ (y|Z (ℓ) )
ℓ|ℓ−1
Dℓ|ℓ [1 − pS ]
[
]
bℓ+1|ℓ
Dℓ+1|k 1 −
Dℓ+1|ℓ
=
=
−bℓ+1|ℓ (y) − Dℓ|ℓ (y|Z (ℓ) )
∫
ℓ|ℓ−1
(1 − pS (x)) · Dℓ|ℓ (x|Z (ℓ) )dx (14.101)
)
∫ (
bℓ+1|ℓ (y)
1−
(14.102)
Dℓ+1|ℓ (y|Z (ℓ) )
·Dℓ+1|k (y|Z (k) )dy.
It is also possible to derive formulas for the mean and variance of pℓ|k (n) ([196],
p. 7, Proposition 3), though these will not be given here.
The PHD forward-backward smoother is employed using a three-step process
similar to that for the single-target forward-backward smoother:
1. Forward recursion: Use the conventional PHD filter and the initial PHD
D0|0 (x) to compute the PHDs Dℓ+1|ℓ (y|Z (ℓ) ) for ℓ = 0, ..., k − 1 and
Dℓ|ℓ (y|Z (ℓ) ) for ℓ = 1, ..., k.
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2. Backward recursion: Starting with ℓ = k − 1, the PHD Dk−1|k (x|Z k ) can
be computed using
Dk−1|k (x|Z (k) )
Dk−1|k−1 (x|Z (ℓ) )
(14.103)
k|k−1
=
1 − pS
(x)
∫
fk|k−1 (y|x) · Dk|k (y|Z (k) )
k|k−1
+ pS
(x)
dy.
Dk|k−1 (y|Z (k−1) )
Given Dk−1|k (x|Z (k) ), Dk−2|k (x|Z (k) ) can be computed as
Dk−2|k (x|Z (k) )
Dk−2|k−2 (x|Z (k−2) )
(14.104)
k−1|k−2
=
1 − pS
(x)
∫
fk−1|k−2 (y|x) · Dk−1|k (y|Z (k) )
k−1|k−2
+pS
(x)
dy.
Dk−1|k−2 (y|Z (k−2) )
Continuing in this fashion, we eventually get to a formula for D1|k (y|Z (k) ),
from which we compute D0|k (y|Z (k) ) as
D0|k (x|Z (k) )
D0|0 (x)
1|0
=
(14.105)
1 − pS (x)
1|0
+pS (x)
∫
f1|0 (y|x) · D1|k (y|Z (k) )
dy.
D1|0 (y)
3. State estimation: At each stage of the backward recursion, apply the usual
PHD filter state-estimation approach to Dℓ|k (x|Z (k) ) to get smoothed
estimates of the multitarget state set X at times tℓ = tk , tk−1 , ..., t0 .
14.5.2
Derivation of the PHD Forward-Backward Smoother
The basic idea behind the derivation is as follows. By definition,
• The forward-backward smoothed PHD is the PHD of the forward-backward
smoothed multitarget distribution, which was defined in (14.58).
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425
It is most easily derived via direct substitution into the p.g.fl. forwardbackward smoother equation, (14.59). The derivation consists of the following
steps:
1. Step 1: Derive the function F˜ℓ+1|ℓ [r, h] for the standard multitarget motion
model:
ℓ+1|ℓ
ℓ+1|ℓ
F˜ℓ+1|ℓ [r, h] = GB
+ pS Mrℓ+1|ℓ )]
ℓ+1|ℓ [r] · Gℓ|ℓ [h(1 − pS
(14.106)
where GB
ℓ+1|ℓ [r] is the p.g.fl. of the target appearance process and where
Mrℓ+1|ℓ (x) =
∫
r(x′ ) · fℓ+1|ℓ (x′ |x)dx′ .
(14.107)
2. Step 2: Into (14.106), substitute the equations
bℓ+1|ℓ [r−1]
GB
,
ℓ+1|ℓ [r] = e
Gℓ|ℓ [h] = eDℓ|ℓ [h−1]
(14.108)
to get
F˜ℓ+1|ℓ [r, h] = exp
(
bℓ+1|ℓ [r − 1] − Nℓ|ℓ
ℓ+1|ℓ
ℓ+1|ℓ
ℓ+1|ℓ
+Dℓ|ℓ [h(1 − pS
+ pS M r
)]
)
. (14.109)
3. Step 3: Construct the functional derivatives of F˜ℓ+1|ℓ [r, h] with respect to
r:
δ F˜ℓ+1|ℓ
X′
[0, h] = F˜ℓ+1|ℓ [0, h] · γℓ+1|ℓ
(14.110)
′
δX
where
ℓ+1|ℓ
γℓ+1|ℓ (x′ ) = bℓ+1|ℓ (x′ ) + Dℓ|ℓ [pS
ℓ+1|ℓ
M x′
].
(14.111)
4. Step 4: Construct the first functional derivative of (δ F˜ℓ+1|ℓ /δX ′ )[0, h] with
respect to h and then set h = 1:
δ F˜ℓ+1|ℓ
[0, 1]
δX ′ δx 
=
X′
θℓ+1|ℓ
·
(14.112)

δ F˜ℓ+1|ℓ
δx [0, 1]
ℓ+1|ℓ
ℓ+1|ℓ
∑
D (x)·pS
(x)·Mx′
(x)
+F˜ℓ+1|ℓ [0, 1] · x′ ∈X ′ ℓ|ℓ
θℓ+1|ℓ (x′ )
.
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5. Step 5: Into (14.59), substitute (14.112) and the equations
fℓ+1|ℓ (X|Z (k) )
′
fℓ+1|k (X |Z
(k)
)
′
=
X
e−Nℓ+1|ℓ · Dℓ+1|ℓ
=
−Nℓ+1|k
e
X′
· Dℓ+1|k
.
(14.113)
(14.114)
6. Step 6: Derive the PHD smoother equation using algebra, based on Campbell’s theorem (4.96), and the formula
∫ (
=
θℓ+1|ℓ · Dℓ+1|k
Dℓ+1|ℓ
){x′ }∪X ′
δX ′
(14.115)
θℓ+1|ℓ (x′ ) · Dℓ+1|k (x′ )
Dℓ+1|ℓ (x′ )
(∫
)
θℓ+1|ℓ (x′ ) · Dℓ+1|k (x′ ) ′
· exp
dx
Dℓ+1|ℓ (x′ )
for the PHD of the function
f˜(X ′ ) =
(
θℓ+1|ℓ · Dℓ+1|k
Dℓ+1|ℓ
)X ′
.
(14.116)
Remark 61 The proof given in [196] is more general than the one just sketched,
in that one can relax the assumption that the intermediary smoothed distributions
fℓ+1|k (X ′ |Z (k) ) are Poisson.
14.5.3
Fast Particle-PHD Forward-Backward Smoother
Mahler, Vo, and Vo proposed a particle implementation of (14.97) in [196]. Subsequently, Nagappa and Clark noted that, because of the backward-smoothing step,
this implementation is computationally expensive: O(n2 ν 2 ), where n is the
current number of tracks and ν is the current number of particles assigned per
target. To remedy this problem, they devised a significantly faster implementation,
one with the following advantages [220]:
• The computational complexity is O(nν 2 ) rather than O(n2 ν 2 ).
• Is independent of the clutter rate.
The Nagappa-Clark SMC-PHD forward-backward smoother is based on two
ideas:
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427
1. Append a label state variable τ to the target state, x̊ = (τ, x).
2. During Markov transitions, never allow states to change labels:
fℓ+1|ℓ (τ, x|τ ′ , x′ ) = fℓ+1|ℓ (x|x′ ) · δτ,τ ′ .
(14.117)
The addition of the label variable τ necessitates the following redefinitions:
pℓD (τ, x)
fℓ (z|τ, x)
=
=
pℓD (x)
fℓ (z|x).
(14.118)
(14.119)
Given this, (14.97) becomes:
Dℓ|k (τ, x|Z (k) )
Dℓ|ℓ (τ, x|Z (ℓ) )
ℓ+1|ℓ
=
1 − pS
+
∫
(14.120)
(x)
ℓ+1|ℓ
pS
(x) · fℓ+1|ℓ (y|x) · Dℓ+1|k (τ, y|Z (ℓ) )
dy
bℓ+1|ℓ (τ, y) + ρℓ+1|ℓ (τ, y)
where
ρℓ+1|ℓ (τ, y)
=
∑∫
ℓ+1|ℓ
pS
(x) · δτ,τ ′ · fℓ+1|ℓ (y|x)
(14.121)
τ′
=
·Dℓ|ℓ (τ ′ , x|Z (ℓ) )dx
∫
ℓ+1|ℓ
pS (x) · fℓ+1|ℓ (y|x) · Dℓ|ℓ (τ, x|Z (ℓ) )dx.(14.122)
For the purpose of the forward recursion, the target-birth PHD bℓ+1|ℓ (τ, y)
is defined as follows. For each new measurement zj a Gaussian component is
created, along with a unique label, resulting in the Gaussian mixture:
mℓ+1
bℓ+1|ℓ (τ, x) =
∑
bℓ+1
· δτ,τ ℓ+1 · NP ℓ+1 (x − xℓ+1
).
j
j
j
(14.123)
j
j=1
Again for the purpose of the forward recursion, this representation can be approximated as a Dirac mixture as follows. Particles are drawn from each NP ℓ+1 (x −
j
xℓ+1
), assigning them the corresponding label. Then
j
m̃ℓ+1
bℓ+1|ℓ (τ, y) ∼
=
∑
l=1
bℓ+1
· δτ,τ̃ ℓ+1 · δx̃ℓ+1 (x)
l
l
l
(14.124)
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where for each l, τ̃lℓ+1 is equal to τjℓ+1 for some j. The forward recursion
of the smoother is then implemented as a conventional SMC-PHD filter (Section
9.6.2) in order to obtain particle representations of the PHDs Dℓ|ℓ (τ, x|Z (ℓ) ) and
Dℓ+1|ℓ (τ, x|Z (ℓ) ).
Two points should be made:
• Once a particle has been created, it and its surviving resampled copies retain
the same label thereafter.
• Thus when the forward recursion terminates with the construction of the
particle representation of Dk|k (τ, x|Z (k) ), the full complement T of
assigned labels has been created.
• Since the particle representation of Dk−1|k (τ, x|Z (k) ) derives from the
particle representation of Dk−1|k−1 (τ, x|Z (k−1) ), the labels of the former
are the same as the labels of the latter. More generally, the labels of
Dℓ|k (τ, x|Z (k) ) are also labels drawn from T .
For the backward recursion, assume that
Dℓ|ℓ (τ, y|Z
(k)
)
νℓ|ℓ
∑
=
ℓ|ℓ
(14.125)
wi · δτ ℓ|ℓ ,τ · δxℓ|ℓ (y)
i
i
i=1
νℓ+1|k
Dℓ+1|k (τ, y|Z (k) )
∑
=
ℓ+1|k
wl
· δτ ℓ+1|k ,τ · δxℓ+1|k (y). (14.126)
l
l
l=1
Then (14.120) becomes
=
Dℓ|k (τ, x|Z (k) )
(14.127)


ℓ+1|ℓ ℓ|ℓ
νℓ|ℓ
1
−
p
(x
)
∑
i
S
ℓ|ℓ
ℓ|ℓ
ℓ|ℓ
ℓ|ℓ

wi ·  ∫ pℓ+1|ℓ
(xi )·fℓ+1|ℓ (y|xi )·Dℓ+1|k (τi ,y|Z (ℓ) )
S
+
dy
ℓ|ℓ
ℓ|ℓ
i=1
bℓ+1|ℓ (τi ,y)+ρℓ+1|ℓ (τi ,y)
·δτ ℓ|ℓ ,τ · δxℓ|ℓ (y)
i
i
where
ρℓ+1|ℓ (τ, y) =
νℓ|ℓ
∑
i=1
ℓ+1|ℓ
ℓ|ℓ
wi · δτ ℓ|ℓ ,τ · pS
i
ℓ|ℓ
ℓ|ℓ
(xi ) · fℓ+1|ℓ (y|xi )
(14.128)
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and where
∫
ℓ+1|ℓ
pS
ℓ|ℓ
ℓ|ℓ
ℓ|ℓ
(xi ) · fℓ+1|ℓ (y|xi ) · Dℓ+1|k (τi , y|Z (ℓ) )
ℓ|ℓ
dy (14.129)
ℓ|ℓ
bℓ+1|ℓ (τi , y) + ρℓ+1|ℓ (τi , y)
ℓ+1|k
νℓ+1|k
=
∑ wi
ℓ+1|ℓ
· δτ ℓ+1|k ,τ ℓ|ℓ · pS
ℓ|ℓ
ℓ+1|k
ℓ+1|ℓ
· pS
wi
=
ℓ|ℓ
ℓ+1|k
ℓ|ℓ
ℓ+1|k
) + ρℓ+1|ℓ (τi , xl
ℓ|ℓ
ℓ+1|k
(xi ) · fℓ+1|ℓ (xi
ℓ+1|k
bℓ+1|ℓ (τi , xi
ℓ|ℓ
|xi )
(14.130)
ℓ+1|k
bℓ+1|ℓ (τi , xl
l=1
ℓ|ℓ
(xi ) · fℓ+1|ℓ (xl
i
l
ℓ|ℓ
|xi )
ℓ|ℓ
)
(14.131)
.
ℓ+1|k
) + ρℓ+1|ℓ (τi , xi
)
ℓ+1|k
The last equation follows from the fact that the label τl
must always be equal
ℓ|ℓ
to one of the labels τi . Thus the particle representation of the backward recursion
equation, (14.120), is
ℓ|k
wi
(14.132)
ℓ|ℓ
wi
ℓ+1|k
ℓ+1|ℓ
=
1 − pS
ℓ|ℓ
ℓ+1|ℓ
· pS
wi
(xi ) +
ℓ|ℓ
ℓ|ℓ
ℓ+1|k
(xi ) · fℓ+1|ℓ (xi
ℓ+1|k
bℓ+1|ℓ (τi , xi
ℓ|ℓ
ℓ|ℓ
|xi )
.
ℓ+1|k
) + ρℓ+1|ℓ (τi , xi
)
The elimination of the summation in (14.130) is what accounts for the improved
computational characteristics of this implementation of the SMC-PHD forwardbackward smoother.
14.5.4
Alternative PHD Forward-Backward Smoother
This is constructed in direct analogy with (14.23) through (14.49), by defining a
recursive backward corrector. For ℓ = 0, ..., k − 1, define the backward corrector
ℓ+1|ℓ
ℓ+1|ℓ
Bℓ|k (x) = 1−pS (x)+pS (x)
∫
Dℓ+1|k (y|Z (k) )
·fℓ+1|ℓ (y|x)dy (14.133)
Dℓ+1|ℓ (y|Z (ℓ) )
and for ℓ = 1, ..., k the forward corrector
Lℓ (Zℓ |x) = 1 − pℓD (x) +
∑ pℓ (x) · fℓ (z|x)
D
κℓ (z) + τℓ (z)
z∈Zℓ
(14.134)
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where
τℓ (z) =
∫
pℓD (x) · fℓ (z|x) · Dℓ|ℓ−1 (x|Z (ℓ−1) )dx.
(14.135)
Then (14.96) can be equivalently replaced by [302], Eqs. (79-83)
Dℓ|k (x|Z (k) )
Bℓ|k (x)
=
Dℓ|ℓ (x|Z (ℓ) ) · Bℓ|k (x)
(14.136)
=
ℓ|ℓ−1
1 − pS (x)
(14.137)
ℓ|ℓ−1
+pS
(x)
∫
Bℓ|k (x) · Lℓ (Zℓ |x) · fℓ|ℓ−1 (y|x)dy
for ℓ = 0, ..., k − 1.
This is easily demonstrated as follows. From (14.136) we get
Dℓ|k (x|Z k )
Dℓ|ℓ−1 (x|Z (ℓ−1) )
=
Dℓ|ℓ (x|Z ℓ )
· Bℓ|k (x)
Dℓ|ℓ−1 (x|Z (ℓ−1) )
(14.138)
=
Lℓ (Zℓ |x) · Bℓ|k (x).
(14.139)
Thus, as claimed,
ℓ|ℓ−1
Bℓ−1|k (x)
(x)
(14.140)
∫
(k)
Dℓ|k (y|Z )
ℓ|ℓ−1
+pS (x)
· fℓ|ℓ−1 (y|x)dy
Dℓ|ℓ−1 (y|Z (ℓ) )
=
1 − pS
=
1 − pS
ℓ|ℓ−1
14.5.5
(x)
(14.141)
∫
ℓ|ℓ−1
+pS (x) Bℓ|k (x) · Lℓ (Zℓ |x) · fℓ|ℓ−1 (y|x)dy.
Gaussian-Mixture PHD Smoother
Vo and Vo have demonstrated that, as with the single-target forward-backward
smoother, the PHD forward-backward smoother, (14.96), can be implemented in
exact closed form using GM methods [314], [302]. The forward recursion is
implemented using the conventional GM-PHD filter formulas (Section 9.5.4). The
key is the alternative form of the PHD backward recursion described in the previous
section.
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This smoother is based on the following assumptions. Suppose that for all
ℓ = 1, ..., k,
ℓ|ℓ−1
pS
ℓ|ℓ−1
(x)
pℓD (x)
fℓ (z|x)
fℓ|ℓ−1 (y|x)
=
pS
(14.142)
=
=
=
pℓD
(14.143)
(14.144)
(14.145)
NRℓ (z − Hℓ x)
NQℓ−1 (y − Fℓ−1 x)
and
νℓ|ℓ−1
Dℓ|ℓ−1 (y|Z ℓ−1 )
=
∑
ℓ|ℓ−1
wiℓ|ℓ−1 · NP ℓ|ℓ−1 (y − xi
)
(14.146)
i
i=1
νℓ|ℓ
Dℓ|ℓ (y|Z ℓ )
=
∑
ℓ|ℓ
wiℓ|ℓ · NP ℓ|ℓ (y − xi )
(14.147)
∑ ℓ|k
ℓ|k
ci · NC ℓ|k (y − ci ).
(14.148)
i
i=1
nℓ|k
Bℓ|k (y)
=
i
i=1
Then the PHD forward-backward smoother equations, (14.136) and (14.137), can
be solved in closed form.
The equations for this implementation will not be further discussed here. For
more complete implementation details, see [302].
As already noted, a major consequence of the Vo-Vo approach is that the
exact closed-form solution is preserved in the single-sensor special case (see Section
14.2.3).
14.5.6
Implementations of the PHD Forward-Backward Smoother
The following experimental implementations will be described in this section:
• Conventional SMC implementation by Mahler, Vo, and Vo [196].
• Fast SMC implementation by Nagappa and Clark [220].
• Exact GM implementation by Vo, Vo, and Mahler [302].
• SMC implementation by Nadarajah and Kirubarajan, including multiple
motion-model implementation [217], [218].
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Conventional SMC implementation by Mahler, Vo, and Vo [196]. Five appearing and disappearing targets, following curvilinear trajectories, were observed by a
single range-bearing sensor with probability of detection pD = 0.98 and clutter
rate λ = 7. The SMC-PHD smoother was applied with a smoother-lag time of
5, and compared to an SMC-PHD filter. The authors reported that the smoother
achieved about 33% better localization accuracy than the SMC-PHD filter, but exhibited approximately the same accuracy in target-number estimation.
Further investigation revealed a curious anomaly: the smoother successfully
removed the effect of false alarms, but did not respond well to missed detections or
to target disappearances. The authors conjectured that this behavior is attributable
to the Poisson approximation made in both the filter and smoother. Because
this approximation attempts to model target number using a single parameter, the
smoother may lack enough degrees of freedom to account for sudden decreases in
target number (whether due to missed detections or target disappearances).
Fast SMC implementation by Nagappa and Clark [220]. Twelve appearing
and disappearing targets, following curvilinear trajectories, were observed by a
single range-bearing sensor with probability of detection pD = 0.98 and a
clutter rate λ = 30. The authors observed the same behavior as reported in
[196]. While exhibiting significantly better localization accuracy than the filter,
the smoother experienced some difficulty with missed detections and disappearing
targets. Because of its greater computational efficiency, however, the smoother was
able to accommodate larger numbers of targets, and in denser clutter. The authors
verified that the fast smoother is linear in the number of targets, and essentially flat
with respect to the clutter rate.
Exact closed-form GM implementation by Vo, Vo, and Mahler [302]. Four
persisting targets appear simultaneously at the origin and follow linear trajectories
along the coordinate axes thereafter. They are observed by a linear-Gaussian sensor
with probability of detection pD = 0.98 and uniformly distributed Poisson clutter
with clutter rate λ = 7. The smoother was run using one-step, two-step, and
three-step lags. Performance, as measured using the OSPA metric (Section 6.2.2),
conformed to expectations: the smoother initialized and terminated the tracks
correctly with all three lags; and state estimation was improved as the lag increased
from 1 to 2 to 3.
SMC implementation by Nadarajah and Kirubarajan [217], [218]. These
authors used SMC techniques to implement two versions of the PHD smoother:
with and without multiple motion models. The simulation results for both versions
will be reported in turn.
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For the simulations involving an a priori motion model, the authors tested their
algorithm in a two-dimensional scenario involving a single range-bearing sensor
and three appearing and disappearing targets, with the first target existing for only a
short period of time. Probability of detection was pD = 0.9, clutter rate was either
λ = 20 or λ = 50, and the smoother lag ranged from 1 to 5. The PHD smoother,
with smoother lags 1, 2, 3, 4, and 5, was compared to a conventional SMC-PHD
filter using the Wasserstein multitarget miss-distance (see Section 6.2.1.2). For
λ = 20, the authors reported that the PHD smoother significantly outperformed
the PHD filter for lags 1, 2, and 3, but with no discernible improvement for lags
4 and 5. Similar results were observed for λ = 50, although—as expected—the
performance of both the smoother and the filter were degraded compared to the
λ = 20 case.
For the simulations involving multiple motion models, two models were
used: a constant-velocity model and a coordinated-turn model. In this case, two
maneuvering targets were present throughout the entire scenario, and the clutter
rate was λ = 20. The multiple-model PHD smoother was compared to a multiplemodel PHD filter. Once again, the smoother significantly outperformed the filter
in regard to tracking performance. The smoother also exhibited somewhat better
behavior in regard to estimating which motion model was in effect at any given
time.
14.6
ZTA-CPHD SMOOTHER
Is there a CPHD forward-backward smoother? It is possible to derive exact
formulas for such a smoother, but they are computationally intractable. Suppose,
however, that target appearances in each filtering cycle are negligible—in which
case we can also neglect the CPHD filter’s target appearance model. Then the
resulting smoother—the “zero target appearances” (ZTA) CPHD forward-backward
smoother—becomes tractable. The purpose of this section is to briefly describe this
smoother (but without proof, since the result appears to be a relatively minor one).
In the multitarget forward-backward smoother equation, (14.58), assume that
the distributions fℓ|ℓ (X|Z (ℓ) ), fℓ+1|ℓ (Y |Z (ℓ) ), fℓ+1|k (Y |Z (k) ) are i.i.d.c. in the
sense of Section 4.3.2. The corresponding cardinality and spatial distributions are
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pℓ|ℓ (n), pℓ+1|ℓ (n), pℓ+1|k (n) and sℓ|ℓ (x), sℓ+1|ℓ (x), sℓ+1|k (x). Define:
ℓ+1|ℓ
ψℓ|ℓ
=
θℓ,k
=
ℓ+1|ℓ
sℓ|ℓ [1 − pS ] = 1 − sℓ|ℓ [pS ]
∑ (1 − ψℓ|ℓ )n · pℓ+1|k (n)
(n+1)
· Gℓ|ℓ (ψℓ|ℓ )
n! · pℓ+1|ℓ (n)
(14.149)
(14.150)
n≥0
+
θℓ,k
∑ (1 − ψℓ|ℓ )n · pℓ+1|k (n + 1)
=
n! · pℓ+1|ℓ (n + 1)
n≥0
(n+1)
· Gℓ|ℓ
(ψℓ|ℓ ).
(14.151)
Then using the p.g.fl. forward-backward smoother equation, (14.59), the ZTACPHD smoother equations can be shown to be:
Gℓ|k (x)
=
n
∑ pℓ+1|k (n) · θℓ,k
n≥0
=
n! · pℓ+1|ℓ (n)
(n)
· Gℓ|ℓ (x · ψℓ|ℓ ) · xn
∑ (1 − ψℓ|ℓ )n · pℓ+1|k (n)
n≥0
n ·p
ψℓ|ℓ
ℓ+1|ℓ (n)
∑
pℓ|ℓ (i) · Ci,n · (ψℓ|ℓ x)i
·
(14.152)
(14.153)
i≥n
and
Dℓ|k (x)
sℓ|ℓ (x)
ℓ+1|ℓ
=
(1 − pS
ℓ+1|ℓ
+pS
(x)) · θℓ,k
+
(x) · θℓ,k
∫
(14.154)
sℓ+1|k (y) · fℓ+1|ℓ (y|x)
dy
sℓ+1|ℓ (y)
where Ci,n is the binomial coefficient as defined in (2.1). It is easily shown that if
pℓ+1|ℓ (n), pℓ+1|k (n), and pℓ|ℓ (n) are Poisson distributions, this formula reduces
to the PHD smoother equation, (14.97), but with no target appearances, that is,
bℓ+1|ℓ (y) = 0.
The ZTA-CPHD smoother can be implemented in exact closed form using
Gaussian mixture techniques, in the same manner as described in Section 14.5.5.
Chapter 15
Exact Closed-Form Multitarget Filter
15.1
INTRODUCTION
The single-sensor, multitarget Bayes filter is the optimal approach for singlesensor, multitarget detection, tracking, and identification. However, it is also
computationally intractable except for very simple tracking problems. Thus far,
a number of filters have been described that tractably approximate the multitarget
Bayes filter. These include PHD filters, CPHD filters, and multi-Bernoulli filters.
Let us briefly review the basic approximation philosophy motivating these filters.
• PHD and CPHD filters: In the theory of PHD and CPHD filters (Chapter 8),
no attempt is made to try to accurately approximate the multitarget posterior
distribution fk|k (X|Z (k) ) itself. Rather, the information in fk|k (X|Z (k) ) is
lossily compressed into multitarget moments of various kinds. Consequently,
this information loss is expected to result in a significant degradation of
performance in comparison to the optimal filter.
• Multi-Bernoulli filters: The theory of multi-Bernoulli filters (Chapter 13)
goes one step further, in that an attempt is being made to approximate
fk|k (X|Z (k) ) with some degree of accuracy. Specifically, fk|k (X|Z (k) ) is
approximated by a multi-Bernoulli distribution. However, this approximation
is not general, since the target-number variance must always be smaller
than the target-number mean. It is also not exact, since several additional
approximations must be assumed in order to achieve closed-form formulas.
One can therefore ask if there is a solution that is both general and exact:
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• Is there an exact closed-form (and computationally tractable) solution of the
multitarget Bayes filter, in the same sense that the Kalman filter is an exact
closed-form solution of the single-target Bayes filter?
As was discovered by B.-T. Vo and B.-N. Vo, the answer is—somewhat
surprisingly—Yes [295], [296]. Their exact closed-form solution is based on the
fact that target tracks must be distinctly labeled if temporal track-continuity is to be
achieved. This, in turn, means that the concept of a random finite set (RFS) must
be generalized to that of a labeled random finite set (labeled RFS, for short).
The discovery of the Vo-Vo exact-closed form multitarget filter has the following significant consequences:
• The Vo-Vo exact closed-form multitarget filter is apparently the very first
tractable, provably Bayes-optimal multitarget detection and tracking algorithm.
• Consequently, it is apparently also the first multitarget detection and tracking
algorithm to have a provably Bayes-optimal track-management scheme.
• The approach is easily extended to permit, in a theoretically rigorous fashion,
the inclusion of target-type tags—and thus to permit provably Bayes-optimal
joint multitarget detection, tracking, and target identification.
• Using suitable approximations, the filter can be made efficient in regard to
both tracking performance and computational throughput.
• The “spookiness” phenomenon, discussed in Section 9.2, affects PHD and
CPHD filters and, to a lesser extent, the CBMeMBer filter of Section 13.4.
Because the Vo-Vo filter is an exact closed-form solution of the general
multitarget Bayes filter, one would expect that it would not exhibit this
phenomenon.1 Vo and Vo have verified that this is indeed the case, that is,
the exact closed-form filter does not exhibit spooky behavior [304].
The purpose of this chapter is to describe in detail the theory and practice of
the Vo-Vo exact closed-form multitarget Bayes filter. Because of the importance of
the result, a significant amount of space will be devoted to the demonstration that
the Vo-Vo filter is indeed exact closed-form and thus Bayes-optimal.
The remainder of this introduction is organized as follows:
1
For, the contrary result would mean that—despite its optimality—the multitarget Bayes filter itself
exhibits spookiness.
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1. Section 15.1.1: The concept of an exact closed-form solution of the singlesensor, single-target Bayes filter.
2. Section 15.1.2: The concept of an exact closed-form solution of the singlesensor, multitarget Bayes filter.
3. Section 15.1.3: Overview of the Vo-Vo filter approach.
4. Section 15.1.4: A summary of the major lessons learned in this chapter.
5. Section 15.1.5: The organization of the chapter.
15.1.1
Exact Closed-Form Solution of the Single-Target Bayes Filter
The Kalman filter is an exact, algebraically closed-form solution of the singlesensor, single-target Bayes filter, in the following three-stage sense.
First, suppose that both target motion and the prior track distribution are
linear-Gaussian:
fk+1|k (x|x′ )
=
NQk (x − Fk x′ )
(15.1)
k
=
NPk|k (x − xk|k ).
(15.2)
fk|k (x|Z )
Then the Bayes filter time-update formula (prediction integral) can be evaluated
exactly, and results in an exact closed-form linear-Gaussian predicted track density
function ([179], pp. 35-36):
∫
k
fk+1|k (x|Z ) =
fk+1|k (x|x′ ) · fk|k (x′ |Z k )dx′
(15.3)
=
NPk+1|k (x − xk+1|k )
(15.4)
where
Pk+1|k
=
Qk + Fk Pk|k FkT
(15.5)
xk+1|k
=
Fk xk|k .
(15.6)
Second, suppose that both the sensor likelihood function and the predicted
track distribution are linear-Gaussian:
fk+1 (z|x)
=
NRk+1 (z − Hk+1 x)
(15.7)
k
=
NPk+1|k (x − xk+1|k ).
(15.8)
fk+1|k (x|Z )
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Then the Bayes filter measurement-update formula (Bayes’ rule) can be evaluated
in exact closed form, resulting in the linear-Gaussian measurement-updated density
function ([179], pp. 37-39):
fk+1k+1 (x|Z k+1 )
fk+1 (zk+1 |x) · fk+1|k (x|Z k )
fk+1 (zk+1 |y) · fk+1|k (y|Z k )dy
=
∫
=
NPk+1|k+1 (x − xk+1|k+1 )
(15.10)
(15.9)
where
−1
Pk+1|k+1
=
−1
−1
Pk+1|k
+ HkT Rk+1
Hk
(15.11)
−1
Pk+1|k+1
xk+1|k+1
=
−1
−1
Pk+1|k
xk+1|k + HkT Rk+1
zk+1 .
(15.12)
The family of linear-Gaussian distributions is thereby said to be a family of conjugate priors for the family of likelihood functions Lz (x) = NRk+1 (z − Hk+1 x).
Third, it follows that the entire Bayes filter can be solved in exact closed form
if:
• The Markov transition densities
fk+1|k (x|x′ ) = NQk (x − Fk x′ )
(15.13)
are linear-Gaussian.
• The sensor likelihood functions
fk+1 (z|x) = NRk+1 (z − Hk+1 x)
(15.14)
are linear-Gaussian.
• The initial distribution
f0|0 (x) = NP0|0 (x − x0|0 )
(15.15)
is linear-Gaussian.
Stated more formally: Let D be the family of linear-Gaussian probability
density functions fp (x) parametrized by the space P of parameters p = (x, P ).
Then this family has the following three properties:
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1. Exact closed-form closure with respect to the prediction integral: if fpk|k ∈
D with pk|k = (xk|k , Pk|k ); and if
f+ (x) =
∫
fk+1|k (x|x′ ) · fpk|k (x′ )dx′
(15.16)
is the Markov prediction of fpk|k using the specified Markov transition
′
density fk+1|k (x|x ) = NQk (x − Fk x′ ); then there exists a fpk+1|k ∈ D
with pk+1|k = (xk+1|k , Pk+1|k ) such that
f+ = fpk+1|k .
(15.17)
2. Exact closed-form closure with respect to Bayes’ rule: if fpk+1|k ∈ D with
pk+1|k = (xk+1|k , Pk+1|k ) and if
f z (x) = ∫
fk+1 (z|x) · fpk+1|k (x)
fk+1 (z|y) · fpk+1|k (y)dy
(15.18)
is the Bayes-rule update of fpk+1|k using the specified likelihood function
fk+1 (z|x) = NRk+1 (z − Hk+1 x) and any specific measurement z, then
there exists a fpk+1|k+1 ∈ D with pk+1|k+1 = (xk+1|k+1 , Pk+1|k+1 ) such
that
f z = fpk+1|k+1 .
(15.19)
(The family D is therefore a class of conjugate priors for the family of
likelihood functions Lz (x) = fk+1 (z|x).)
3. Exact closed-form closure with respect to Bayes-optimal state estimation.
Suppose that some Bayes-optimal state estimator is applied to fpk+1|k+1 —
two examples being the expected a posteriori (EAP) estimator and the maximum a posteriori (MAP) estimator:
EAP [fpk+1|k+1 ]
=
∫
M AP [fpk+1|k+1 ]
=
sup fpk+1|k+1 (x).
x · fpk+1|k+1 (x)dx
(15.20)
(15.21)
x
Suppose that, among all of these Bayes-optimal estimators, there is at least
one such that its state estimate can be exactly constructed from the parameter
pk+1|k+1 without knowledge of fpk+1|k+1 . In our case, both the EAP and
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MAP estimators satisfy this property:
EAP [fpk+1|k+1 ] = M AP [fpk+1|k+1 ] = π(pk+1|k+1 ) = xk+1|k+1 (15.22)
where π : (x, P ) ?→ x is the projection operator.
If the first two properties are satisfied, then the family D of distributions is
said to be:
• An exact closed-form solution of the single-target Bayes filter. In this
case it follows that the Bayes filter can be replaced, in exact closed form,
by a filter that propagates parameters in the smaller (and hopefully more
computationally advantageous) space P:
... →
pk|k
→
pk+1|k
→
pk+1|k+1
→ ...
If the third property is also satisfied, then we are justified in saying that the
family D of distributions is also:
• Exact closed-form with respect to Bayes-optimal state estimation. In this
case, the filter has the form
... →
15.1.2
pk|k
→
pk+1|k
→
π(pk+1|k+1 )
↑
pk+1|k+1
→ ...
Exact Closed-Form Solution of the Multitarget Bayes Filter
At the outset, assume that the following specifications for the multitarget measurement and motion models:
• The multitarget likelihood function fk+1 (Z|X) is the one corresponding to
the standard multitarget measurement model—that is, the superposition of a
multi-Bernoulli target-detection RFS and a Poisson clutter RFS (7.21).
• The multitarget Markov density fk+1|k (X|X ′ ) is the one corresponding
to the modified standard multitarget motion model. By this is meant the
superposition of a multi-Bernoulli target-survival RFS and a multi-Bernoulli
target appearance RFS. That is, the p.g.fl. of this model is
appearing targets
surviving targets
??
?
? ?? ? ?
X′
Gk+1|k [h|X ′ ] = GB
k+1|k [h] · (1 − pS + pS Mh )
(15.23)
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441
where the p.g.fl. GB
k+1|k [h] of the target appearance RFS is multi-Bernoulli
rather than Poisson, as in the standard multitarget motion model of (7.66):
B
νk+1|k
GB
k+1|k [h] =
∏
l
l
(1 − qk+1|k
+ qk+1|k
· ˆblk+1|k [h]).
(15.24)
l=1
Given this, an exact closed-form solution of the multitarget Bayes filter is a
family D of parametrized multitarget probability distributions fp (X) with p ∈ P
for some parameter space P, which satisfies the following conditions:
1. Exact closed-form closure with respect to the multitarget prediction integral:
if fpk|k ∈ D and if
f+ (X) =
∫
fk+1|k (X|X ′ ) · fpk|k (X ′ )δX ′
(15.25)
is the Markov prediction of fpk|k using the multitarget Markov transition
density fk+1|k (X|X ′ ) for the modified standard multitarget motion model,
then there is an fpk+1|k ∈ D such that
f+ = fpk+1|k .
(15.26)
2. Exact closed-form closure with respect to multitarget Bayes’ rule: if fpk+1|k ∈
D and
fk+1 (Z|X) · fpk+1|k (X)
f Z (X) = ∫
(15.27)
fk+1 (Z|Y ) · fpk+1|k (Y )δY
is the Bayes-rule update of fpk+1|k using the multitarget likelihood function fk+1 (Z|X) for the standard multitarget measurement model, and any
specified measurement set Z, then there is an fpk+1|k+1 ∈ D such that
f Z = fpk+1|k+1 .
(15.28)
In this case the multitarget Bayes filter can be replaced by an equivalent filter
on the parameter space P.
Mahler showed that Condition 1 is satisfied if D is the class of multiBernoulli distributions, as defined in Section 4.3.4 ([179], pp. 675-677). Thus,
suppose that the initial multitarget distribution f0|0 (X|Z (0) ) is multi-Bernoulli.
Then so is the first time-updated distribution f1|0 (X|Z (0) ).
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However, this progression fails with the next step. The first measurementupdated distribution f1|1 (X|Z (1) ) can be only approximately multi-Bernoulli—as
is the case, for example, with the CBMeMBer filter.
One might then ask: Can the class of multi-Bernoulli distributions be generalized to a larger class D of multitarget distributions that is exact closed-form under
time-update and measurement-update? Better yet, is it potentially computationally
tractable?
The answer is, Yes. The crucial insight, due to B.-T. Vo, is the following. Let
D be the class of generalized labeled multi-Bernoulli distributions, to be defined in
Section 15.3.4.1. Then D turns out to be the desired solution class.
15.1.3
Overview of the Vo-Vo Filter Approach
The core idea underlying the Vo-Vo approach is this:
• If target-tracks are distinctly labeled, then computationally tractable exact
closed-form closure becomes possible.
That is, kinematic states x must be replaced by labeled states x̊ = (x, ℓ)
where ℓ is a discrete label variable that is uniquely associated with this track as
time progresses. Thus conventional multitarget states X = {x1 , ..., xn } must
˚ = {x̊1 , ...,x̊n } with x̊i = (xi , ℓi )
be replaced by labeled multitarget states X
2
where ℓ1 , ..., ℓn are distinct. Given this, an RFS multitarget state Ξ becomes a
labeled RFS multitarget state ˚
Ξ—that is, an RFS whose instantiations are labeled
multitarget states.
Recall (Section 4.3.2) that an (unlabeled) i.i.d.c. RFS Ξ has a distribution of
the form
∏
fΞ (X) = |X|! · p(|X|)
s(x).
(15.29)
x∈X
As we shall see in (15.85), the multitarget probability density function of a labeled
multi-Bernoulli RFS ˚
Ξ generalizes the concept of an i.i.d.c. RFS:
˚
˚
f˚
˚ X
˚L | · ω(XL )
Ξ (X) = δ|X|,|
∏
s(x, ℓ)
(15.30)
˚
(x,ℓ)∈X
where:
2
It is unknown at this time if it there exists a practically useful exact closed-form solution of the
multitarget Bayes filter with the standard multitarget measurement model, but with unlabeled target
states.
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443
˚L denotes the set of labels of the targets in X.
˚ 3
• X
• sℓ (x) = s(x, ℓ) is the spatial probability density function of the track that
has label ℓ.
∑
• ω(L) ≥ 0 with
L ω(L) = 1 is the weight (probability) of the following
hypothesis: there are |L| targets whose labels are the elements of L, and
such that the sℓ (x) with ℓ ∈ L are their respective track distributions.
Equation (15.30) defines a labeled multi-Bernoulli distribution, hereafter
˚ is the labeled version of
abbreviated as an “LMB distribution.” If fk+1 (Z|X)
the standard multitarget likelihood function, then the Bayes’ rule update of a LMB
distribution has the form
∫
˚ · f (X)
˚
fk+1 (Zk+1 |X)
= δ|X|,|
˚ X
˚L |
˚
˚
˚
fk+1 (Zk+1 |Y ) · f (Y )δ Y
∑
˚
θ∈TZk+1
∏
˚
˚L )
ω θ (X
˚
sθ (x, ℓ)
˚
(x,ℓ)∈X
(15.31)
˚
where sθ (x, ℓ) are probability distributions in x for each fixed ℓ; and where
∑ ∑ ˚
ω θ (L) = 1;
(15.32)
˚
θ∈TZk+1
L
and where the first summation is taken over all measurement-to-track associations
˚
θ ∈ TZk+1 between the measurements in Zk+1 and the set of labels in L, where
TZk+1 denotes the set of all such associations (see Section 15.3.4 for more details).
If we replace TZk+1 by an arbitrary index set O, we get a generalized LMB
distribution—or “GLMB distribution” or “Vo-Vo prior” for short. Given this, it can
further be shown that:
• The Bayes’ rule update of a GLMB distribution, using the multitarget likelihood function for the standard multitarget measurement model of (7.21), is
also a GLMB distribution.
• The time-update of a GLMB distribution, using the labeled version of the
multitarget Markov density for the modified standard multitarget motion
model of (15.23), is a GLMB distribution.
In other words: the family D of GLMB distributions forms an exact closedform and potentially tractable solution of the multitarget Bayes filter, hereafter
called the “Vo-Vo filter.”
3
˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} then X
˚L = {ℓ1 , ..., ℓn }.
For example, if X
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Furthermore, GLMB distributions are easily extended to include target features and target identification (see Section 15.4.4).
Remark 62 (Another exact closed-form solution) As will be seen in Section 20.4,
there exists another closed-form solution of the multitarget Bayes filter—but not
for the standard multitarget measurement model. Rather, this one applies to a
measurement model, due to B.-N. Vo, in which measurements are pixelized images
with pixel-to-pixel independence—see Section 20.2. In this case, the solution-class
D also turns out to be a family of multi-Bernoulli distributions.
15.1.4
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• The Vo-Vo filter is the first provably Bayes-optimal, but computationally
tractable, multitarget filter for the standard multitarget motion and measurement models.
• In particular, its track-management scheme is provably Bayes-optimal—
apparently the first tractable multitarget tracking filter for which such a claim
can be made.
• As a consequence, the track management approaches in the multihypothesis
tracker (MHT)—as well as in its many offshoots and “target-existence probability” generalizations—appear to be heuristic. Unless, that is, it can someday be demonstrated that MHT, or any of its generalizations, also constitutes
an exact closed-form solution of the multitarget Bayes filter.
• Stated differently: The Vo-Vo filter can be thought of as the first theoretically rigorous, Bayes-optimal formulation of an MHT-type (that is, data
association-based) tracking algorithm.
• It is also the first theoretically rigorous, Bayes-optimal formulation of an
association-based tracking algorithm that includes target-identification capability (see Section 15.4.4).
• Practical implementation of the Vo-Vo solution is based on δ-GLMB distributions of the form
∑
˚ =
˚
f˚O (X)
ωo · f˚o (X),
(15.33)
o∈O
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445
˚ are component functions indexed by o ∈ O. The number
where f˚o (X)
of components grows combinatorially, so components with small weights ωo
must be pruned. The loss due to pruning can be characterized exactly. That
is, if components with index set O′ are to be discarded, then:
∑
∥f˚O − f˚O′ ∥1 =
ωo .
(15.34)
o∈O−O′
where ∥f˚∥1 denotes the L1 norm. See Section 15.6.3.
15.1.5
Organization of the Chapter
The chapter is organized as follows:
• Section 15.2: Introduction to the theory of labeled random finite sets (labeled
RFSs).
• Section 15.3: Examples of labeled RFSs: labeled i.i.d.c. RFSs, labeled
Poisson RFSs, labeled multi-Bernoulli (LMB) RFSs, and generalized LMB
(GLMB) RFSs.
• Section 15.4: Modeling assumptions for the Vo-Vo filter, including GLMB
forms of the modified standard multitarget Markov density and the standard
multitarget likelihood function. This section includes an overview of the VoVo filter (Section 15.4.2).
• Section 15.5: Demonstration that the family of GLMB distributions solves
the multitarget Bayes filter, with respect to the standard multitarget measurement model and the modified standard multitarget motion model.
• Section 15.6: A sketch of the practical implementation of the Vo-Vo filter.
• Section 15.7: Performance results.
15.2
LABELED RFSS
In the typical theory and practice of multitarget tracking, the state x of a single
target is purely kinematic. Its state variables consist of position x, y, z and velocity
vx , vy , νz and perhaps also acceleration variables and body-frame orientation
variables. In the real world, however, every target inherently possesses an additional
state variable: its identity. This identity-variable can be:
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• Explicit: a class identifier (for example, jet fighter versus transport plane) or
a specific identifier (for example, an aircraft tail number).
• Implicit: a track label ℓ that usually does not tell us anything about identity,
but which serves the purpose of distinguishing the time-evolving tracks in
the scene from each other. This is the common practice in purely kinematic
multitarget tracking.
The purpose of this section is to examine the concept and the statistics of
target labeling in greater depth. The following topics are considered:
1. Section 15.2.1: Labeling of single targets.
2. Section 15.2.2: Distinct labeling of multiple targets.
3. Section 15.2.3: Set integrals for labeled targets.
15.2.1
Target Labels
It is sometimes asserted that the RFS approach inherently cannot deal with track
continuity. This is supposedly because the elements of finite sets are orderindependent, and thus are indistinguishable except for their dynamic behavior.
Thus—or again so it is often asserted—one cannot determine which track at time
tk+1 evolved from which track at time tk .
These assertions are untrue. As was noted in [179], pp. 506-507, track
management becomes possible—indeed, Bayes-optimally possible—if individual
target states are distinctly labeled. In this case any complete state specification of a
single target must have the form
x̊ = (x, ℓ) ∈ X × L
(15.35)
where x is in the kinematic state space X and ℓ is an element of a discrete space
L of labels. Since the number of distinct targets is unbounded, the label space L
must have the general form
L = {ϖ1 , ..., ϖi , ...}
(15.36)
where ϖ1 , ..., ϖi , ... is a countable number of distinct labels drawn, once and for
all, from some convenient alphabet.
Remark 63 (The Vo-Vo labeling convention) For purposes of implementation of
the Vo-Vo multitarget filter, in (15.146) the label space will be given the specific
Exact Closed-Form Multitarget Filter
447
form L = {0, 1, ...} × {1, 2, ...}. Specifically, a label will have the form ℓ = (k, i)
where k is the time that a given track was created, and i is an index that uniquely
distinguishes this track from all other tracks created at that time.
For all i = 1, 2, ... define
L(i) = {ϖ1 , ..., ϖi }.
(15.37)
Fn (L) = {L ⊆ L| |L| = n}
(15.38)
Also, let the symbol
denote the class of finite subsets of L of cardinality n.
15.2.2
Labeled Multitarget State Sets
Suppose that there are multiple targets to be considered. Then the state set of labeled
targets will have the form
˚ = {x̊1 , ...,x̊n } = {(x1 , ℓ1 ), ..., (xn , ℓn )} ⊆ X × L.
X
(15.39)
Pairs of the form (x1 , ℓ), (x2 , ℓ) with x1 ̸= x2 are physically unrealizable.
This is because it is not possible for the same target to simultaneously have two
different kinematic states (for example, two different positions) at the same time.
Thus {(x1 , ℓ1 ), ..., (xn , ℓn )} is not a physically well defined multitarget state
representation unless the ℓ1 , ..., ℓn are distinct.4
Define the projection operators x̊ ?→ x̊X and x̊ ?→ x̊L on points x̊ = (x, ℓ)
by
x̊X
x̊L
=
=
x
ℓ.
(15.40)
(15.41)
˚ ?→ X
˚X and X
˚ ?→ X
˚L on finite sets
Likewise, define the projection operators X
˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} by
X
˚X
X
˚L
X
4
=
=
{x1 , ..., xn }
{ℓ1 , ..., ℓn }.
(15.42)
(15.43)
In this case, one must verify that the resulting space of distinctly labeled target states has appropriate
topological properties—see [94], pp. 196-198.
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˚ ⊆ X × L that has distinct
Then a labeled multitarget state set is a finite subset X
labels, i.e.:
˚L | = |X|.
˚
|X
(15.44)
Consequently, a labeled RFS state set is a random labeled multitarget state set
˚
Ξ ⊆ X × L, in which case
|˚
ΞL | = |˚
Ξ|.
(15.45)
Thus:
• It is possible for two different targets to have the same kinematic state—that
is, (x, ℓ1 ), (x, ℓ2 ) with ℓ1 ̸= ℓ2 is allowed.
• A target cannot have different kinematic states—that is, (x1 , ℓ), (x2 , ℓ) with
x1 ̸= x2 is not allowed.
Remark 64 (Bayes-optimal track management) It follows that the multitarget
Bayes filter will—inherently—optimally maintain track continuity. For, suppose
˚′ is the track set at time tk and (xk|k , ℓ) ∈ X
˚′ is the state of one of
that X
˚
its tracks. Further suppose that X is the track set at time tk+1 . Then after
˚ is the state
a single recursion of the multitarget Bayes filter, (xk+1|k+1 , ℓ) ∈ X
of the same track at the next time tk+1 ([94], pp. 196-198). This results in a
Bayes-optimal approach to track management. The difficulty, of course, is that this
optimal approach to target labeling will be computationally intractable for all but
the simplest practical problems.
15.2.3
Set Integrals for Labeled Multitarget States
˚ be a function of a labeled state set X
˚ that has the following properties:
Let f˚(X)
˚
˚ are u−|X|
1. The units of measurement of f˚(X)
where u are the units of
measurement of x ∈ X.
∫
2. f˚({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn exists for all ℓ1 , ..., ℓn ∈ L and
n ≥ 1.
3. For each n ≥ 1,
∫
f˚({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn = 0
for all but a finite number of n-tuples (ℓ1 , ..., ℓn ) ∈ Ln .
(15.46)
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449
˚ exists and is defined to be
Then the set integral of f˚(X)
∫
˚ X
˚=
f˚(X)δ
∑ 1
n!
n≥0
∫
∑
f˚({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn .
(ℓ1 ,...,ℓn )∈Ln
(15.47)
If ˚
h(x, ℓ) is any unitless test function taking values in [0, 1], the probability
˚ is
generating function (p.g.fl.) of f˚(X)
˚˚[˚
G
f h] =
∫
˚
˚
˚ X
˚
hX · f˚(X)δ
(15.48)
˚
where the power functional notation ˚
hX was defined in (3.5).
15.3
EXAMPLES OF LABELED RFSS
The purpose of this section is to consider the following examples of labeled RFSs:
1. Section 15.3.1: Labeled i.i.d.c. RFSs.
2. Section 15.3.2: Labeled Poisson RFSs.
3. Section 15.3.3: Labeled multi-Bernoulli (LMB) RFSs.
4. Section 15.3.4: Generalized labeled multi-Bernoulli (GLMB) RFSs.
15.3.1
Labeled i.i.d.c. RFSs
The multitarget distribution of a labeled i.i.d.c. RFS ˚
Ξ is defined to be
˚X
X
˚
˚
f˚˚
˚
˚L · p(|X|) · s
Ξ (X) = δL(|X|),
X
(15.49)
where the notation L(n) was defined in (15.37), the power functional notation sX
was defined in (3.5), and where:
• p(n) is a cardinality distribution—that is, a probability distribution on target
number n.
• s(x) is a probability density on x—that is, the spatial distribution of a target
track.
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• For any subsets L, L′ ⊆ L, the Kronecker delta δL,L′ is defined by
δL,L′ = 1 if L = L′ and δL,L′ = 0 otherwise.
˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} with |X|
˚ = n then (15.49) can be
Thus if X
more concretely written as:
˚
f˚˚
Ξ (X)
=
=
δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn } · p(n) · s(x1 ) · · · s(xn )

 p(n) · s(x1 ) · · · s(xn ) if {ϖ 1 , ..., ϖn }
= {ℓ1 , ..., ℓn }

0
if
otherwise
(15.50)
(15.51)
where the ϖ1 , ..., ϖn were defined in (15.36). Consequently, for each n > 1 we
have f˚˚
Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )}) = 0 for all (ℓ1 , ..., ℓn ) except for the selections
(15.52)
(ℓ1 , ..., ℓn ) = (ϖπ1 , ..., ϖπn )
where π are the n! permutations on 1, ..., n. Thus the set integral
exists and is finite.
15.3.1.1
∫
˚ ˚
f˚˚
Ξ (X)δ X
The p.g.fl. of a Labeled i.i.d.c. RFS
For any label ℓ ∈ L and any test function ˚
h(x, ℓ) with 0 ≤ ˚
h(x, ℓ) ≤ 1, define
the linear functional
∫
sℓ [˚
h] def.
= ˚
h(x, ℓ) · s(x)dx.
(15.53)
˚
Then the p.g.fl. of f˚˚
Ξ (X), as defined in (15.48), is
˚˚[˚
G
Ξ h]
=
∑
p(n)
∑
n≥0
sϖi [˚
h]
(15.54)
i=1
n≥0
=
n
∏
∏
p(n)
sℓ [˚
h].
(15.55)
ℓ∈L(n)
In particular if ˚
h = 1 identically then sℓ [˚
h] = 1 for any ℓ, and so
˚˚[1] =
G
Ξ
∫
˚ ˚
f˚˚
Ξ (X)δ X =
∑
n≥0
p(n) = 1.
(15.56)
Exact Closed-Form Multitarget Filter
˚
Thus f˚˚
Ξ (X) is a labeled multitarget probability distribution.
To see why (15.54) is true, note that
∫
˚
˚
˚
˚
˚ ˚
G˚
hX · f˚˚
Ξ [h] =
Ξ (X)δ X
∫
∑ 1
∑
˚
=
h(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
n!
n
n≥0
=
451
(15.57)
(15.58)
(ℓ1 ,...,ℓn )∈L
·f˚˚
Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn
∫
∑ 1
∑
˚
h(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
n!
n
n≥0
(15.59)
(ℓ1 ,...,ℓn )∈L
·δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn } · p(n) · s(x1 ) · · · s(xn )dx1 · · · dxn
=
∑ p(n)
∑
n!
n≥0
=
∑
(ℓ1 ,...,ℓn
∑
p(n)
n≥0
sℓ1 [˚
h] · · · sℓn [˚
h] · δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn }(15.60)
)∈Ln
sℓ1 [˚
h] · · · sℓn [˚
h]
(15.61)
{ℓ1 ,...,ℓn }⊆Fn (L)
·δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn }
=
∑
n≥0
p(n)
n
∏
sϖi [˚
h]
(15.62)
i=1
where Fn (L), the class of finite subsets of L of cardinality n, was defined in
(15.38).
15.3.1.2
PHD, p.g.f., and Cardinality Distribution of a Labeled i.i.d.c. RFS
The PHD, p.g.f., and cardinality distribution of a labeled i.i.d.c. RFS ˚
Ξ are
∑
˚˚(x, ℓ) = s(x)
D
p(n) · 1L(n) (ℓ)
(15.63)
Ξ
n≥0
˚˚(x)
G
Ξ
=
∑
p(n) · xn
(15.64)
n≥0
p̊˚
Ξ (n)
=
p(n)
where 1L(n) is the indicator function of the set L(n).
(15.65)
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To prove (15.63), first note that
δ
sℓ′ [˚
h] = δℓ,ℓ′ · s(x).
δ(x, ℓ)
(15.66)
˚˚[˚
Then note that, from (15.54), the first functional derivative of G
Ξ h] is
˚˚
δG
Ξ ˚
[h]
δ(x, ℓ)
=
=


n
∑
′ · s(x)
δ
ℓ,ℓ
 (15.67)
p(n) 
sℓ′ [˚
h] 
′ [˚
s
h]
ℓ
′
′
n≥0
ℓ ∈L(n)
ℓ ∈L(n)


∑
∏
1L(n) (ℓ)
s(x)
p(n) 
sℓ′ [˚
h] ·
.
(15.68)
sℓ [˚
h]
′
∑

∏
n≥0
ℓ ∈L(n)
Therefore, from (4.75),
˚
∑
Ξ
˚˚(x, ℓ) = δ G˚
D
[1]
=
s(x)
p(n) · 1L(n) (ℓ).
Ξ
δ(x, ℓ)
(15.69)
n≥0
Equation (15.64) follows immediately from (4.66) by substituting ˚
h = x into
(15.54). Equation (15.65) follows immediately from that.
Remark 65 (De-labeling) If the labels of a labeled RFS ˚
Ξ are stripped away,
resulting in an unlabeled RFS Ξ, then the multitarget distribution of Ξ is the
marginal distribution defined by
f˚
Ξ ({x1 , ..., xn }) =
∑
f˚˚
Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )}).
(15.70)
ℓ1 ,...,ℓn ∈L
˚˚[˚
˚
Using this equation, it is easily shown that the p.g.fl. G
Ξ h] of Ξ and the p.g.fl.
GΞ [h] of Ξ are related by
˚˚[h]
GΞ [h] = G
(15.71)
Ξ
for all test functions h(x) on unlabeled states x.
Remark 66 (The de-labeling of a labeled i.i.d.c. RFS is i.i.d.c.) A labeled i.i.d.c.
RFS is not i.i.d.c., but its corresponding delabeled RFS is i.i.d.c. If ˚
Ξ is a labeled
i.i.d.c. RFS then the projection Ξ = ˚
ΞX into unlabeled state sets, as defined in
(15.42), is an i.i.d.c. RFS. For, according to the previous remark and (15.55), the
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453
p.g.fl. of the delabeled RFS is
GΞ [h]
=
˚˚[h] =
G
Ξ
∑
p(n)
n≥0
=
∑
p(n)
n≥0
∏
∏
sℓ [h]
(15.72)
ℓ∈L(n)
s[h] =
ℓ∈L(n)
∑
p(n) · s[h]n ,
(15.73)
n≥0
which is the p.g.fl. of an i.i.d.c. RFS.
15.3.2
Labeled Poisson RFSs
These are labeled i.i.d.c. RFSs whose cardinality distributions have the following
form:
˚
|X|
˚
˚ = δ ˚ ˚ · e−N · N
f˚(X)
· sXX
(15.74)
L(|X|),XL
˚
|X|!
where N is the Poisson parameter. A labeled Poisson RFS is not Poisson, but its
corresponding delabeled RFS is Poisson.
15.3.3
Labeled Multi-Bernoulli (LMB) RFSs
These are defined as follows, using the following four steps.
Step 1: Let us be given a fixed number ν of target-tracks with:
• Track probability densities s1 (x), ..., sν (x).
• Track probabilities of existence q 1 , ..., q ν .
• Track labels ϖτ 1 , ..., ϖτ ν ∈ L where the “track labeling function” τ :
{1, ..., ν} → {1, 2, ...} is a one-to-one function that selects specific labels
for the tracks.
As with a conventional multi-Bernoulli RFS (Section 4.3.4), the number q i
is interpreted as the probability that the ith track actually exists—that is, actually
is a target. The primary difference between a LMB RFS and a multi-Bernoulli RFS
is the fact that the track indices i = 1, ..., ν must be associated with specific labels.
This is the purpose of the track labeling function τ .
Step 2: Let the set of labels that have been assigned to the tracks be denoted
as
ν
L = {ϖτ 1 , ..., ϖτ ν }.
(15.75)
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ν
˚
Step 3: Note that, for any ℓ ∈ L and any labeled multitarget state set X,
˚L (that is, ℓ is not the label of any target in X),
˚ or ℓ ∈ X
˚L —in
either ℓ ∈
/ X
which case ℓ = ϖτ i for some unique i ∈ {1, ..., ν}. Thus define the function
σ : L → {0, 1, ..., ν} by
{
i if ℓ = ϖτ i
σℓ abbr.
= σ(ℓ) =
.
(15.76)
0 if otherwise
Also define the function s̊(x, ℓ) on labeled states by
s̊(x, ℓ) = sσℓ (x)
(15.77)
ν
for all x ∈ X and ℓ ∈ L.
Step 4: Define the set function ω(L) by


(
)
∏
∏


ω(L) = 
(1 − q σℓ )
q σℓ · 1 ν (ℓ)
L
ν
(15.78)
ℓ∈L
ℓ∈L−L
=
Q
∏ q σℓ · 1 ν (ℓ)
L
(15.79)
1 − q σℓ
ℓ∈L
for all L ⊆ L, where 1S (ℓ) is the set indicator function for the subset S ⊆ L;
and where
ν
∏
Q=
(1 − q i ).
(15.80)
i=1
ν
Note that ω(L) = 0 unless L ⊆ L. Also note that
∑
∑
ω(L) =
ω(L) = 1.
(15.81)
ν
L⊆L
L⊆L
For,
∑
ω(L)
=
Q
∑∏
ν
ν ℓ∈L
L⊆L
q σℓ
1 − q σℓ
(15.82)
L⊆L

=

∏
ν
ℓ∈L
=
1

σℓ 
(1 − q )
∏(
ν
q σℓ
1+
1 − q σℓ
)
(15.83)
ℓ∈L
(15.84)
Exact Closed-Form Multitarget Filter
455
where (15.83) follows from the power-functional identity, (3.7).
15.3.3.1
Definition of a LMB RFS
Given this, the probability distribution for a labeled multi-Bernoulli (LMB) RFS ˚
Ξ
is defined to be [295]:
˚
f˚˚
Ξ (X)
=
∏
˚
δ|X|,|
˚ X
˚L | · ω(XL )
(15.85)
s̊(x, ℓ)
˚
(x,ℓ)∈X
=
˚
X
˚
δ|X|,|
˚ X
˚L | · ω(XL ) · s̊ .
(15.86)
This generalizes the concept of a labeled i.i.d.c. RFS, in that the cardinality weight˚ in (15.49) is replaced by the label weighting factor ω(X
˚L );
ing factor p(|X|)
and in that the spatial distribution s(x) is replaced by a labeled track distribution
s̊(x, ℓ):
˚X
˚
X
X
˚
˚
δL(|X|),
?→ δL(|X|),
˚
˚L · p(|X|) · s
˚
˚L · ω(XL ) · s̊ .
X
X
˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} with |X|
˚ = n, then (15.85) becomes
If X
˚
f˚˚
Ξ (X) = δn,|{ℓ1 ,...,ℓn }| ·
( n
∏
1{ϖτ 1 ,...,ϖτ ν } (ℓi )
)
·Q·
i=1
( n
)
∏ q σℓi sσℓi (xi )
i=1
.
1 − q σℓi
(15.87)
This is mathematically well defined. First, the factor δn,|{ℓ1 ,...,ℓn }| ensures that
˚
˚
f˚X
˚ (X) vanishes whenever the labels of X are not distinct. Second, the product
˚
of the factors 1{ϖτ 1 ,...,ϖτ ν } (ℓi ) ensures that f˚X
˚ (X) vanishes whenever any
˚ is not one of the preassigned track labels ϖτ 1 , ..., ϖτ ν . Third and
label of X
˚ actually are distinct and preassigned track labels,
thereby, since all labels ℓ of X
then ℓ = ϖτ i for some i = 1, ..., ν and so σℓ = i is a well defined track-index.
˚
For nonzero values of f˚
Ξ (X) we therefore get
˚
f˚˚
Ξ (X) = Q
n
∏
q σℓi · sσℓi (xi )
i=1
1 − q σℓi
.
(15.88)
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15.3.3.2
The p.g.fl. of a LMB RFS
The p.g.fl. of the LMB RFS ˚
Ξ is
)
∫
∏(
σℓ
σℓ
˚
˚˚[˚
G
h]
=
1
−
q
+
q
h(x,
ℓ)
·
s̊(x,
ℓ)dx
Ξ
(15.89)
ν
=
ℓ∈L
ν (
∏
i
1−q +q
i
∫
˚
h(x, ϖτ i ) · si (x)dx
)
(15.90)
i=1
where ˚
h(x, ℓ) is a test function with values in [0, 1]; and where, from (15.75),
ν
L = {ϖτ 1 , ..., ϖτ ν }. Setting ˚
h = 1, it is easily verified that
∫
˚ ˚
f˚˚
(15.91)
Ξ (X)δ X = 1.
To prove (15.89), note from (15.87) that
∫
˚
˚
˚
˚
˚ ˚
G˚
hX · f˚˚
Ξ [h] =
Ξ (X)δ X
∫
∑ 1
∑
˚
=
h(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
n!
n
n≥0
=
(15.92)
(15.93)
(ℓ1 ,...,ℓn )∈L
·f˚˚
Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn
∫
∑ 1
∑
˚
Q
h(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
n!
n
n≥0
(ℓ1 ,...,ℓn )∈L
( n
)
∏
1{ϖτ 1 ,...,ϖτ ν } (ℓi )
·δn,|{ℓ1 ,...,ℓn }| ·
(15.94)
i=1
=
q σℓ1 sσℓ1 (x1 )
q σℓn sσℓn (xn )
·
·
·
·
dx1 · · · dxn
1 − q σℓ1
1 − q σℓn
∑ 1
∑
Q
s̃σℓ1 [˚
h] · · · s̃σℓn [˚
h] · δn,|{ℓ1 ,...,ℓn }|(15.95)
n!
ν
n≥0
(ℓ1 ,...,ℓn )∈Ln
where
ℓ˚
abbr.
s̃ [h] =
∫
q σℓ sσℓ (x)
˚
h(x, ℓ) ·
dx.
1 − q σℓ
(15.96)
Exact Closed-Form Multitarget Filter
ν
457
ν
If Fn (L) denotes the set of all finite subsets of L of cardinality n, then
˚˚[˚
G
Ξ h]
=
Q
∑
∑
s̃ℓ1 [˚
h] · · · s̃ℓn [˚
h]
(15.97)
ν
n≥0
{ℓ1 ,...,ℓn }∈Fn (L)
=
Q
∑∏
s̃ℓ [˚
h].
(15.98)
ν ℓ∈L
L⊆L
Because of the identity for the power functional, (3.7) and since σϖτ i = i for
i = 1, ..., ν then because of (15.76), we get the desired result:
˚˚[˚
G
Ξ h]
=
Q
∏(
1 + s̃ℓ [˚
h]
)
(15.99)
ν
ℓ∈L
=
∏(
1 − q σℓ + q σℓ
∫
˚
h(x, ℓ) · sσℓ (x)dx
)
(15.100)
ν
=
=
ℓ∈L
ν (
∏
i=1
ν (
∏
1 − q σϖτ i + q σϖτ i
i
1−q +q
i
∫
∫
)
˚
h(x, ϖτ i ) · sσϖτ i (x)dx (15.101)
)
i
˚
h(x, ϖτ i ) · s (x)dx .
(15.102)
i=1
A labeled multi-Bernoulli RFS is not multi-Bernoulli, but its corresponding
delabeled RFS is multi-Bernoulli.
15.3.3.3
PHD and Cardinality Distribution of a LMB RFS
The PHD, p.g.f., and cardinality distribution of a LMB RFS ˚
Ξ are given by
˚˚(x, ℓ)
D
Ξ
=
˚˚(x)
G
Ξ
=
p̊˚
Ξ (n)
=
1 ν (ℓ) · q σℓ · s̊(x, ℓ)
(15.103)
(1 − q i + q i · x)
(15.104)
L
ν
∏
i=1
( ν
∏
i=1
i
(1 − q )
)
· σν,n
(
q1
qν
,
...,
1 − q1
1 − qν
)
(15.105)
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where σν,n (x1 , ..., xν ) is the elementary homogeneous symmetric function of
degree n in ν variables. Equation (15.103) can be rewritten as:
{ i i
q · s (x) if ℓ = ϖτ i , 1 ≤ i ≤ ν
˚˚(x, ℓ) =
D
.
(15.106)
Ξ
0
if
otherwise
To prove (15.103), note that from (15.89)
˚˚
δG
Ξ ˚
[h]
δ(x, ℓ)

=

∏(
′
1 − q σℓ + q σℓ
)
˚
h(x′ , ℓ′ ) · s̊(x′ , ℓ′ )dx′ 
∫
ν
ℓ′ ∈ L
′
∑
·
′
(15.107)

δℓ′ ,ℓ · q σℓ · s̊(x, ℓ′ )
∫
′
σℓ + q σℓ′ ˚
h(x′ , ℓ′ ) · s̊(x′ , ℓ′ )dx′
ν 1 − q
ℓ′ ∈ L

=

·
∏(
1−q
σℓ′
+q
σℓ′
∫
′
′
′
′
˚
h(x , ℓ ) · s̊(x , ℓ )dx
ν
ℓ′ ∈ L
′
)


(15.108)
1 ν (ℓ) · q σℓ · s̊(x, ℓ)
L
.
∫
1 − q σℓ + q σℓ ˚
h(x′ , ℓ) · s̊(x′ , ℓ)dx′
Therefore, from (4.75),
˚
Ξ
˚˚(x, ℓ) = δ G˚
D
[1] = 1 ν (ℓ) · q σℓ · s̊(x, ℓ).
Ξ
L
δ(x, ℓ)
(15.109)
Equation (15.104) follows immediately from (4.66) by substituting ˚
h = x into
(15.89). Equation 15.126) follows from (4.130).
15.3.4
Generalized Labeled Multi-Bernoulli (GLMB) RFSs
This section introduces a generalization of the LMB RFS, the generalized LMB
(GLMB) RFS. The section is organized as follows:
• Section 15.3.4.1: Definition of a GLMB RFS.
Exact Closed-Form Multitarget Filter
459
• Section 15.3.4.2: Intuitive interpretation of a GLMB distribution.
• Section 15.3.4.3: The p.g.fl. of a GLMB RFS.
• Section 15.3.4.4: The PHD and cardinality distribution of a GLMB RFS.
• Section 15.3.4.5: Example: labeled i.i.d.c. RFSs are GLMB RFSs.
• Section 15.3.4.6: Example: LMB RFSs are GLMB RFSs.
• Section 15.3.4.7: Approximate multitarget state estimation for GLMB distributions.
15.3.4.1
Definition of a GLMB RFS
Let the following be given:
• A set O of indices. (This is required to achieve exact closed-form closure
for the multitarget Bayes filter, and will itself evolve with time.)
o
• For
∫ o each o ∈ O, a function s̊ (x, ℓ) on (x, ℓ) ∈ X × L such that
s̊ (x, ℓ)dx = 1 for all ℓ ∈ L.
• For each o ∈ O, a unitless set function ω o (L) defined on L ⊆ L.
• ω o (L) = 0 except for a finite number of pairs (o, L) with finite L ⊆ L, in
which case it is required that
∑∑
ω o (L) = 1.
(15.110)
o∈O L⊆L
Given this, the probability distribution of a GLMB RFS (also known as Vo-Vo
prior) is defined to be ([295], Eq. (13)):
˚
f˚
Ξ (X)
=
=
δ|X|,|
˚ X
˚L |
δ|X|,|
˚ X
˚L |
∑
˚L )
ω o (X
∏
s̊o (x, ℓ)
o∈O
˚
(x,ℓ)∈X
∑
˚L ) · (s̊o )X .
ω o (X
(15.111)
˚
(15.112)
o∈O
Note that this distribution is well defined with respect to units of measurement, since
˚
˚L ) is unitless and the units of (s̊o )X
˚ for every
ω o (X
are the same as those of X
o.
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Advances in Statistical Multisource-Multitarget Information Fusion
15.3.4.2
Intuitive Interpretation of Vo-Vo Priors
˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} be a labeled multitarget state set with |X|
˚ = n,
Let X
˚
and let L = XL = {ℓ1 , ..., ℓn } be the corresponding set of distinct labels. Then
for each o ∈ O, the number ω o (L) is the weight of the hypothesis that:
• There are n targets present with labels ℓ1 , ..., ℓn .
• s̊o (x, ℓ1 ), ...,s̊o (x, ℓn ) are their respective track distributions.
The larger the value of ω o (L), the more likely it is that the corresponding
hypothesis is true. The s̊o (x1 , ℓ1 ), ...,s̊o (xn , ℓn ) are the probabilities (probability densities) that the respective kinematic states of the tracks are, respectively,
x1 , ..., xn . The summation
∑
ω o ({ℓ1 , ..., ℓn }) · s̊o (x1 , ℓ1 ) · · ·s̊o (xn , ℓn )
(15.113)
o∈O
˚
is the total probability (probability density) f˚˚
Ξ (X) that targets with state set
{(x1 , ℓ1 ), ..., (xn , ℓn )} are present.
15.3.4.3
The p.g.fl. of a GLMB RFS
Define the linear functional s̊oℓ [˚
h] by
∫
s̊oℓ [˚
h] abbr.
= ˚
h(x, ℓ) · s̊o (x, ℓ)dx.
(15.114)
˚
Then the p.g.fl. of a Vo-Vo prior f˚˚
Ξ (X) is
∑∑
∏
˚˚[˚
G
ω o (L)
s̊oℓ [˚
h].
Ξ h] =
(15.115)
o∈O L⊆L
ℓ∈L
Setting ˚
h = 1 we get
∫
∑∑
˚ X
˚=G
˚˚[1] =
f˚˚(X)δ
ω o (L) = 1
Ξ
Ξ
(15.116)
o∈O L⊆L
˚
where the last equation is true because of (15.110). Thus f˚
Ξ (X) is a multitarget
probability distribution on labeled targets. Note that, by the definition of a GLMB
RFS, both the summation and product in (15.115) are finite.
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461
To prove (15.115), note that
˚˚[˚
G
Ξ h]
∫
=
˚
˚
˚ ˚
hX · f˚˚
Ξ (X)δ X
∑
∑ 1
=
n!
n≥0
(15.117)
∫
(ℓ1 ,...,ℓn
˚
h(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
·f˚˚
Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn
∫
∑ 1
∑
˚
h(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
n!
n≥0
(ℓ1 ,...,ℓn )∈Ln
∑
·δn,|{ℓ1 ,...,ℓn }|
ω o ({ℓ1 , ..., ℓn })
=
(15.118)
)∈Ln
(15.119)
o∈O
o
o
·s̊ (x1 , ℓ1 ) · · ·s̊ (xn , ℓn )dx1 · · · dxn
and thus that
˚˚[˚
G
Ξ h]
=
∑∑ 1
∑
n!
s̊oℓ1 [˚
h] · · ·s̊oℓn [˚
h]
(15.120)
(ℓ1 ,...,ℓn )∈Ln
·δn,|{ℓ1 ,...,ℓn }| · ω o ({ℓ1 , ..., ℓn })
o∈O n≥0
=
∑∑
∑
s̊oℓ1 [˚
h] · · ·s̊oℓn [˚
h]
(15.121)
o∈O n≥0 {ℓ1 ,...,ℓn }∈Fn (L)
=
·ω o ({ℓ1 , ..., ℓn })
∑∑
∏
ω o (L)
s̊oℓ [˚
h].
o∈O L⊆L
15.3.4.4
(15.122)
ℓ∈L
PHD and Cardinality Distribution of a GLMB RFS
The PHD of a GLMB RFS ˚
Ξ is given by:
˚˚(x, ℓ)
D
Ξ
=
∑
o∈O
=
∑
o∈O


(
∑
L⊆L
∑
L∋ℓ

o
ω (L) · 1L (ℓ) · s̊o (x, ℓ)
ω o (L)
)
· s̊o (x, ℓ).
(15.123)
(15.124)
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The p.g.f. of ˚
Ξ is:
˚˚(x) =
G
Ξ
∑∑
ω o (L) · x|L| .
(15.125)
o∈O L⊆L
The cardinality distribution of ˚
Ξ is ([295], Eq. (16)):
p̊˚
Ξ (n) =
∑
∑
ω o (L)
(15.126)
o∈O L∈Fn (L)
where Fn (L) was defined in (15.38). Note that the expected number of targets in
˚
Ξ is
∑∫
∑∑
˚˚ = G
˚(1) (1) =
˚˚(x, ℓ)dx =
N
D
ω o (L) · |L|.
(15.127)
˚
Ξ
Ξ
Ξ
o∈O L⊆L
ℓ∈L
To prove (15.123), first note that
δ
s̊o′ [˚
h] = δℓ′ ,ℓ · s̊o (x, ℓ′ ).
δ(x, ℓ) ℓ
(15.128)
˚˚[˚
Then note that the first functional derivative of G
Ξ h] in (15.115) is
˚˚
δG
Ξ ˚
[h]
δ(x, ℓ)
=
∑∑
ω o (L)
(
·
∑ δℓ′ ,ℓ · s̊o (x, ℓ′ )
ℓ′ ∈L
=
s̊oℓ′ [˚
h]
ℓ′ ∈L
o∈O L⊆L
(
∏
∑∑
o
s̊oℓ′ [˚
h]
(
ω (L)
o∈O L⊆L
∏
ℓ′ ∈L
)
(15.129)
)
s̊oℓ′ [˚
h]
)
·
1L (ℓ) · s̊o (x, ℓ)
(15.130)
s̊o [˚
h]
ℓ
and therefore, from (4.75),
˚
∑∑
Ξ ˚
˚˚(x, ℓ) = δ G˚
D
[h] =
ω o (L) · 1L (ℓ) · s̊o (x, ℓ).
Ξ
δ(x, ℓ)
o∈O L⊆L
(15.131)
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463
Equation (4.66) follows immediately from (15.115) by substituting ˚
h = x
into (15.115). Equation (15.126) follows from (4.62):
[
]
1 dn ˚
p̊˚
G˚(x)
(15.132)
Ξ (n) =
n! dxn Ξ
x=0
[
]
1 ∑∑ o
=
ω (L) · n!C|L|,n · x|L|−n
(15.133)
n!
o∈O L
x=0
∑ ∑
=
ω o (L).
(15.134)
o∈O L∈Fn (L)
15.3.4.5
Labelled i.i.d.c. RFSs Are GLMB RFSs
From (15.54) the p.g.fl. of a labeled i.i.d.c. process is
∑
˚
G˚
Ξ [h] =
p(n)
n (∫
∏
)
˚
h(x, ϖi ) · s(x)dx .
(15.135)
n≥0
i=1
ω(L)
=
p(|L|) · δL,L(|L|)
(15.136)
s̊(x, ℓ)
=
s(x)
(15.137)
Let |O| = 1 and define
where L(n) = {ϖ1 , ..., ϖn } was defined in (15.37). Then from (15.115) we
see that the p.g.fl. of the corresponding GLMB RFS is identical to the p.g.fl. of the
labeled i.i.d.c. RFS:
∑
∏
˚˚[˚
G
ω(L)
s̊ℓ [˚
h]
(15.138)
Ξ h] =
L⊆L
=
∑
ℓ∈L
p(|L|) · δL,L(|L|)
L⊆L
∏ (∫
˚
h(x, ℓ) · s̊(x, ℓ)dx
)
(15.139)
ℓ∈L
and so
˚˚[˚
G
Ξ h]
=
∑ ∑
p(n) · δL,{ϖ1 ,...,ϖn })
n≥0 |L|=n
·
∏ (∫
ℓ∈L
˚
h(x, ℓ) · s(x)dx
)
(15.140)
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Advances in Statistical Multisource-Multitarget Information Fusion
and so
˚˚[˚
G
Ξ h]
=
∑
n≥0
=
∑
n≥0
15.3.4.6
(∫
∏
p(n)
˚
h(x, ℓ) · s(x)dx
)
(15.141)
ℓ∈{ϖ1 ,...,ϖn }
p(n)
n
∏
sϖi [˚
h].
(15.142)
i=1
LMB RFSs Are GLMB RFSs
Let |O| = 1. Then it is clear that the distribution of a LMB RFS, (15.85), is a
special case of the distribution of a GLMB RFS, (15.111), with |O| = 1.
15.3.4.7
Approximate Multitarget State Estimation for GLMB Distributions
The MaM multitarget state estimator (5.10) and the JoM multitarget state estimator
(5.9) can be applied to GLMB distributions. However, they will usually not
be computationally tractable. In their stead, Vo and Vo proposed the following
intuitively appealing estimator.5
Let us be given a GLMB distribution
∑
∏
o ˚
˚ =δ˚ ˚
f˚˚
(
X)
ω
(
X
)
s̊o (x, ℓ).
(15.143)
L
Ξ
|X|,|XL |
o∈O
˚
(x,ℓ)∈X
Then:
ˆ so that ω ô (L)
ˆ is maximal:
1. Step 1: Choose ô and L
ˆ = arg max ω o (L).
(ô, L)
(15.144)
o,L
ˆ = {ℓˆ1 , ..., ℓˆn̂ } where n̂ = |L|
ˆ is the estimate of the number
2. Step 2: Let L
of tracks.
3. Step 3: For i = 1, ..., n̂, define the state estimates of the tracks to be the
ˆ
means of the track distributions corresponding to the labels in L:
∫
x̂i = x · s̊ô (x, ℓˆi )dx.
(15.145)
5
As of the time of writing, it is not known whether or not this estimator is Bayes-optimal or even
approximately so.
Exact Closed-Form Multitarget Filter
15.4
465
MODELING FOR THE VO-VO FILTER
The purpose of this section is to begin discussion of the Vo-Vo exact closed-form
multitarget filter by specifying its models. The section is organized as follows:
1. Section 15.4.1: Labeling conventions for the Vo-Vo filter.
2. Section 15.4.2: An overview of the Vo-Vo filter.
3. Section 15.4.3: Basic motion and measurement models.
4. Section 15.4.4: Motion and measurement models for joint multitarget detection, tracking, and identification/classification.
5. Section 15.4.5: The multitarget likelihood function for the labeled standard
multitarget measurement model.
6. Section 15.4.6: The multitarget Markov density for the labeled version of the
standard multitarget motion model.
7. Section 15.4.7: The multitarget Markov density for the labeled version of the
modified standard multitarget motion model.
15.4.1
Labeling Conventions
Thus far, the space L of all possible track labels has been treated abstractly as an
unspecified, countably infinite set. For the Vo-Vo filter, it is assumed to have the
specific form
L = I+ × N
(15.146)
where I+ = {0, 1, ...} is the set of nonnegative integers and N = {1, 2, ...} is the
set of natural numbers. Thus the set of all possible labeled target states x̊ is
˚
X = X × I+ × N.
(15.147)
We must first define some particular label spaces and state spaces (Section
15.4.1.1), and then describe the restrictions that they implicitly impose on the
labeled multitarget Bayes filter (Section 15.4.1.2).
15.4.1.1
Special Label and State Spaces
The following definitions are required in what follows, for k = 0, 1, ...:
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• Possible labels of all target-tracks at time tk :
L0:k = {0 ≤ l ≤ k} × N.
(15.148)
• Possible labels of new tracks at time tk+1 :
LB
k+1 = {k + 1} × N
(15.149)
in which case it follows that
L0:k+1 = L0:k ⊎ LB
k+1 .
(15.150)
• State space of all possible labeled tracks at time tk :
˚
X0:k = X × L0:k .
(15.151)
• State-space of all possible newly appearing labeled tracks at time tk+1 :
˚
XB = X ⊎ LB
k+1
(15.152)
˚
X0:k+1 = ˚
X0:k ⊎ ˚
XB .
(15.153)
in which case
The evolution of track labels from one time-step to the next is specified as
follows. Let (xk|k , l, i) be the state of a track at time tk with label ℓ = (l, i). If
the target does not disappear at time tk+1 , its state will be (xk+1|k , l, i)—that is,
its label does not change.
Thus, in general, if (x, l, i) ∈ ˚
X0:k then, at time tk , the target with this
label:
• Originally appeared at time tl .
• Was at that time assigned identifying index i.
• Currently has the kinematic state x.
We therefore have the following strictly nested sequence of labeled state
spaces:
˚
X0:0 ⊂ ˚
X0:1 ⊂ ... ⊂ ˚
X0:k ⊂ ˚
X0:k+1 ⊂ ...
Exact Closed-Form Multitarget Filter
467
Remark 67 (Labeling and statistical independence) At first, it might appear that
the necessity of labeling new tracks results in a theoretical problem: namely, that it
is not possible for new target tracks to be statistically independent of existing ones.
That is, suppose that at time tk , Lk|k ⊆ L is the finite subset of labels for the
currently-existing tracks. Then, at time tk+1 , we must choose labels for the new
tracks. If we are to distinguish between tracks, then the labels for these new tracks
cannot be in Lk|k and so they must be in L − Lk|k . Ergo, one might conclude
that the labeling of new tracks is a procedure that is statistically dependent on prior
knowledge of the labels of existing tracks. The labeling convention proposed by Vo
and Vo avoids this seeming difficulty. At time tk , the labels of the existing tracks
are in L0:k , whereas the labels of the new tracks are in LB
k+1 . Thus it is indeed the
case that the new labels will be in L − Lk|k . However, they are not chosen by first
determining Lk|k and then choosing labels in L − Lk|k . Rather, selection of the
new labels requires no prior knowledge of Lk|k . The only information employed
is the fact that the new tracks must exist at time tk+1 but cannot exist prior to
that—and therefore must reside in LB
k+1 .
15.4.1.2
Properties of the Labeled Multitarget Bayes Filter
The Vo-Vo filter is an exact, closed-form solution of the multitarget Bayes filter for
labeled targets:
... →
˚ (k) )
f˚k|k (X|Z
→
˚ (k) )
f˚k+1|k (X|Z
→
˚ (k+1) )
f˚k+1|k+1 (X|Z
→ ...
defined by the time-update and measurement-update equations
∫
˚ (k) ) =
˚X
˚′ ) · f˚k|k (X
˚′ |Z (k) )δ X
˚′(15.154)
f˚k+1|k (X|Z
f˚k+1|k (X|
˚ (k+1) )
f˚k+1|k+1 (X|Z
f˚k+1 (Zk+1 |Z (k) )
˚ · f˚k+1|k (X|Z
˚ (k) )
f˚k+1 (Zk+1 |X)
=
(15.155)
f˚k+1 (Zk+1 |Z (k) )
∫
˚ · f˚k+1|k (X|Z
˚ (k) )δ X
˚ (15.156)
= f˚k+1 (Zk+1 |X)
˚ X
˚′ ⊆ ˚
and where X,
X. Because of the definition of a labeled multitarget state, the
˚L | ̸= |X|
˚ then
following must be true: if |X
˚ (k) ) = f˚k+1|k+1 (X|Z
˚ (k+1) ) = f˚k+1|k (X|
˚X
˚′ ) = 0.
f˚k+1|k (X|Z
That is,
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Advances in Statistical Multisource-Multitarget Information Fusion
˚ (k) ) of a physically unrealizable state
• The probability (density) f˚k+1|k (X|Z
˚ is nil.
set X
In addition,
˚
f˚k+1 (Z|X)
˚X
˚′ )
f˚k+1|k (X|
=
κk+1 (Z)
˚
= ˚
bk+1|k (X)
˚L | ̸= |X|
˚
|X
′
˚L | ̸= |X
˚′ |.
if |X
if
(15.157)
(15.158)
˚L | ̸= |X|
˚
˚ is
The first equation follows from the fact that if |X
then X
physically unrealizable, and thereby cannot generate measurements. Thus the only
measurements are those due to the clutter process κk+1 (Z). The second equation
˚′ is unrealizable then it cannot persist into the next
follows from the fact that if X
time-step. Thus the only targets in the next time-step are those due entirely to the
˚
target appearance process ˚
bk+1|k (X).
In addition, for any k ≥ 0,
˚ (k+1) ) = 0 unless X
˚ ⊆ ˚
˚
• f˚k+1|k+1 (X|Z
X0:k+1 . For if otherwise, X
contains a target that cannot yet exist at time tk+1 .
˚X
˚′ ) = 0 unless X
˚ ⊆˚
˚ contains a
• f˚k+1k (X|
X0:k+1 . For if otherwise, X
target that cannot exist at time tk+1
˚X
˚′ ) = ˚
˚ if it is not the case that X
˚′ ⊆ ˚
• f˚k+1k (X|
bk+1|k (X)
X0:k . For if
′
˚
otherwise, X contains a target that cannot exist at time tk .
˚X
˚′ ) = 0 unless
• If there is no model for target appearance, then f˚k+1k (X|
˚⊆˚
˚ contains a target that did not originate with
X
X0:k . For if otherwise, X
′
˚
a target in X .
˚X
˚′ ) = 0 unless X
˚L ⊆ X
˚′ . For, the only possible
• Furthermore, f˚k+1k (X|
L
′
˚ are those targets in X
˚ that survived from time tk to time
targets in X
tk+1 .
˚ = κk+1 (Z) if it is not the case that X
˚ ⊆ ˚
• f˚k+1 (Z|X)
X0:k+1 . For if
˚
otherwise, X contains a target that cannot exist at time tk+1 and thus it
cannot generate measurements.
15.4.2
Overview of the Vo-Vo Filter
The purpose of this section is to provide a “road map” of the Vo-Vo filter, before we
plunge into its technical details in the sections that follow.
Exact Closed-Form Multitarget Filter
469
First, suppose that the initial labeled multitarget distribution is a LMB distribution:
˚
˚ = δ ˚ ˚ · ω0|0 (X
˚L ) · (s̊0|0 )X
f˚0|0 (X)
(15.159)
|X|,|XL |
where ω0|0 (L) = 0 for all but a finite number of finite L ⊆ L0:0 .
˚X
˚′ )
Second, suppose that the labeled multitarget Markov density f˚k+1|k (X|
has the following generalized form (to be described more fully in Section 15.4.7):
• Persisting targets are governed by the labeled version of the modified standard
multi-Bernoulli motion model; and this means that the target appearance
process is a LMB RFS.
Then (as will be shown in Section 15.5.3) the predicted distribution is LMB:
∫
˚
˚X
˚′ ) · f˚0|0 (X
˚′ )δ X
˚′
f˚1|0 (X|Z)
=
f˚k+1|k (X|
(15.160)
=
˚
X
˚
δ|X|,|
˚ X
˚L | · ω1|0 (XL ) · (s̊1|0 )
(15.161)
for some s̊1|0 (x, ℓ) and some ω1|0 (L) with ω1|0 (L) = 0 for all but a finite
number of finite L ⊆ L1:0 .
Third, suppose that the multitarget likelihood function is the labeled version of
the standard multitarget measurement model (to be described more fully in Section
15.4.5). Let Z1 be the measurement set collected at time t1 and let m1 = |Z1 |.
Then (as will be shown in Section 15.5.2) the Bayes-updated distribution is a GLMB
distribution:
˚ (1) )
f˚1|1 (X|Z
=
=
˚ · f˚1|0 (X)
˚
f˚1 (Z1 |X)
˚ ) · f˚1|0 (Y
˚ )δ Y
˚
f˚1 (Z1 |Y
∑ ˚
˚
θ1 X
˚L ) · (s̊˚
δ˚ ˚
ω θ1 ( X
)
∫
|X|,|XL |
1|1
1|1
(15.162)
(15.163)
˚
θ1 ∈TZ1
˚1
˚
˚
θ1
θ1
for some s̊θ1|1
(x, ℓ) and some ω1|1
(L) with, for each ˚
θ1 , ω1|0
(L) = 0
for all but a finite number of finite L ⊆ L1:0 . Also, the summation is taken
over all measurement-to-track associations—that is, all functions ˚
θ1 : L0:1 →
{0, 1, ..., m1 } such that ˚
θ1 (ℓ) = ˚
θ1 (ℓ′ ) implies ℓ = ℓ′ .
Fourth, apply the next time-update step, in which case we will get a GLMB
distribution of the form
∑ ˚
˚1 X
˚
θ1 ˚
˚ (1) ) = δ ˚ ˚
f˚2|1 (X|Z
ω2|1
(XL ) · (s̊θ2|1
)
(15.164)
|X|,|XL |
˚
θ1 ∈TZ1
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Advances in Statistical Multisource-Multitarget Information Fusion
˚
θ1
with, for each ˚
θ1 , ω2|1
(L) = 0 for all but a finite number of finite L ⊆ L2:0 .
Fifth, let Z2 be the measurement set collected at time t2 and let m2 = |Z2 |.
Then the result of the next Bayes’ rule update will be a GLMB distribution
∑
˚
˚1 ,˚
˚
θ1 ,˚
θ2 ˚
θ2 X
˚ (2) ) = δ ˚ ˚
f˚2|2 (X|Z
ω2|2
(XL ) · (s̊θ2|2
)
(15.165)
|X|,|XL |
(˚
θ1 ,˚
θ2 )∈TZ1 ×TZ2
˚
˚1 ,˚
θ2
θ1 ,˚
θ2
for some s̊θ2|2
(x, ℓ) and ω2|2
(L) and where, for each ˚
θ1 , ˚
θ2 , it must be the
˚ ˚
θ 1 ,θ 2
case that ω2|2
(L) = 0 for all but a finite number of finite L ⊆ L2:0 . Also,
˚
θ2 : L0:2 → {0, 1, ..., m2 } is such that ˚
θ2 (ℓ) = ˚
θ2 (ℓ′ ) implies ℓ = ℓ′ .
Proceeding in this fashion, after k + 1 recursions of the time- and
measurement-update steps, we will end up with GLMB distributions of the form
˚ (k+1) )
f˚k+1|k+1 (X|Z
=
δ|X|,|
˚ X
˚L |
∑
·
(15.166)
˚
˚
˚
θ1 ,...,˚
θk+1 ˚
θ1 ,...,˚
θk+1 X
ωk+1|k+1
(XL ) · (s̊k+1|k+1
)
(˚
θ1 ,...,˚
θk+1 )
˚ (k) )
f˚k+1|k (X|Z
=
δ|X|,|
˚ X
˚L |
∑
˚
˚1 ,...,˚
˚
θ1 ,...,˚
θk ˚
θk X
(XL ) · (s̊θk+1|k
)
·
ωk+1|k
(15.167)
(˚
θ1 ,...,˚
θk )
where, for j = 1, ..., k, the functions ˚
θj : L0:j → {0, 1, ..., |Zj |} are such that
˚
˚
′
′
˚
˚
θj (ℓ) = θ(ℓ ) implies ℓ = ℓ . Because ω θ1 ,...,θk (L) = 0 for all but a finite
k|k
˚ (k) ) = 0
number of finite L ⊆ Lk:0 , for each ˚
θ1 , ..., ˚
θk , it follows that f˚k|k (X|Z
˚ ⊆ Xk:0 .
unless X
Equations (15.166) and (15.167) can be intuitively interpreted as follows:
˚ = n and L = X
˚L =
• Interpretation of the measurement-update: Let |X|
˚
θ ,...,˚
θ
1
k+1
{ℓ1 , ..., ℓn }. Then ωk+1|k+1
(L) is the weight of the hypothesis that:
– There are n tracks with distinct labels ℓ1 , ..., ℓn .
˚
θ ,...,˚
θ
˚
θ ,...,˚
θ
1
1
k+1
k+1
– Their track distributions are s̊k+1|k+1
(x, ℓ1 ), ...,s̊k+1|k+1
(x, ℓn ).
– These distributions arose as a result of the time history ˚
θ1 , ..., ˚
θk , ˚
θk+1
of measurement-to-track associations, including the latest association
˚
θk+1 .
Exact Closed-Form Multitarget Filter
471
˚
θ ,...,˚
θ
1
k+1
– For each i = 1, ..., n, s̊k+1|k+1
(x, ℓi ) will be of two types (see
(15.253)):
∗ The distribution of a track that was not detected, and which therefore arose from the previous time history ˚
θ1 , ..., ˚
θk .
∗ The distribution of a track that was detected, and which therefore
arose from the current time history ˚
θ1 , ..., ˚
θk , ˚
θk+1 .
˚ = n and L = X
˚L = {ℓ1 , ..., ℓn }.
• Interpretation of the time-update: Let |X|
˚
˚
1 ,...,θk
Once again, s̊θk+1|k
(x, ℓi ) will be of two types (see (15.281)):
– The distribution of a track that persisted from the previous time tk , and
which therefore arose from the previous time history ˚
θ1 , ..., ˚
θk .
– the distribution of a newly appearing track, and which will therefore be
independent of ˚
θ1 , ..., ˚
θk .
Now let D be the space of all Vo-Vo priors. At time tk these distributions
are parametrized by the space P of a finite list of parameters of the form
˚
p
=
˚
θ1 ,...,θk
( ωk|k
({ℓ1 , ..., ℓn }),
˚
˚
˚
˚
(15.168)
θ1 ,...,θk
s̊k|k
(x, ℓ1 ), ...,
1 ,...,θk
s̊θk|k
(x, ℓn ) )k,n,ℓ1 ,...,ℓn ,˚
θ1 ,...,˚
θk .
It then follows that the GLMB distributions constitute an exact closed-form
solution of the labeled multitarget Bayes filter, in the sense of Section 15.1.2.
Consequently, the labeled multitarget Bayes filter can be replaced by a filter that
propagates these parameters rather than their associated GLMB distributions:
... →
pk|k
→
pk+1|k
→
pk+1|k+1
→ ...
If one further employs the multitarget state-estimator of Section 15.3.4.7, the
˚k+1|k+1 can be constructed directly from the parameter
multitarget state estimate X
pk+1|k+1 . Thus this filter has the further form
... →
pk|k
→
pk+1|k
→
˚k+1|k+1
X
↑
pk+1|k+1
→ ...
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15.4.3
Basic Motion and Measurement Models
The Vo-Vo filter requires the following basic models:
• Probability of target survival: p̊S (x′ , ℓ′ ) abbr.
= p̊S,k+1|k (x′ , ℓ′ ). Since a target
cannot persist if it has not yet appeared, for completeness we must define
p̊S,k+1|k (x′ , l′ , i′ ) = 0
l′ > k.
(15.169)
f˚k+1|k (x, ℓ|x′ , ℓ′ ) = fk+1|k (x|x′ , ℓ′ ) · δℓ,ℓ′
(15.170)
if
• Single-target Markov transition density:
where fk+1|k (x|x′ , ℓ′ ) is the conventional single-target Markov density for
a target with label ℓ′ and where the Kronecker delta δℓ,ℓ′ specifies that
transitioning targets retain their labels in the manner described at the end of
Section 15.4.1.1.6
• Single-target probability of detection: p̊D (x, ℓ) abbr.
= p̊D,k+1|k (x, ℓ). Since
no measurement can be generated by a target that has not yet appeared, for
completeness we must define
p̊D,k+1|k (x, l, i) = 0
if
l > k + 1.
(15.171)
• Single-target likelihood function: ˚
Lz (x, ℓ) abbr.
= f˚k+1 (z|x, ℓ).7
• Poisson clutter process with clutter rate λ abbr.
= λk+1 and clutter spatial
distribution ck+1 (z) and intensity function κ(z) abbr.
= λk+1 · ck+1 (z):
κk+1 (Z) = e−λ
∏
κ(z) = e−λ · κZ .
z∈Z
6
Since f˚k+1 (x, ℓ|x′ , ℓ′ ) will not appear in formulas except in the products
p̊S (x′ , ℓ′ ) · f˚k+1 (x, ℓ|x′ , ℓ′ )
7
it is not necessary to define f˚k+1 (x, ℓ|x′ , l′ , i′ ) for l′ > k.
Since f˚k+1 (z|x, ℓ) will not appear in formulas except in products
p̊D (x, ℓ) · f˚k+1 (z|x, ℓ)
it is not necessary to define f˚k+1 (z|x, l, i) for l > k + 1.
(15.172)
Exact Closed-Form Multitarget Filter
15.4.4
473
Motion and Measurement Models with Target ID
Vo and Vo have shown that the Vo-Vo filter is easily extended to incorporate
target identity [307]. As a consequence, this filter provides an exact closed-form
and Bayes-optimal approach to joint target detection, tracking, localization, and
identification/classification. The purpose of this section is to briefly describe this
approach.
Let T = {τ1 , ..., τN } be a finite set of target-identity types. Replace the
kinematic state space X with the joint space X × T, so that the labeled state space
is X × T × L. Also, the sensor measurement space will have the form Z × F,
where Z is the kinematic measurement space and F is a space of feature vectors
ϕ.
Given this, the basic models in the previous section take the following forms:
• Probability of target survival: p̊S (x′ , τ ′ , ℓ′ ) abbr.
= p̊S,k+1|k (x′ , τ ′ , ℓ′ ).
• Single-target Markov transition density:
f˚k+1|k (x, τ, ℓ|x′ , τ ′ , ℓ′ )
=
=
′
′
′
(15.173)
′
′
′
fk+1|k (x|x , τ, τ , ℓ ) · pk+1|k (τ |x , τ , ℓ ) · δℓ,ℓ′
fk+1|k (x|x′ , τ ′ , ℓ′ ) · pk+1|k (τ |τ ′ ) · δℓ,ℓ′ .
(15.174)
If target identity does not change with time, then pk+1|k (τ |τ ′ ) = δτ,τ ′ .
• Single-target probability of detection: p̊D (x, τ, ℓ) abbr.
= p̊D,k+1|k (x, τ, ℓ).
• Single-target likelihood function:
˚
Lz,ϕ (x, τ, ℓ)
(15.175)
=
f˚k+1 (z, ϕ|x, τ, ℓ)
f˚k+1 (z|x, τ, ℓ, ϕ) · f˚k+1 (ϕ|x, τ, ℓ)
=
f˚k+1 (z|x, τ, ℓ) · f˚k+1 (ϕ|x, τ, ℓ).
(15.177)
=
(15.176)
If target features do not depend on target kinematics, then f˚k+1 (ϕ|x, τ, ℓ) =
f˚k+1 (ϕ|τ, ℓ) (see Remark 33 in Section 9.5.8.1).
• Poisson clutter process:
κk+1 (Z) = e−λ
∏
z∈Z
κ(z) = e−λ · κZ .
(15.178)
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15.4.5
Advances in Statistical Multisource-Multitarget Information Fusion
The Labeled Multitarget Likelihood Function
The purpose of this section is to:
1. Provide the specific formula for the multitarget likelihood function for the
labeled version of the standard multitarget measurement model.
2. Derive the GLMB reformulation of it that is necessary for achieving exact
closed-form closure with respect to the multitarget version of Bayes’ rule.
˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} with |X| = n and let |Z| = m.
Let X
Then from (7.21) it follows that the multitarget likelihood function for the labeled
standard multitarget measurement model is
˚
f˚k+1 (Z|X)

=
e
(15.179)

1 − δ|X|,|
˚ X
˚L |


˚
X

+δ ˚ X
˚L | · (1 − p̊D )
κ ·
 ∑ ∏ |X|,|

p̊D (xi ,ℓi )·f˚k+1 (zθ(ℓ) |xi ,ℓi )
· θ i:θ(i)>0 (1−p̊D (xi ,ℓi ))·κ(zθ(i) )
−λ Z
∏
where as usual κZ =
z∈Z κ(z), and where the summation is taken over all
functions θ : {1, ..., n} → {0, 1, ..., m} such that θ(i) = θ(i′ ) > 0 implies
i = i′ .
˚ is not physically realizable—that is, if |X|
˚ ̸= | X
˚L |—then (15.179)
If X
reduces to
˚ = e−λ κZ .
f˚k+1 (Z|X)
(15.180)
That is, there can be no target-generated measurements, and so the only measurements are those due to clutter.
˚ is physically realizable, then (15.179) reduces to a distinct-label version
If X
of the likelihood function for the standard multitarget measurement model, (7.21):
˚
˚ = e−λ κZ (1 − p̊D )X
f˚k+1 (Z|X)
∑
∏
θ
i:θ(i)>0
p̊D (xi , ℓi ) · f˚k+1 (zθ(ℓ) |xi , ℓi )
.
(1 − p̊D (xi , ℓi )) · κ(zθ(i) )
(15.181)
˚∩˚
˚ contains a target with a state (xj , lj , ij )
Note that if X
X0:k+2 ̸= ∅ then X
with lj > k + 1. Thus p̊D (xj , lj , ij ) = 0 and so the product in (15.179) vanishes
and so
˚ = e−λ κZ .
f˚k+1 (Z|X)
(15.182)
Exact Closed-Form Multitarget Filter
475
Now define
˚
˚
LθZ (x, ℓ) = δ0,˚
θ(ℓ) · (1 − p̊D (x, ℓ)) + (1 − δ0,˚
θ(ℓ) ) ·
p̊D (x, ℓ) · f˚k+1 (z˚
θ(ℓ) |x, ℓ)
κ(z˚
θ(ℓ) )
(15.183)
where ˚
θ : L → {0, 1, ..., m} is any function such that (1) ˚
θ(ℓ) = ˚
θ(ℓ′ ) > 0
implies ℓ = ℓ′ and (2) ˚
θ(ℓ) > 0 for only a finite number of ℓ ∈ L.
Then (15.179) can be equivalently written in GLMB-like form:8


∏
∑ ˚
˚
˚ = e−λ κZ ·1 − δ ˚ ˚ + δ ˚ ˚
˚L )
˚
f˚k+1 (Z|X)
λθk+1 (X
LθZ (x, ℓ)
|X|,|XL |
|X|,|XL |
˚
θ
˚
(x,ℓ)∈X
(15.184)
where9
˚
˚
λθk+1 (L) =
{ ∏
ℓ∈L0:k+1 −L δ˚
θ(ℓ),0
0
if
if
L ⊆ L0:k+1 is finite
.
otherwise
(15.185)
˚L | = |X|
˚ = n, in which
To see why (15.184) is true, first assume that |X
case ℓ1 , ..., ℓn are distinct. Then (15.184) becomes
˚ = e−λ κZ
f˚k+1 (Z|X)
∑
˚
˚
˚L )
λθk+1 (X
˚
θ
n
∏
˚
˚
LθZ (xi , ℓi ).
(15.186)
i=1
Second, note that, because of (15.185), the summation in (15.186) is finite. For,
˚
˚L ) ̸= 0—that is, for all ˚
it is actually taken over all ˚
θ such that λθk+1 (X
θ that
˚
˚
have the following additional property: θ(ℓ) = 0 for all ℓ not in XL . There
is a one-to-one correspondence between such ˚
θ and the finite set of all functions
˜ = θ(ℓ
˜ ′ ) implies ℓ = ℓ′ . Third, notice that
˚L → {0, 1, ..., m} such that θ(ℓ)
θ˜ : X
˜
˚
LθZ (x, ℓ) =
8
9
It is not true that
{
1 − pD (x, ℓ)
if
pD (x,ℓi )·f˚k+1 (zθ(ℓ)
|x,ℓ)
˜
κ(zθ(ℓ)
)
˜
if
˜ =0
θ(ℓ)
˜ >0 .
θ(ℓ)
(15.187)
∑ ∑
L
˚
θ
˚
θ λk+1 (L) = 1, so the expression in (15.184) is, strictly speaking, not
˚ need not be a GLMB distribution in the
GLMB. It need not be so, however, since fk+1 (Z|X)
˚
variable X.
˚
Note: It will always be clear from context that ˚
λθk+1 (L) does not mean the same thing as the
clutter rate λk+1 of the Poisson clutter process.
476
Advances in Statistical Multisource-Multitarget Information Fusion
Thus (15.186) becomes
˚
f˚k+1 (Z|X)
=
∑
e−λ κZ
θ˜

˜ i )=0
i:θ(ℓ
e
(1 − pD (xi , ℓi ))
(15.188)
κ(zθ(ℓ
˜ i))
˜ i )>0
i:θ(ℓ
=

∏
pD (xi , ℓi ) · f˚k+1 (zθ(ℓ
˜ i ) |xi , ℓi )
∏
·

−λ Z
κ ·
( n
∏
(1 − pD (xi , ℓi ))
)
(15.189)
i=1
·
∑
θ˜


∏
˜ i )>0
i:θ(ℓ
pD (xi , ℓi ) · f˚k+1 (zθ(ℓ
˜ i ) |xi , ℓi )
(1 − p̊D (xi , ℓi )) · κ(zθ(ℓ
˜ i))


and so
˚
f˚k+1 (Z|X)
˚
=
e−λ κZ · (1 − p̊D )X
(15.190)


∑
∏ pD (xi , ℓi ) · f˚k+1 (zθ(ℓ
˜ i ) |xi , ℓi )


·
(1 − p̊D (xi , ℓi )) · κ(zθ(ℓ
˜ i))
θ˜
˜ i )>0
i:θ(ℓ
˚
=
e−λ κZ · (1 − p̊D )X
(15.191)


∑
∏ pD (xi , ℓi ) · f˚k+1 (zθ(i) |xi , ℓi )


·
(1 − p̊D (xi , ℓi )) · κ(zθ(i) )
θ
i:θ(i)>0
which is identical to (15.181).
15.4.6
The Labeled Multitarget Markov Density—Standard Version
The purpose of this section is to:
1. Provide the specific formula for the multitarget Markov density for the labeled
version of the standard multitarget motion model.
2. Provide the specific formula for the special case of this density in which there
is no target appearance process.
Exact Closed-Form Multitarget Filter
477
3. Derive the GLMB reformulation of this latter Markov density (necessary for
achieving exact closed-form closure with respect to the multitarget prediction
integral).
Let
˚
X
˚′
X
=
{(x1 , ℓ1 ), ..., (xn , ℓn )}
(15.192)
=
{(x′1 , ℓ′1 ), ..., (x′n′ , ℓ′n′ )}
(15.193)
with |X| = n and |X ′ | = n′ . Then from (7.66) it follows that the Markov density
for the labeled version of the standard multitarget motion model is
˚X
˚′ )
f˚k+1|k (X|

=
e
B
−Nk+1|k

˚

bX
k+1|k · 
(15.194)

1 − δn′ ,|X
˚′ |
L
˚′
X
+δn′ ,|X
˚′ | · (1 − p̊S )
·
∑ ∏
θ
L
p̊S (x′i ,ℓ′i )·f˚k+1 (xθ(i) |x′i ,ℓ′i )·δℓ
′
θ(i) ,ℓi
i:θ(i)>0
(1−p̊S (x′i ,ℓ′i ))·bk+1|k (xθ(i) )



where the summation is taken over all functions θ : {1, ..., n′ } → {0, 1, ..., n}
such that θ(i) = θ(i′ ) implies i = i′ .
Suppose that no new targets appear, and thus that the only targets are those
that survive from one time-step to the next. Then because of (7.69), we can
equivalently write (15.194) as follows. If n ≤ n′ then
−
˚X
˚′ )
f˚k+1|k
(X|
˚′
=
X
δX,∅
(15.195)
˚ · (1 − δn′ ,|X
˚′ | ) + δn′ ,|X
˚′ | · (1 − p̊S )
L
·
L
n
∑ ∏
p̊S (x′ , ℓ′ ) · f˚k+1 (xi |x′ , ℓ′ ) · δi,τ i
τi
τ
i=1
τi
τi
τi
1 − p̊S (x′τ i , ℓ′τ i )
where the summation is taken over all τ : {1, ..., n} → {1, ..., n′ }
τ i = τ i′ implies i = i′ . Otherwise, if n > n′ ,
−
˚X
˚′ ) = δ ˚ .
f˚k+1|k
(X|
X,∅
such that
(15.196)
Given this, (15.195) can be written in a GLMB-like form, as follows. If
˚ ≤ |X
˚′ | then
|X|
∏
−
˚X
˚′ ) = δ ˚ ·(1− δ ˚′ ˚′ )+ δ ˚ ˚ ·β ˚′ (X
˚L )
˜ ˚ (x′ , ℓ′ )
f˚k+1|k
(X|
M
X,∅
|X |,|X |
|X|,|XL |
X
X
L
L
˚′
(x′ ,ℓ′ )∈X
(15.197)
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Advances in Statistical Multisource-Multitarget Information Fusion
where10
βL′ (L)
=
∏
(15.198)
1L′ (ℓ)
ℓ∈L
˜ ˚ (x′ , ℓ′ )
M
X
=
′
′
(1 − 1X
(15.199)
˚ (ℓ )) · (1 − p̊S (x , ℓ))
∑L
′ ′
′ ′
δℓ,ℓ′ · p̊S (x , ℓ ) · fk+1|k (x|x , ℓ ).
+
˚
(x,ℓ)∈X
˚ > |X
˚′ | then
Otherwise, if |X|
−
˚X
˚′ ) = δ ˚ .
f˚k+1|k
(X|
X,∅
(15.200)
˚ ≤ |X
˚′ | and then that |X
˚′ | = | X
˚′ |.
To prove (15.197), first assume that |X|
L
Then (15.195) becomes
˚′
−
˚X
˚′ ) = (1 − p̊S )X
f˚k+1|k
(X|
n
∑ ∏
p̊S (x′ , ℓ′ ) · f˚k+1 (xi |x′ , ℓ′ ) · δi,τ i
τi
τ
τi
τi
τi
1 − p̊S (x′τ i , ℓ′τ i )
i=1
.
(15.201)
Because of the factor δi,τ i , the only term in the summation that survives is that
corresponding to the unique function τ such that τ i = i for all i = 1, ..., n. Thus
(15.201) becomes
˚′
˚X
˚′ ) = (1 − p̊S )X
f˚k+1|k (X|
n
∏
p̊S (x′ , ℓ′ ) · f˚k+1 (xi |x′ , ℓ′ )
i
i=1
i
i
1 − p̊S (x′i , ℓ′i )
τi
.
(15.202)
10 That is, βL′ (L) = 0 unless L ⊆ L′ . The notation βL′ (L) in (15.198) follows from the fact
that βL′ (L) is the belief-mass function of the deterministic RFS Λ whose instantiations are the
subsets of L′ . In this case the belief-mass function is
∏
βΛ (L) = Pr(Λ ⊆ L) = Pr(L′ ⊆ L) = 1L′ ⊆L =
1L′ (ℓ).
ℓ∈L
Exact Closed-Form Multitarget Filter
479
˚L ⊆ X
˚′ —that is, the label-set for
Under the same conditions and assuming that X
L
surviving targets remains unchanged— (15.197) becomes
′
−
˚X
˚′ )
f˚k+1|k
(X|
=
n
∏
˜ ˚ (x′i , ℓ′i )
M
X
(15.203)
i=1
=
( n
∏
=
( n
∏
i=1
i=1
)
˜ ˚ (x′i , ℓ′i ) 
M
X
)
˜ ˚ (x′i , ℓ′i ) 
M
X

′
n
∏
i=n+1
˜ ˚ (x′i , ℓ′i ) (15.204)
M
X

′
n
∏
i=n+1
˜ ˚ (x′i , ℓ′i ) . (15.205)
M
X
But
n
∏
˜ ˚ (x′i , ℓ′i )
M
X
i=1
n (
∏
=
i=1
n
∏
=
(
(15.206)
′
′
(1 − 1X
˚L (ℓi )) · (1 − p̊S (xi , ℓi ))
∑n
′
′
+ j=1 δℓj ,ℓ′i · p̊S (xi , ℓi ) · fk+1|k (xj |x′i , ℓ′i )
p̊S (x′i , ℓ′i ) · fk+1|k (xi |x′i , ℓ′i )
)
)
(15.207)
i=1
and
′
′
n
∏
˜ ˚ (x′i , ℓ′i ) =
M
X
i=n+1
n
∏
(1 − p̊S (x′i , ℓi ))
(15.208)
i=n+1
and so
˚X
˚′ ) =
f˚k+1|k (X|
( n
∏(
i=1


)  n′
∏
)
p̊S (x′i , ℓ′i ) · fk+1|k (xi |x′i , ℓ′i ) ·
(1 − p̊S (x′i , ℓ′i ))
i=n+1
 (
)
n
′ ′
′ ′
∏
p̊
(x
,
ℓ
)
·
f
(x
|x
,
ℓ
)
S
i
k+1|k
i
i
i
i
=  (1 − p̊S (x′i , ℓi )) ·
1 − p̊S (x′i , ℓi )
i=1
i=1
′
n
∏
= (1 − p̊S )
n
′ ′
˚k+1 (xi |x′ )
˚′ ∏ p̊S (xi , ℓi ) · f
X
i
′ , ℓ′ )
1
−
p̊
(x
S
i
i
i=1
which, as claimed, is just (15.202).
(15.209)
480
15.4.7
Advances in Statistical Multisource-Multitarget Information Fusion
Labeled Multitarget Markov Density—Modified
The multitarget Markov density for the standard multitarget motion model is based
on the presumption that the target appearance RFS is Poisson. For the modified
standard multitarget motion model of (15.23), the target appearance RFS is LMB in
the sense of (15.85):
˚
˚
bk+1|k (X)
=
B
˚
δ|X|,|
˚ X
˚L | · ωk+1|k (XL )
∏
s̊B
k+1|k (x, ℓ) (15.210)
˚
(x,ℓ)∈X
˚
=
B
B
X
˚
δ|X|,|
˚ X
˚L | · ωk+1|k (XL ) · (s̊k+1|k ) .
(15.211)
The purpose of this section is to derive the formula for the multitarget Markov
density that incorporates this more general target appearance model.
The standard multitarget Markov density is also based on the presumption that
the persisting-target process is the same as that for the standard multitarget motion
model with no target appearances, that is, (15.197):
∏
−
˚X
˚′ ) = δ ˚ ·(1−δ ˚′ ˚′ )+δ ˚ ˚ ·β ˚′ (X
˚L )
f˚k+1|k
(X|
X,∅
|X |,|X |
|X|,|XL | X
L
L
˜ ˚ (x′ , ℓ′ ).
M
X
˚′
(x′ ,ℓ′ )∈X
(15.212)
Given that target appearances occur independently of existing targets, the total
Markov transition density is given by the convolution rule, (4.18):
˚X
˚′ ) =
f˚k+1|k (X|
∑
˚
˚−W
˚ ) · f˚− (W
˚ |X
˚′ ).
bk+1|k (X
k+1|k
(15.213)
˚ ⊆X
˚
W
Equation (15.213) can be greatly simplified as follows. Using the notation
˚ can be partitioned into
of Section 15.2.1, the time-updated multitarget state X
−
+
˚
˚
surviving targets X and appearing targets X :
˚=X
˚− ⊎ X
˚+
X
(15.214)
where
˚− = X
˚∩˚
X
X0:k ,
˚+ = X
˚∩˚
X
XB
k+1 .
(15.215)
Given this, (15.213) reduces to the following simple factored form:
˚− | X
˚′ ).
˚X
˚′ ) = ˚
˚+ ) · f˚− (X
f˚k+1|k (X|
bk+1|k (X
k+1|k
(15.216)
Exact Closed-Form Multitarget Filter
481
To demonstrate (15.216), we want to show that the only nonzero term in the
˚ =X
˚− and thus
summation on the right side of (15.213) is the one for which W
−
+
˚
˚
˚
˚
˚
X − W = X − X = X . To see this, note from the discussion in Section 15.4.1
−
˚ |X
˚′ ) = 0 unless W
˚ ⊆ X
˚− and thus W
˚ = W
˚ −, W
˚ + = ∅.
that f˚k+1|k
(W
˚−W
˚ ) = 0 unless X
˚−W
˚ ⊆X
˚+ . Given this,
Similarly, ˚
bk+1|k (X
˚+ ⊇ X
˚−W
˚ = (X
˚− − W
˚ − ) ⊎ (X
˚+ − W
˚ + ) = (X
˚− − W
˚ −) ⊎ X
˚+ . (15.217)
X
˚− − W
˚ − = ∅ since it cannot be in X
˚+ , and thus
From this it follows that X
˚ =W
˚− = X
˚− , as desired.
W
Thus the total Markov density has the form
˚X
˚′ )
f˚k+1|k (X|
=
(15.218)
B
˚+
δ |X
˚B |,|X
˚B | · ωk+1|k (XL )
L

·
·

∏
˚B
(x,ℓ)∈X

sB
k+1|k (x, ℓ)
(
δX
˚S ,∅ · (1 − δ|X
˚′ ˚′ )
∏ |,|XL | ˜
S
′ ′
˚
·
β
(
X
)
·
′ ′
˚S
˚′
˚′ M ˚S (x , ℓ )
+δ|X
˚S |,|X |
L
XL
L
(x ,ℓ )∈X
)
X
or
˚X
˚′ )
f˚k+1|k (X|
=
(
˚+
B
B
X
˚+
δ |X
˚B |,|X
˚+ | · ωk+1|k (XL ) · (sk+1|k )
L
(
)
˚′
˚−
˜X
· δ |X
˚− |,|X
˚− | · βX
˚′ ( X L ) · M X
−
˚
L
)
(15.219)
L
or
˚X
˚′ )
f˚k+1|k (X|
=
(
)
˚+
+
B
X
˚+
δ|X|,|
(15.220)
˚ X
˚L | · ωk+1|k (XL ) · (sk+1|k )
(
)
˚′
˚−
˜X
· βX
˚′ (XL ) · MX
˚− .
L
15.5
CLOSURE OF MULTITARGET BAYES FILTER
The purpose of this section is to demonstrate that the set of all Vo-Vo priors (GLMB
multitarget distributions) provides an exact closed-form solution of the multitarget
Bayes filter, in the sense described in Section 15.1.2. The section is organized as
follows:
482
Advances in Statistical Multisource-Multitarget Information Fusion
1. Section 15.5.1: A “road map” of the derivations of the time-update and
measurement-update equations.
2. Section 15.5.2: A demonstration that GLMB distributions have exact closedform closure under the multitarget Bayes filter measurement-update step
(multitarget Bayes’ rule).
3. Section 15.5.3: A demonstration that GLMB distributions have exact closedform closure under the multitarget Bayes filter time-update (multitarget prediction integral).
15.5.1
A “Road Map” for the Derivations
The purpose of this section is to provide a “road map” of the proof that the family of
GLMB distributions provides an exact closed-form solution of the multitarget Bayes
filter. This roadmap is has two parts, one for the time-update (Section 15.5.1.1) and
one for the measurement-update (Section 15.5.1.2). Both derivations require the
following lemma (Lemma 3 of [295]):
Lemma 1 Suppose that ω(L) ̸= 0 for only a finite number of finite subsets L ⊆ L.
∫
Let s̊(x, ℓ) be any function such that s̊(x, ℓ)dx = 1 for all ℓ; and let ˚
h(x, ℓ)
∫
˚
be any function such that h(x, ℓ) · s̊(x, ℓ)dx exists for all ℓ. Then:
∫
˚ ˚
˚ X
˚
δ|X|,|
˚ X
˚L | · ω(XL ) · (hs̊) δ X =
∑
ω(L)
L⊆L
∏∫
˚
h(x, ℓ) · s̊(x, ℓ)dx. (15.221)
ℓ∈L
To prove (15.221), note that
∫
˚ ˚
˚
˚
(˚
hX s̊)X · δ|X|,|
(15.222)
˚ X
˚L | · ω(XL )δ X
∫
∑ 1
∑
˚
h(x1 , ℓ1 ) · s̊(x1 , ℓ1 ) · · · ˚
h(xn , ℓn )
=
n!
n
n≥0
(ℓ1 ,...,ℓn )∈L
·s̊(xn , ℓn ) · δn,|{ℓ1 ,...,ℓn }| · ω({ℓ1 , ..., ℓn })dx1 · · · dxn
=
∑ 1
n!
n≥0
∑
n (∫
∏
˚
h(x, ℓi ) · s̊(x, ℓi )dx
(ℓ1 ,...,ℓn )∈Ln i=1
·δn,|{ℓ1 ,...,ℓn }| · ω({ℓ1 , ..., ℓn })
)
(15.223)
Exact Closed-Form Multitarget Filter
=
∑
∑
ω(L)
n≥0 L∈Fn (L)
=
∑
ω(L)
L⊆L
15.5.1.1
∏ (∫
483
˚
h(x, ℓ) · s̊(x, ℓ)dx
)
(15.224)
ℓ∈L
∏ (∫
)
˚
h(x, ℓ) · s̊(x, ℓ)dx .
(15.225)
ℓ∈L
Time Update Derivation: Road Map
A major reason that the GLMB distributions (Vo-Vo priors) provide a computationally tractable closed-form solution to the Bayes filter is the following fact:
• the convolutional formula for the labeled multitarget Markov density (15.213),
˚X
˚′ ) =
f˚k+1|k (X|
∑
˚
˚−W
˚ ) · f˚− (W
˚ |X
˚′ ),
bk+1|k (X
k+1|k
(15.226)
˚ ⊆X
˚
W
reduces to the simple factored form of (15.216):
˚X
˚′ ) = ˚
˚+ ) · f˚− (X
˚− | X
˚′ ).
f˚k+1|k (X|
bk+1|k (X
k+1|k
(15.227)
with specific models substituted, (15.227) becomes (15.220):
˚+
˚′
B
X
˚X
˚′ ) = δ ˚ ˚ ·ω B
˚+
˚− ˜ X
f˚k+1|k (X|
·βX
˚′ (XL )· MX
˚− . (15.228)
|X|,|XL | k+1|k (XL )·(sk+1|k )
L
Given this, let the prior distribution be GLMB:
˚ =δ˚ ˚
f˚k|k (X)
|X|,|XL |
∑
˚
o
˚L ) · (s̊o )X .
ωk|k
(X
k|k
(15.229)
o∈Ok|k
Substituting (15.228) and (15.229) into the prediction integral,
∫
˚ = f˚k+1|k (X|
˚X
˚′ ) · f˚k|k (X
˚′ )δ X
˚′ ,
f˚k+1|k (X)
(15.230)
we get
˚
f˚k+1|k (X)
˚+
=
B
B
X
˚+
δ|X|,|
˚ X
˚L | · ωk+1|k (XL ) · (s̊k+1|k )
∫
˚′
˚ ˚′ ˚′
˚−
˜X
· βX
˚′ ( X L ) · M X
˚− · fk|k (X )δ X .
L
(15.231)
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Advances in Statistical Multisource-Multitarget Information Fusion
An application of Lemma 1—that is, (15.221)—to the integral on the right leads to
(see (15.304))
∑
˚ = δ ˚ ˚ · ωB
˚+
f˚k+1|k (X)
k+1|k (XL )
|X|,|XL |
′
˚+
′
˚−
o
X
˚− ) · (s̊B
ω̃k+1|k
(X
· (ŝo )X
k+1|k )
L
o′ ∈Ok|k
(15.232)
which can then be rewritten as (see (15.275))
∑
˚ =δ˚ ˚
f˚k+1|k (X)
|X|,|XL |
˚
o
X
˚L ) · (s̊o
ωk+1|k
(X
k+1|k ) .
(15.233)
o∈Ok+1|k
15.5.1.2
Measurement Update Derivation: Road Map
We are given the multitarget likelihood function of (15.186):

˚ = e−λ κZ · 1 − δ ˚ ˚ + δ ˚ ˚
f˚k+1 (Z|X)
|X|,|XL |
|X|,|XL |
∑
˚
θ∈TZ

˚
˚ ˚
˚
˚L ) · (˚
λθk+1 (X
LθZ )X  .
(15.234)
Substitute this and the GLMB predicted distribution
∑
˚
o
X
˚L ) · (s̊o
ωk+1|k
(X
k+1|k )
(15.235)
˚ · fk+1|k (X)
˚
f˚k+1 (Zk+1 |X)
˚
f˚k+1|k+1 (X|Z)
=
.
f˚k+1 (Z)
(15.236)
˚ =δ˚ ˚
f˚k+1|k (X)
|X|,|XL |
o∈Ok+1|k
into multitarget Bayes’ rule:
Applying Lemma 1—i.e, (15.221)—to the normalization factor
f˚k+1 (Z) =
∫
˚ · f˚k+1|k (X)δ
˚ X
˚
f˚k+1 (Zk+1 |X)
(15.237)
we get
f˚k+1 (Z) = e−λ κZ
∑∑∑
L⊆L ˚
θ o∈O
˚
θ
o
ωk+1
(L) · ωk+1|k
(L)
∏
ℓ∈L
˚
˚θ
s̊o,ℓ
k+1|k [LZ ] (15.238)
Exact Closed-Form Multitarget Filter
485
and thus the posterior distribution becomes
=
˚
f˚k+1|k+1 (X|Z)
(15.239)
∑ ∑
˚
˚
˚
o
o
˚θ
˚
˚
˚θ X
δ|X|,|
˚ X
˚L | ·
˚
o∈O λk+1 (XL ) · ωk+1|k (XL ) · (s̊k+1|k LZ )
θ
.
∑
∑ ∑
∏
′
′
o ,ℓ
θ′
o′
θ′
′
˚˚
˚
˚˚
˚
L′ ⊆L
o′ ∈O λk+1 (XL ) · ωk+1|k (L )
ℓ′ ∈L′ s̊k+1|k [LZ ]
θ′
After suitable algebraic grouping, we end up with a GLMB distribution:
∑
˚
o,˚
θ
θ
X
˚
˚L ) · (s̊o,˚
f˚k+1|k+1 (X|Z)
= δ|X|,|
ωk+1|k+1
(X
˚ X
˚L |
k+1|k+1 ) .
(o,˚
θ)∈Ok+1|k+1
(15.240)
15.5.2
Closure Under Measurement Update with Respect to Vo-Vo Priors
Let Zk+1 be the new measurement set. Then Vo and Vo prove the following result
([295], Proposition 7). Let the predicted multitarget distribution be GLMB as in
(15.186):
∑
˚
o
X
˚ =δ˚ ˚
˚L ) · (s̊o
f˚k+1|k (X)
ωk+1|k
(X
(15.241)
k+1|k )
|X|,|XL |
o∈Ok+1|k
o
where ωk+1|k
(L) = 0 except for a finite number of finite L ⊆ L0:k+1 . Let the
labeled multitarget likelihood for Zk+1 be defined as in (15.186):
˚
f˚k+1 (Zk+1 |X)

=

e−λ κZk+1 · 1 − δ|X|,|
˚ X
˚L | + δ|X|,|
˚ X
˚L |
∑
˚
θ∈TZk+1
(15.242)

˚
˚
˚
˚
˚L ) · (˚
λθk+1 (X
LθZk+1 )X 
where TZk+1 is the set of all functions ˚
θ : L0:k+1 → {0, 1, ..., |Zk+1 |} such that
˚
θ(ℓ) = ˚
θ(ℓ′ ) implies ℓ = ℓ′ ; and where
∏
˚
˚
λθk+1 (L) =
δ˚
(15.243)
θ(ℓ),0
ℓ∈L0:k+1 −L
˚
˚
LθZ (x, ℓ)
=
(15.244)
δ0,˚
θ(ℓ) · (1 − p̊D (x, ℓ))
p̊D (x, ℓ) · f˚k+1 (z˚
θ(ℓ) |x, ℓ)
+(1 − δ0,˚
θ(ℓ) ) ·
.
κ(z˚
θ(ℓ) )
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Advances in Statistical Multisource-Multitarget Information Fusion
Let the posterior multitarget distribution be
˚
˚
˚
˚ k+1 ) = fk+1 (Zk+1 |X) · fk+1|k (X)
f˚k+1|k+1 (X|Z
f˚k+1 (Zk+1 )
(15.245)
with normalization factor
f˚k+1 (Zk+1 ) =
∫
˚ · f˚k+1|k (X)δ
˚ X.
˚
f˚k+1 (Zk+1 |X)
(15.246)
˚
Then f˚k+1|k+1 (X|Z)
is a GLMB distribution of the form
˚
f˚k+1|k+1 (X|Z)
=
∑
δ|X|,|
˚ X
˚L |
˚
o,θ
˚L ) (15.247)
ωk+1|k+1
(X
(o,˚
θ)∈Ok+1|k+1
·
∏
˚
θ
s̊o,
k+1|k+1 (x, ℓ)
˚
(x,ℓ)∈X
=
(15.248)
δ|X|,|
˚ X
˚L |
∑
·
˚
˚
˚
o,θ
X
˚L ) · (s̊o,θ
(X
ωk+1|k+1
k+1|k+1 )
(o,˚
θ)∈Ok+1|k+1
where:
• Measurement updated index set:
(15.249)
Ok+1|k+1 = Ok+1|k × TZk+1 .
• Measurement updated weight functions:
( ˚
)
o
˚
λθk+1 (L) · ωk+1|k
(L)
∏
θ
˚˚
· ℓ∈L s̊o,ℓ
k+1|k [LZk+1 ]
o,˚
θ
ωk+1|k+1 (L) = ( ∑
)
∑ ∑
θ′
′
˚˚
˚
L′ ⊆L
o′ ∈Ok+1|k λk+1 (L )
θ′
′ ′
∏
′
˚′
·ω o
(L′ ) ′ ′ s̊o ,ℓ [˚
Lθ
]
k+1|k
ℓ ∈L
k+1|k
(15.250)
Zk+1
where
θ
˚˚
s̊o,ℓ
k+1|k [LZk+1 ] =
∫
˚
˚
LθZk+1 (x, ℓ) · s̊ok+1|k (x, ℓ)dx.
(15.251)
Exact Closed-Form Multitarget Filter
487
• Measurement updated spatial distributions:
˚
s̊ok+1|k (x, ℓ) · ˚
LθZk+1 (x, ℓ)
o,˚
θ
s̊k+1|k+1 (x, ℓ) =
θ
˚˚
s̊o,ℓ
k+1|k [LZk+1 ]
(15.252)
o
δ0,˚
θ(ℓ) · (1 − p̊D (x, ℓ)) · s̊k+1|k (x, ℓ)
=
o,ℓ
δ0,˚
θ(ℓ) · s̊k+1|k [1 − p̊D ] + (1 − δ0,˚
θ(ℓ) ) ·
p̊D (x,ℓ)·˚
Lz˚
(1 − δ0,˚
θ(ℓ) ) ·
θ(ℓ)
˚
s̊o,ℓ
[p̊ L
k+1|k D z˚
(15.253)
]
θ(ℓ)
κ(z˚
θ(ℓ) )
(x,ℓ)·s̊ok+1|k (x,ℓ)
κ(z˚
θ(ℓ) )
+
o,ℓ
δ0,˚
θ(ℓ) · s̊k+1|k [1 − p̊D ] + (1 − δ0,˚
θ(ℓ) ) ·
˚
s̊o,ℓ
[p̊ L
k+1|k D z˚
.
]
θ(ℓ)
κ(z˚
θ(ℓ) )
Equations (15.248) through (15.252) are proved as follows. First, abbreviate
Z = Zk+1 and compute f˚k+1 (Z):
∫
˚ · f˚k+1|k (X)δ
˚ X
˚
f˚k+1 (Z) =
f˚k+1 (Z|X)
(15.254)


∫
∑ ˚
˚
˚
θ
θ
X
−λ Z 
˚
˚L ) · (˚
=
e κ
λk+1 (X
LZ ) 
(15.255)
˚
θ
(
· δ|X|,|
˚ X
˚L |
∑
o
˚L ) · (s̊o
ωk+1|k
(X
k+1|k )
˚
X
)
˚
δX
o∈O
=
e
−λ Z
κ
∑∑∫
˚ ˚
LθZ )X · δ|X|,|
(sok+1|k ˚
˚ X
˚L |
(15.256)
˚
θ o∈O
˚
˚L ) · ω o
˚
˚
·˚
λθk+1 (X
k+1|k (XL )δ X.
Now apply Lemma 1—that is, (15.221)—in which case (15.256) becomes
∑∑∑ ˚
∏ o,ℓ
˚
θ
o
f˚k+1 (Z) = e−λ κZ
ωk+1
(L) · ωk+1|k
(L)
s̊k+1|k [˚
LθZ ] (15.257)
L⊆L ˚
θ o∈O
where
θ
˚˚
s̊o,ℓ
k+1|k [LZ ] =
∫
ℓ∈L
˚
˚
LθZ (x, ℓ) · s̊ok+1|k (x, ℓ)dx.
(15.258)
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Advances in Statistical Multisource-Multitarget Information Fusion
So the posterior distribution is
=


˚
f˚k+1|k+1 (X|Z)
(
)
∑ ∑
θ
˚˚
˚
δ|X|,|
˚ X
˚L | ·
˚
o∈O λk+1 (XL )
θ
˚
o
θ X
˚L ) · (s̊o
˚˚
·ωk+1|k
(X
k+1|k LZ )
)
( ∑
∑ ∑
θ′
˚˚
˚
˚
L′ ⊆L
o′ ∈O λk+1 (XL )
θ′
′ ′
∏
˚′
,ℓ
o′
·ωk+1|k
(L′ ) ℓ′ ∈L′ s̊ok+1|k
[˚
LθZ ]
∑ ∑
θ
o
˚˚
˚
˚
o∈O λk+1 (XL ) · ωk+1|k (XL )
)
(∏
˚
˚
o
θ X
˚
(s̊k+1|k LZ )
o,ℓ
θ
˚˚
·
˚
o,ℓ
ℓ∈L s̊k+1|k [LZ ] · ∏
˚
s̊
[L θ ]
˚
θ
ℓ∈L


Z
( ∑
)
∑ ∑
˚
θ′
′
˚
λ
(L
)
′
′
′
˚
L ⊆L
θ
∏ o ∈O ′k+1
′
,ℓ′ ˚˚
o′
·ωk+1|k
(L′ ) ℓ′ ∈L′ s̊ok+1|k
[LθZ ]
= δ|X|,|
˚ X
˚L | ·
=
k+1|k
(15.259)
δ|X|,|
˚ X
˚L |
∑∑
(15.260)
˚
o,θ
˚L )
ωk+1|k+1
(X
(15.261)
˚
θ o∈O
·
˚
s̊ok+1|k (x, ℓ) · ˚
LθZ (x, ℓ)
∏
θ
˚˚
s̊o,ℓ
k+1|k [LZ ]
∑ ∑ o,˚
θ
θ
˚L ) · s̊o,˚
δ|X|,|
ωk+1|k+1
(X
˚ X
˚L |
k+1|k+1 (x, ℓ)
˚
(x,ℓ)∈X
=
(15.262)
˚
θ o∈O
where
(
˚
o,θ
ωk+1|k+1
(L)
=
)
˚
˚
˚L ) · ω o
λθk+1 (X
k+1|k (L)
∏
θ
˚˚
· ℓ∈L s̊o,ℓ
k+1|k [LZ ]
( ∑
)
∑ ∑
˚
θ′
′
˚
λ
(L
)
′
′
′
˚
k+1
L ⊆L
θ
∏ o ∈O ′ ,ℓ′ ˚˚
′
o′
·ωk+1|k
(L′ ) ℓ′ ∈L′ s̊ok+1|k
[LθZ ]
(15.263)
˚
˚
θ
s̊o,
k+1|k+1 (x, ℓ)
s̊ok+1|k (x, ℓ) · ˚
LθZ (x, ℓ)
=
˚
˚θ
s̊o,ℓ
k+1|k [LZ ]
.
(15.264)
Exact Closed-Form Multitarget Filter
15.5.3
489
Closure Under Time Update with Respect to Vo-Vo Priors
Vo and Vo prove the following result ([295], Proposition 8). Let the prior multitarget
distribution be GLMB as in (15.186):
∑
˚
o
˚ =δ˚ ˚
˚L ) · (s̊o )X
f˚k|k (X)
ωk|k
(X
(15.265)
k|k
|X|,|XL |
o∈Ok|k
o
where ωk|k
(L) = 0 except for a finite number of finite L ⊆ L0:k ; and where
∑ ∑
o
ωk|k
(L) = 1.
(15.266)
L⊆L0:k o∈Ok|k
Let the labeled multitarget Markov density be defined as in (15.220):
˚+
˚′
B
X
˚X
˚′ ) = δ ˚ ˚ ·ω B
˚+
˚− ˜ X
f˚k+1|k (X|
·βX
˚′ (XL )· MX
˚− (15.267)
k+1|k (XL )·(sk+1|k )
|X|,|XL |
L
where
˚−
X
˚+
X
=
=
˚∩X
˚0:k (persisting targets in X)
˚
X
B
˚∩X
˚k+1 (appearing targets in X)
˚
X
B
ωk+1|k
(L)
=
LMB weight function for target appearances (15.270)
sB
k+1|k (x, ℓ)
=
βL′ (L)
=
LMB spatial density for target appearances
∏
1L′ (ℓ)
(15.268)
(15.269)
(15.271)
(15.272)
ℓ∈L
˜ ˚− (x′ , ℓ′ )
M
X
=
′
′ ′
(1 − 1X
(15.273)
˚− (ℓ )) · (1 − p̊S (x , ℓ ))
L
∑
+
δℓ,ℓ′ · p̊S (x′ , ℓ′ ) · fk+1|k (x|x′ , ℓ′ )
˚−
(x,ℓ)∈X
where, note, βL′ (L) = 0 unless L ⊆ L′ .
Let the predicted multitarget distribution be defined by the prediction integral
∫
˚
˚
˚X
˚′ ) · f˚k|k (X
˚′ )δ X
˚′ .
fk+1|k (X) = f˚k+1|k (X|
(15.274)
˚
Then f˚k+1|k (X|Z)
is a GLMB distribution of the form
∑
˚
o
X
˚ =δ˚ ˚
˚L ) · (s̊o
f˚k+1|k (X)
ωk+1|k
(X
k+1|k )
|X|,|XL |
o∈Ok+1|k
(15.275)
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Advances in Statistical Multisource-Multitarget Information Fusion
where:
• Time updated index set:
(15.276)
Ok+1|k = Ok|k .
• Time updated weight functions:
o
B
o
ωk+1|k
(L) = ωk|k
(L ∩ Lk+1 ) · ω̃k|k
(L ∩ L0:k )
where for finite J ⊆ L0:k ,
(
o
ω̃k+1|k
(J )
=
∏
s̊oℓ [p̊S ]
)
(15.277)
(15.278)
ℓ∈J
·
(
∑
o
βL (J ) · ωk|k
(L)
L
s̊oℓ [p̊S ]
s̊oℓ [1 − p̊S ]
∏
s̊oℓ [1 − p̊S ]
)
ℓ∈L−J
=
∫
p̊S (x, ℓ) · s̊ok|k (x, ℓ)dx
(15.279)
=
∫
(1 − p̊S (x, ℓ)) · s̊ok|k (x, ℓ)dx.
(15.280)
• Time updated spatial distributions:
s̊ok+1|k (x, ℓ) = 1L0:k (ℓ) · ŝo (x, ℓ) + (1 − 1L0:k (ℓ)) · s̊B
k+1|k (x, ℓ) (15.281)
where
ŝo (x, ℓ)
=
˚x ]
s̊oℓ [p̊S M
=
˚x ]
s̊oℓ [p̊S M
o
s̊ℓ [p̊S ]
 ∑

′
∫
˚ p̊S (x , ℓ)
(x,ℓ)∈X
 ·fk+1|k (x|x′ , ℓ)  dx′
·s̊ok|k (x′ , ℓ)
and where the expression
∑
p̊S (x, ℓ) · fk+1|k (x|x′ , ℓ) · s̊ok|k (x′ , ℓ)
˚
(x,ℓ)∈X
is a shorthand way of writing the expression
p̊S (x′ , ℓ) · fk+1|k (xi |x′ , ℓ) · s̊ok|k (x′ , ℓ)
(15.282)
(15.283)
Exact Closed-Form Multitarget Filter
491
˚
where xi corresponds to the unique value of i such that (xi , ℓ) ∈ X.
To prove (15.275) through (15.283), note that the predicted labeled multitarget
distribution is given by the prediction integral:
∫
˚ =
˚X
˚′ ) · f˚k|k (X
˚′ )δ X
˚′
f˚k+1|k (X)
f˚k+1|k (X|
(15.284)
˚+
=
B
B
X
˚+
δ|X|,|
(15.285)
˚ X
˚L | · ωk+1|k (XL ) · (s̊k+1|k )
∫
˚′
˚−
˜X
· ωX
˚′ ( X L ) · M X
˚−
L


∑
′
′
′
˚
o
˚L′ ) · β ˚′ (X
˚− ) · (s̊o )X
˚′
 δX
·  δ |X
ωk|k
(X
˚′ |,|X
˚′ |
k|k
L
X
L
L
o′ ∈Ok|k
˚+
=
B
B
X
˚+
δ|X|,|
(15.286)
˚ X
˚L | · ωk+1|k (XL ) · (s̊k+1|k )
∫
∑
˚′ ˚′
o′
X
o′ ˜
˚′
˚−
·
δ |X
˚′ |,|X
˚′ | · ωk|k (XL ) · βX
˚− ) δ X .
˚′ (XL ) · (s̊k|k MX
L
L
o′ ∈Ok|k
By Lemma 1—that is, (15.221)—the integral becomes
∫
˚′ ˚′
o′
o′ ˜
X
˚′
˚−
δ |X
˚′ |,|X
˚′ | · ωk|k (XL ) · βX
˚′ (XL ) · (s̊k|k MX
˚− ) δ X (15.287)
L
L
∑
∏
′
−
o′
˚
˜ ˚− ]
=
ωk|k (L) · βL (XL )
(s̊ok|k )ℓ′ [M
X
ℓ′ ∈L′
˚′
L⊆X
L
where
′
˜ ˚− ] =
(s̊ok|k )ℓ′ [M
X
and where the product
′
∏
∫
˜ ˚− (x′ , ℓ′ ) · s̊o′ (x′ , ℓ′ )dx′
M
k|k
X
o′
˜ ˚− ]
ℓ′ ∈L (s̊k|k )ℓ′ [MX
is finite since, by assumption,
o
ωk|k
(L) vanishes for nonfinite L. Thus (15.287) becomes
∫
˚′ ˚′
o′
o′ ˜
X
˚′
˚−
δ |X
˚′ |,|X
˚′ | · ωk|k (XL ) · βX
˚′ (XL ) · (s̊k|k MX
˚− ) δ X
L
L


∑
∏
∏
′
′
o
˜ ˚− ] 
=
ωk|k
(L) · 
(s̊ok|k )ℓ′ [M
X
˚′ ⊇L⊇X
˚−
L:X
L
L
˚−
ℓ′ ∈ X
L
(15.288)
˚−
ℓ′ ∈L−X
L
(15.289)

′
˜ ˚− ] 
(s̊ok|k )ℓ′ [M
X
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Advances in Statistical Multisource-Multitarget Information Fusion
∑
=
′
o
ωk|k
(L)
(15.290)
˚′ ⊇L⊇X
˚−
L:X
L
L
(
∏
·
˚−
ℓ′ ∈ X
L

·
p̊S (x′ , ℓ′ )
˚ δℓ,ℓ′ ·
(x,ℓ)∈X
′ ′
o′
·fk+1|k (x|x , ℓ ) · s̊k|k (x′ , ℓ′ )dx′
∫
∏
)

′
(1 − p̊S (x′ , ℓ′ )) · s̊ok|k (x′ , ℓ′ )dx′ 
˚−
ℓ′ ∈L−X
L
∑
=
∫
∑
′
o
ωk|k
(L)
(15.291)
˚′ ⊇L⊇X
˚−
L:X
L
L
( ∫∑
∏
·
˚−
ℓ′ ∈ X
L

·
∫
∏
˚−
ℓ′ ∈L−X
L
′ ′
′ ′
˚ p̊S (x , ℓ ) · fk+1|k (x|x , ℓ )
(x,ℓ′ )∈X
′
′
·s̊ok|k (x′ , ℓ )dx′
)

′
(1 − p̊S (x′ , ℓ′ )) · s̊ok|k (x′ , ℓ′ )dx′  .
Let
′
=
˚x ]
s̊oℓ′ [p̊S M

∫
∑

˚
(x,ℓ′ )∈X
(15.292)
′

p̊S (x′ , ℓ′ ) · fk+1|k (x|x′ , ℓ′ ) · s̊ok|k (x′ , ℓ′ ) dx′
and
′
s̊oℓ′ [p̊S ]
=
∫
′
p̊S (x′ , ℓ′ ) · s̊ok|k (x′ , ℓ′ )dx
(15.293)
′
′
ŝo (x, ℓ′ )
=
′
s̊oℓ′ [1 − p̊S ]
=
˚x ]
s̊oℓ′ [p̊S M
′
o
s̊ ′ [p̊S ]
∫ ℓ
′
(1 − p̊S (x′ , ℓ′ )) · s̊ok|k (x′ , ℓ′ )dx′ .
(15.294)
(15.295)
Exact Closed-Form Multitarget Filter
493
Then (15.291) becomes
∫
˚′ ˚′
o′
o′ ˜
X
˚′
˚−
δ |X
˚′ |,|X
˚′ | · ωk|k (XL ) · βX
˚′ (XL ) · (s̊k|k MX
˚− ) δ X (15.296)
L
L


∑
∏
′
′
o
˚x ]
=
ωk|k
(L) 
s̊oℓ′ [p̊S M
˚′ ⊇L⊇X
˚−
L:X
L
L

·
∏
˚−
ℓ′ ∈ X
L
s̊oℓ′ [1 − p̊S ]
˚−
ℓ′ ∈L−X
L
∑
=
·
=
∏
∑

=


·
˚−
(x′ ,ℓ′ )∈X

′
ŝo (x, ℓ′ )) · s̊oℓ′ [p̊S ] (15.297)

s̊oℓ′ [1 − p̊S ]

∏
′
˚−
ℓ′ ∈L−X
L
˚−
ℓ′ ∈ X
L
·
′
o
˚− ) · 
ωk|k
(L) · βL (X
L
∏

∏
′
˚′
L⊆X
L

o
ωk|k
(L) 
′
˚−
ℓ′ ∈L−X
L
·

′
˚′ ⊇L⊇X
˚−
L:X
L
L


′

soℓ′ [1 − p̊S ] 
∏
˚−
ℓ′ ∈ X
L

′
soℓ′ [p̊S ]
∏
′
˚−
(x′ ,ℓ′ )∈X
(15.298)

ŝo (x, ℓ)

′
s̊oℓ′ [p̊S ]
∑
(15.299)
′
o
˚− )
ωk|k
(L) · βL (X
L
˚−
ℓ′ ∈L−X
L
˚′
L⊆X
L
∏
˚−
(x′ ,ℓ′ )∈X
∏
′

ŝo (x′ , ℓ′ )
′

s̊oℓ′ [1 − p̊S ]
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Advances in Statistical Multisource-Multitarget Information Fusion


=
∏
˚−
ℓ′ ∈ X
L

·
s̊oℓ′ [p̊S ]
∑
(15.300)
′
˚− ) · ω o (L)
β L (X
k|k
L
∏
˚−
ℓ′ ∈L−X
L
˚′
L⊆X
L

·

′
∏
˚−
(x′ ,ℓ′ )∈X

′
s̊oℓ′ [1 − p̊S ]

′
ŝo (x′ , ℓ′ )
and so
∫
˚′ ˚′
o′
o′ ˜
X
˚′
˚−
δ |X
˚′ |,|X
˚′ | · ωk|k (XL ) · βX
˚′ (XL ) · (s̊k|k MX
˚− ) δ X (15.301)
L
L
′
=
′
o
˚− ) · (ŝo )
ω̃k+1|k
(X
L
˚−
X
where
′
o
ω̃k+1|k
(J ) =
(
∏
ℓ′ ∈J

)
∑
∏
′
′
o
s̊oℓ′ [p̊S ] 
βL (J ) · ωk|k
(L)
s̊oℓ′ [1 − p̊S ] .
′
˚′
L⊆X
L
ℓ′ ∈L−J
(15.302)
Note that this vanishes for all but a finite number of finite J ⊆ L, because the same
o′
is true of ωk|k
(J ).
Thus the entire predicted multitarget distribution in (15.286) becomes
˚
f˚k+1|k (X)
˚+
=
B
B
X
˚+
δ|X|,|
˚ X
˚L | · ωk+1|k (XL ) · (s̊k+1|k )
∑
˚−
o′
˚− ) · (ŝo′ )X
·
ω̃k+1|k
(X
L
(15.303)
o′ ∈Ok|k
=
B
˚+
δ|X|,|
(15.304)
˚ X
˚L | · ωk+1|k (XL )
∑
′
+
′ ˚−
˚
o
X
˚− ) · (s̊B
·
ω̃k+1|k
(X
· (ŝo )X .
k+1|k )
L
o′ ∈Ok|k
Now, as in (15.278), define
s̊ok+1|k (x, ℓ) = 1L0:k (ℓ) · ŝo (x, ℓ) + (1 − 1L0:k (ℓ)) · s̊B
k+1|k (x, ℓ)
(15.305)
Exact Closed-Form Multitarget Filter
and note that
∫
495
s̊ok+1|k (x, ℓ)dx = 1L0:k (ℓ) + 1 − 1L0:k (ℓ) = 1.
(15.306)
Then
∏
˚
(s̊ok+1|k )X =
s̊ok+1|k (x, ℓ)
(15.307)
˚
(x,ℓ)∈X


=
˚−
(x,ℓ)∈X

=

∏

s̊ok+1|k (x, ℓ) 

∏
˚−
(x,ℓ)∈X
ŝo (x, ℓ) 
˚−
=

∏
˚+
(x,ℓ)∈X
s̊ok+1|k (x, ℓ)
(15.308)

∏
˚+
(x,ℓ)∈X

s̊B
k+1|k (x, ℓ)
(15.309)
˚+
X
(ŝo )X · (s̊B
.
k+1|k )
(15.310)
Thus (15.304) becomes
˚
f˚k+1|k (X)
=
B
˚+
δ|X|,|
˚ X
˚L | · ωk|k (XL )
∑
o′
˚− )
ω̃k+1|k
(X
L
(15.311)
o′ ∈Ok|k
′
˚
·(s̊ok+1|k )X
=
∑
δ|X|,|
˚ X
˚L |
′
′
˚
o
X
˚L ) · (s̊o
ωk+1|k
(X
k+1|k )
(15.312)
o′ ∈Ok|k
where
′
o
ωk+1|k
(L)
Now, since
∫
′
=
B
o
ωk+1|k
(L ∩ Lk+1 ) · ω̃k|k
(L ∩ L0:k )
(15.313)
=
B
o′
ωk+1|k
(L − L0:k ) · ω̃k|k
(L ∩ L0:k ).
(15.314)
˚ X
˚ = 1 by construction, (15.116) tells us that
f˚k+1|k (X)δ
∑
∑
L⊆L0:k+1 o′ ∈Ok|k
and so we are done.
′
o
ωk+1|k
(L) = 1
(15.315)
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Advances in Statistical Multisource-Multitarget Information Fusion
15.6
IMPLEMENTATION OF THE VO-VO FILTER: SKETCH
One could try to implement the multitarget Bayes filter using the time-update and
measurement-update steps of Sections 15.5.3 and 15.5.2, respectively. However,
Vo and Vo have shown that it is computationally advantageous to reformulate these
steps by replacing GLMB distributions with “δ-GLMB distributions.”
The implementation approach is discussed in detail in [296]. This section
presents a sketch, organized as follows:
1. Section 15.6.1: δ-GLMB distributions.
2. Section 15.6.2: δ-GLMB version of the Vo-Vo filter.
3. Section 15.6.3: An exact L1 characterization of the effect of pruning δGLMB components.
15.6.1
δ-GLMB Distributions
As was explained in Section 15.4.2, after k recursions of the time-update and
measurement-update steps of Sections 15.5.3 and 15.5.2, we will end up with
GLMB distributions of the form
∑
˚
θ1 ,...,˚
θk ˚
˚ (k) ) = δ ˚ ˚
f˚k|k (X|Z
ωk|k
(X L )
(15.316)
|X|,|XL |
(˚
θ1 ,...,˚
θk )
˚
˚
θ1 ,...,˚
θk X
·(s̊k|k
)
where, for j = 1, ..., k, the functions ˚
θj : L0:j → {0, 1, ..., |Zj |} are such that
′
′
˚
˚
θj (ℓ) = θ(ℓ ) implies ℓ = ℓ .
Now, (15.316) can be rewritten as
˚ (k) )
f˚k|k (X|Z
=
(15.317)
δ|X|,|
˚ X
˚L |
∑
·
J,˚
θ1 ,...,˚
θk
ωk|k
· δJ,X
˚L
(J,˚
θ1 ,...,˚
θk )∈F(L0:k )×Ak|k
˚
˚
˚
1 ,...,θk
·(s̊θk+1|k+1
)X
=
∑
δ|X|,|
˚ X
˚L |
(J,αk )∈F(L0:k )×Ak|k
˚
X
k
·(s̊α
k+1|k+1 )
J,αk
ωk|k
· δJ,X
˚L (15.318)
Exact Closed-Form Multitarget Filter
497
where F(L0:k ) is the class of all finite subsets of L0:k and where
Ak|k
=
αk
J
=
⊆
J,˚
θ1 ,...,˚
θk
ωk|k
=
TZ1 × ... × TZk ,
(˚
θ1 , ..., ˚
θk ),
L0:k
˚
(15.319)
(15.320)
(15.321)
˚
θ1 ,...,θk
ωk|k
(J ).
(15.322)
This leads to the following definition, in which the set-parameter J is
absorbed into the index set Ok|k ([295], Definition 9). A δ-GLMB RFS is a GLMB
RFS with the following specific form:
1. The index space O has the form O = F(L) × A and o = (J, α) ∈ O,
where A is a set of association-sequences α.
2. The GLMB weight ω o (L) = ω J,α (L) has the form ω J,α (L) = ω J,α · δJ,L .
3. The GLMB density so (x, ℓ) = sJ,α (x, ℓ)
s̊J,α (x, ℓ) = sα (x, ℓ).
does not depend on
J:
Intuitively speaking:
• The pair (J, αk ) is a hypothesis about how the measurements in the sequence
Z1 , ..., Zk have been successively assigned to those tracks whose labels are
in J ⊆ L0:k .
J,αk
• The number ωk|k
is the degree of confidence in this hypothesis.
Vo and Vo have shown that the class Dδ of δ-GLMB distributions also
solves the multitarget Bayes filter in exact closed form. The demonstration of this
fact will not be reproduced here. Interested readers are referred to [295].
This seemingly minor reformulation results in a significant computational
savings. When implementing the time-updates and measurement-updates of Sec˚
˚1 ,...,˚
θ1 ,...,˚
θk
θk
(L) and s̊θk|k
(x, ℓ)
tions 15.5.3 and 15.5.2, we must construct ωk|k
for every choice of (L, ˚
θ1 , ..., ˚
θk ). This requires storage and computation of
|F(L0:k ) × TZ1 × ... × TZk | and |F(L0:k ) × TZ1 × ... × TZk | items, respectively.
For the δ-GLMB formulation of the Vo-Vo filter we still must store and compute the
L,˚
θ1 ,...,˚
θk
quantities ωk|k
for every choice of (L, ˚
θ1 , ..., ˚
θk ). But we need store and
˚
˚
1 ,...,θk
compute the sθk+1|k+1
(x, ℓ) only for every choice of ˚
θ1 , ..., ˚
θk —which involves
only |TZ1 × ... × TZk | items.
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15.6.2
Advances in Statistical Multisource-Multitarget Information Fusion
δ-GLMB Version of the Vo-Vo Filter
The δ-GLMB version of the Vo-Vo filter propagates labeled multi-Bernoulli mixtures of the form
∑
˚
˚
J,˚
θ1 ,...,˚
θk
θ1 ,...,˚
θk X
˚ (k) ) = δ ˚ ˚
f˚k|k (X|Z
ωk|k
· δJ,X
˚L · (s̊k+1|k+1 ) . (15.323)
|X|,|XL |
(J,˚
θ1 ,...,˚
θk )
As time progresses the number of components in this mixture greatly increases,
and so components must be pruned to keep it down to a computationally feasible
level. After each measurement-update, a certain number Mk|k of components
˚
˚
J,θ1 ,...,θk
are discarded and the weights ωk|k
are renormalized. This elimination
process is accomplished using Murty’s algorithm, which can determine the Mk|k
most significant components without evaluating the entire set of weights. Since
additional components are added because of the target appearance model, additional
elimination of components is required during the time-update step.
Multitarget state estimation is accomplished using the heuristic approach
described in Section 15.3.4.7. For greater detail, see [296].
15.6.3
Characterization of Pruning
Because the number of terms in a δ-GLMB distribution grows super-exponentially
with time, it is necessary to prune small-weight terms. Similar pruning is required
in tracking algorithms such as MHT and JPDA, but the effect of hypothesis-pruning
on the probability law of the multitarget state is unknown.
By way of contrast, the effect of pruning terms from δ-GLMB distributions
can be characterized not only exactly, but by a simple formula. The purpose of this
section is to summarize this (somewhat amazing) result.
Let us be given an unnormalized δ-GLMB distribution
∑
˚ =δ˚ ˚
f˚O (X)
|X|,|XL |
˚
α X
ω J,α · δJ,X
˚L · (s̊ )
(15.324)
(J,α)∈O
defined for a given set O of indices. Suppose that we wish to eliminate the terms
in this sum corresponding to a subset O′ ⊆ O of indices. Let
∥f˚∥1 =
∫
˚ X
˚
|f˚(X)|δ
(15.325)
Exact Closed-Form Multitarget Filter
499
˚ on X∞ . Then the error caused by the
denote the L1 norm on functions f˚(X)
truncation is given by ([296], Proposition 5):
∥f˚O − f˚O′ ∥1 =
∑
ω J,α .
(15.326)
(J,α)∈O−O′
That is, the L1 norm between the pruned and unpruned distributions is just the sum
of the weights of the pruned terms.
Furthermore, the error between the corresponding normalized distributions is
bounded as follows:
?
?
? f˚
f˚O′ ?
∥f˚O ∥1 − ∥f˚O′ ∥1
? O
?
−
.
(15.327)
?
? ≤2
? ∥f˚O ∥1
∥f˚O′ ∥1 ?
∥f˚O ∥1
1
15.7
PERFORMANCE RESULTS
Vo and Vo report performance evaluations of Gaussian mixture and sequential
Monte Carlo (SMC) implementations of the δ-GLMB version of the Vo-Vo filter.
These are now described.
15.7.1
Gaussian Mixture Implementation of Vo-Vo Filter
In this implementation, the single-target motion and measurement models are
˚1 ,...,˚
θk
linear-Gaussian and the spatial distributions sθk|k
(x, ℓ) are approximated as
Gaussian mixtures [305]. The authors have given this implementation of the δGLMB version of the Vo-Vo filter the name “para-Gaussian multi-target filter.”
The test scenario includes up to 10 appearing and disappearing targets, which
follow linear trajectories in a rectangular region 2000 meters on a side. The
targets are observed by a single linear-Gaussian sensor with constant probability of
detection of 0.98, and with uniformly distributed Poisson clutter having a constant
clutter rate λ = 60.
The authors reported that the filter initiated and terminated tracks with a very
small delay, and accurately estimates target number and the target states. The filter
also exhibited only a very small number of spurious or dropped tracks, especially
considering the fairly large clutter rate.
The authors also compared the new filter with a GM-CPHD filter. The GM
implementation of the Vo-Vo filter was significantly better at estimating target
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Advances in Statistical Multisource-Multitarget Information Fusion
number than the GM-CPHD filter, while also doing a somewhat better job of
estimating target states.
15.7.2
Particle Implementation of the Vo-Vo Filter
In this implementation [295], the single-target motion and measurement models
˚
θ1 ,...,˚
θk
are nonlinear and the sk|k
(x, ℓ) are approximated as Dirac mixtures and
propagated using particle methods. Up to 10 appearing and disappearing targets
are present at any given time, and follow curvilinear trajectories within a half-disc
of 2000 meter radius. The targets are observed by a single range-bearing sensor
located at the origin. Clutter is uniformly distributed and Poisson, with constant
clutter rate λ = 20. The probability of detection pD (x) is state-dependent and
is circular-Gaussian in shape, peaking at 0.98 at the origin and tapering to 0.92 at
the edge of the surveillance region. The single-target motion model is a nonlinear
coordinated-turn model. The target appearance process is a labeled Poisson process.
The track distributions are approximated and propagated using particle methods.
The authors report that this implementation accurately estimated the target
states, while initiating and terminating tracks with a small delay. There were a small
number of dropped tracks and false tracks. Most crucially, no track-switching was
observed—meaning that track labels were consistently estimated and propagated
throughout the entire scenario.
The authors also compared their filter with an SMC-CPHD filter. The SMC
implementation of the SMC-δ-GLMB filter estimated target number significantly
more accurately than the SMC-CPHD filter, and with significantly smaller variance.
The two filters were also compared using the OSPA metric (Section 6.2.2), using
100 Monte Carlo trials. The OSPA error of the SMC-δ-GLMB filter was less than
half that of the SMC-CPHD filter. However, this performance gain was achieved at
the cost of an order of magnitude increase in computational load.
Part III
RFS Filters for Unknown
Backgrounds
501
Chapter 16
Introduction to Part III
In Section 7.2, the following multitarget measurement model was considered:
all measurements
target measurements
clutter measurements
? ?? ?
Σk+1
?
??
?
= Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪
? ?? ?
Ck+1
(16.1)
where
• X = {x1 , ..., xn } with |X| = n is the multitarget state of the targets present
at time tk+1 .
• Υk+1 (x) is the RFS of measurements generated by a target with state x.
• Ck+1 is the clutter RFS.
• Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are assumed to be statistically independent.
The “standard” multitarget measurement model was further distinguished by
the following additional two requirements: Ck+1 is Poisson and Υk+1 (x) is
Bernoulli. That is, the p.g.fl. of Υk+1 (x) is
GΥk+1 (x) [g] = 1 − pD (x) + pD (x)
503
∫
g(z) · fk+1 (z|x)dz
(16.2)
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where pD (x) is the probability of detection and fk+1 (z|x) is the sensor likelihood
function. Equivalently, the probability distribution of Υk+1 (x) is

1 − pD (x)
if
Z=∅

pD (x) · fk+1 (z|x) if Z = {z} .
fΥk+1 (x) (Z) =
(16.3)

0
if otherwise
In what follows, the term measurement background will refer to:
• The clutter background, as specified by Ck+1 or its probability distribution
κk+1 (Z); together with
• The background detection profile, as specified by the state-dependent probability of detection pD (x).
All major multitarget detection and tracking algorithms are based on the
assumption that both of these models are known a priori. Typically, these models
have two forms:
• Explicit clutter and detection-profile models, such as those presumed for all
of the RFS multitarget tracking algorithms considered thus far (such as the
CPHD and CPHD filters).
• Implicit background models, such as those used in traditional multitarget
tracking algorithms such as multihypothesis trackers (MHTs). At a purely
theoretical level, MHTs are usually based on the assumption that probability
of detection is constant and that the clutter process is a spatially uniform
Poisson RFS. But at a practical implementation level, an MHT’s a priori
clutter model and detection profile models are usually implicit, in the form of
track-initiation and track-termination rules. For example:
– Track initiation: Assume that the probability of detection is large. If
the clutter rate is small, then a newly appearing track can be declared
quickly. This is because any new measurements can be presumed
to be probably target-generated. If the clutter rate is large, however,
then track initiation must be accomplished more cautiously, requiring
possibly many time-steps before a new track can be declared.
– Track termination: Assume that the clutter rate is small. If in addition the probability of detection is small, then caution is required before
eliminating a track. This is because it is possible that, at least momentarily, no measurements are being collected from it. If the probability
Introduction to Part III
505
of detection is large, then a track can be eliminated nearly as soon as
one stops observing it.
In real-world applications, a priori background models (whether explicit or
implicit) are often not available. The clutter background will typically be both
unknown and unpredictably varying over time. The detection profile will typically
be even more unpredictable and dynamic, since it often varies with a target’s bodyframe orientation and surface characteristics.
In such cases, trackers with a priori models will typically exhibit degraded
performance, because of mismatch between the presumed and the actual background models. What does one do, then, when the background is unknown and/or
dynamic? The traditional approach, “clutter rejection,” falls roughly into two categories:
• Background modeling: Physics-based modeling of the physical environment
is employed to predict what clutter background one would expect to observe
at any given moment. Typically, this is a very expensive and time-consuming
approach, and must be accomplished anew for each new sensor. It also
requires very accurate a priori environmental information, such as threedimensional terrain maps.
• Empirical training: Learning algorithms are used to statistically characterize
the background using a set of training data. This approach is based on the
assumption that the training data is statistically diverse enough to address all
of the scenarios that might be encountered at all times. Neural nets are one
commonly employed technique. Another one is “clutter mapping,” in which
the region of interest is subdivided into disjoint cells and the clutter density or
its inverse is estimated in each cell using histogram or other methods [216].
The purpose of the chapters in Part III is to describe PHD, CPHD, and
multi-Bernoulli filters that do not fall into either of these categories. They are
“background-agnostic” in the following sense:
• They implicitly estimate the clutter and/or detection background, on-the-fly,
while simultaneously detecting and tracking the targets that may be obscured
within the background.
16.1
INTRODUCTION
This introductory chapter is organized as follows:
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1. Section 16.2: Overview of the approach:
unknown backgrounds.
multitarget Bayes filters for
2. Section 16.3: General and specific models for unknown clutter and unknown
probability of detection.
3. Section 16.4: The organization of Part III.
16.2
OVERVIEW OF THE APPROACH
In the approach taken throughout Part III, the unknown background is taken into
account in three ways:
1. Modeling an unknown detection profile (see Section 16.3.1 for more details):
single-target states x are replaced by augmented states x̊ = (a, x), where
0 ≤ a ≤ 1 is the unknown probability of detection of x. Thus (16.1) takes
the form
Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1
(16.4)
where the RFS Υk+1 (a, x) is Bernoulli with
GΥk+1 (a,x) [g] = 1 − a + a
∫
g(z) · fk+1 (z|x)dz
(16.5)
and where, as usual, fk+1 (z|x) is the sensor likelihood function.
2. Modeling an unknown clutter process (see Section 16.3.2 for more details):
Clutter measurements are presumed to be caused by an unknown number ν
of unknown clutter generators. Like targets, clutter generators are characterized by states c belonging to a clutter state space C. Just as a target with
state x has an associated Bernoulli measurement-generation RFS Υk+1 (x),
so a clutter generator with state c has an associated measurement-generation
RFS Ck+1 (c). In this case, (16.1) takes the form
Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 (c̊1 ) ∪ ... ∪ Ck+1 (c̊ν ) (16.6)
where the notation c̊ will be explained momentarily.
Introduction to Part III
507
3. Modeling an unknown detection profile and unknown clutter process: In this
most general case, (16.1) takes the form
Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1 (c̊1 ) ∪ ... ∪ Ck+1 (c̊ν ).
(16.7)
In 2009, Mahler proposed that the clutter generator measurement-generation
processes Ck+1 (c̊) be Poisson [155], with c̊ = (c, c) and with unknown Poisson
parameter c > 0. Equation (16.1) then takes the form
Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (c1 , cν )
(16.8)
where the p.g.fl. of the RFS Ck+1 (c, c) has the form
( ∫
)
κ
GCk+1 (c,c) [g] = exp c (g(z) − 1) · fk+1 (z|c)dz
(16.9)
with unknown clutter rate c > 0 and clutter-generator spatial distribution
κ
fk+1
(z|c). This approach is the subject of Section 16.3.3.
Chen Xin, Kirubarajan, et al. [37], [39] in 2009 and Mahler [189], [199],
[194] in 2010 independently proposed that Ck+1 (c̊) be Bernoulli. There are two
possibilities.
First, one can assume that the probability of detection c = pκD (c) of
c̊ = (c, c) is known a priori,
∫
κ
GCk+1 (c) [g] = 1 − pκD (c) + pκD (c) g(z) · fk+1
(z|c)dz
(16.10)
with clutter probability of detection pκD (c) and clutter likelihood function
κ
fk+1
(z|c); and, more restrictively, that it is constant:
∫
κ
κ
κ
GCk+1 (c) [g] = 1 − pD + pD g(z) · fk+1
(z|c)dz.
(16.11)
This latter case will be described in more detail in Section 16.3.5.
Second, one can assume that the probability of detection for c is an unknown
quantity 0 ≤ c ≤ 1. In this case the clutter state has the form c̊ = (c, c) and
(16.1) takes the form
Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (cν , cν )
(16.12)
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with
GCk+1 (c,c) [g] = 1 − c + c
∫
κ
g(z) · fk+1
(z|c)dz.
(16.13)
This case is described in more detail in Section 16.3.4.
Given these background models, in principle it becomes possible to detect and
track multiple targets in unknown backgrounds. Let ˚
X be the space of augmented
target-states (a, x) and ˚
C the space of augmented clutter-generator states (c, c)
(where either c > 0 in the Poisson case or 0 ≤ c ≤ 1 in the Bernoulli case). Let
¨ denote a finite subset of the joint target-clutter state space
¨ ⊆X
X
¨ =˚
X
X ⊎˚
C.
(16.14)
Then given the models, the optimal solution to the unknown-background problem
is the following generalization of the multitarget Bayes recursive filter:
¨ (k) ) → fk+1|k (X|Z
¨ (k) ) → fk+1|k+1 (X|Z
¨ (k+1) ) → ...
... → fk|k (X|Z
where
16.3
¨ (k) )
fk+1|k (X|Z
=
¨ (k+1) )
fk+1|k+1 (X|Z
=
fk+1 (Z|Z (k) )
=
∫
¨ X
¨ ′ ) · fk|k (X
¨ ′ |Z (k) )δ X
¨ ′ (16.15)
fk+1|k (X|
¨ · fk+1|k (X|Z
¨ (k) )
fk+1 (Zk+1 |X)
(16.16)
fk+1 (Zk+1 |Z (k) )
∫
¨ · fk+1|k (X|Z
¨ (k) )δ X.
¨ (16.17)
fk+1 (Z|X)
MODELS FOR UNKNOWN BACKGROUNDS
The purpose of this section is to describe, in greater detail, the unknown-background
models just introduced. The section is organized as follows:
1. Section 16.3.1: A general model for unknown detection profile.
2. Section 16.3.2: A general model for unknown clutter.
3. Section 16.3.3: A Poisson-mixture model for unknown clutter.
4. Section 16.3.4: A general multi-Bernoulli model for unknown clutter.
5. Section 16.3.5: A simplified multi-Bernoulli model for unknown clutter.
Introduction to Part III
16.3.1
509
A Model for Unknown Detection Profile
The following simple example demonstrates that it is possible, at least in principle,
to recursively estimate the probability of detection pD (x) at the state x of a given
target track. Suppose that a single sensor observes a single static target with state
x0 ,1 and that there is no clutter. Suppose that, over k time-steps, the sensor collects
the measurement sets Z1 , ..., Zk , where by assumption |Zk | = 0 or |Zk | = 1.
Let
k
∑
νk =
|Zk |
(16.18)
l=1
be the cumulative number of target detections at time tk . Then the probability of
detection at x0 is, approximately,
νk
pD (x0 ) ∼
.
=
k
(16.19)
The same reasoning holds if the target is dynamic and pD is known to be
approximately constant in the region of interest. At any instant, we can estimate
both the state-vector xk|k of the target and the probability of detection pD (xk|k )
of the track. The reasoning still holds if there are multiple targets that are not too
close to each other (with respect to sensor resolution).
Stated differently:
• The probability of detection at a track is an unknown that can be statistically
estimated using a recursive filter.
This fact can be expressed more formally as follows [190]. Replace the
kinematic state x by the augmented state
x̊ = (a, x)
(16.20)
where 0 ≤ a ≤ 1 is the unknown probability of detection of x. Thus the space of
augmented target states is
˚
X = [0, 1] × X
(16.21)
where [0, 1] is the unit interval. The integral of a function f˚(x̊) with arguments
in this space is
∫
∫ ∫ 1
˚
f (x̊)dx̊ =
f˚(a, x)dadx.
(16.22)
0
1
The term “static” means that the state x of the target does not vary with time.
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The multitarget state space ˚
X∞ is the hyperspace of all finite subsets of ˚
X.
A multitarget state set has the form
˚ = {x̊1 , ...,x̊n } = {(a1 , x1 ), ..., (an , xn )}.
X
The corresponding set integral has the form
∫
∑ 1 ∫
˚
˚
˚
f (X)δ X =
f˚({x̊1 , ...,x̊n })dx̊1 · · · dx̊n .
n!
(16.23)
(16.24)
n≥0
Because the state space has been changed from X to ˚
X, we must correspondingly change any modeling formulas, occurring earlier in the book, that involve state
variables. Thus the usual probability of detection pD (x) and usual single-target
likelihood function Lz (x) are replaced by the augmented probability of detection
and augmented likelihood function
p̊D (x̊)
˚
Lz (x̊)
=
=
p̊D (a, x) def.
=a
˚
Lz (a, x) def.
= Lz (x).
(16.25)
(16.26)
(Here it is being assumed, as an approximation, that a target will generate the same
measurement, regardless of its detectability.)
Given this, the multitarget measurement model of (16.4) can be written as
Σk+1 = Υk+1 (x̊1 ) ∪ ... ∪ Υk+1 (x̊n ) ∪ Ck+1
(16.27)
where Ck+1 is the a priori clutter process, and where Υk+1 (x̊) is Bernoulli with
∫
GΥk+1 (a,x) [g] = 1 − a + a g(z) · fk+1 (z|x)dz.
(16.28)
From (4.126), the p.g.fl. of Σk+1 is
˚
˚ = (1 − p̊D + p̊D ˚
Lg )X · Gκk+1 [g]
Gk+1 [g|X]
(16.29)
where the functional-power notation hX was defined in (3.5); where Gκk+1 [g] is
the p.g.fl. of Ck+1 ; and where
∫
∫
˚
˚
˚
Lg (x̊) = Lg (a, x) = g(z) · Lz (a, x)dz = g(z) · Lz (x)dz.
(16.30)
Introduction to Part III
16.3.2
511
A General Model for Unknown Clutter
This has the form of (16.8):
Σk+1 = Υk+1 (x̊1 ) ∪ ... ∪ Υk+1 (x̊n ) ∪ Ck+1 (c̊1 ) ∪ ... ∪ Ck+1 (c̊ν ).
(16.31)
Here, x̊i = (ai , xi ) with 0 ≤ ai ≤ 1, and Υk+1 (x̊) is Bernoulli with p.g.fl.
GΥk+1 (a,x) [g] = 1 − a + a
∫
g(z) · fk+1 (z|x)dz.
(16.32)
Also, c̊i = (ci , ci ) where ci ∈ C, and where there are four cases of interest:
1. Case 1: Ck+1 (c̊) is Poisson with unknown clutter rate c > 0:
˚κ
GCk+1 (c,c) [g] = eLg−1 (c.c)
where
˚
Lκg−1 (c, c) = c
∫
κ
(g(z) − 1) · fk+1
(z|c)dz.
(16.33)
(16.34)
2. Case 2: Ck+1 (c̊) is Bernoulli with unknown probability of detection 0 ≤
c ≤ 1:
∫
κ
GCk+1 (c,c) [g] = 1 − c + c g(z) · fk+1
(z|c)dz,
(16.35)
in which case the clutter probability of detection and clutter-generator likelihood function are
p̊κD (c, c)
κ
fk+1
(z|c, c)
=
=
c
κ
˚
Lκz (c, c) = Lκz (c) = fk+1
(z|c).
(16.36)
(16.37)
3. Case 3: Ck+1 (c̊) = Ck+1 (c) is Bernoulli with known probability of
detection pκD (c):
GCk+1 (c) [g] = 1 − pκD (c) + pκD (c)
∫
κ
g(z) · fk+1
(z|c)dz.
(16.38)
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4. Case 4: Ck+1 (c̊) = Ck+1 (c) is Bernoulli with known and constant
probability of detection c = pκD , and where the clutter spatial distribution is
independent of c and thus is known a priori:
κ
fk+1
(z|c) = ck+1 (z).
(16.39)
In this case
GCk+1 (c) [g] = 1 − pκD + pκD
∫
g(z) · ck+1 (z)dz
(16.40)
and thus only the clutter rate λk+1 is unknown and must be determined.
In all of these cases, let us package the unknowns—those involving both
targets and the clutter generators—into an unknown state set of the form
¨
X
=
˚⊎C
˚
{ẍ1 , ..., ẍn+ν } = X
(16.41)
=
=
{x̊1 , ...,x̊n } ⊎ {c̊1 , ...,c̊ν }
{x̊1 , ...,x̊n ,c̊1 , ...,c̊ν }
(16.42)
(16.43)
where ẍ denotes an element of the joint target-clutter state space
¨ =˚
X
X ⊎˚
C
(16.44)
and where ‘⊎’ denotes disjoint union. Thus ẍ = x̊ or ẍ = c̊.
Remark 68 (Notational convention) Using a slight abuse of notation, if a =
pD (x) or c = pκD (c), then it will be understood that
x̊ = (pD (x), x) ?→ x
respectively
c̊ = (pκD (c), c) ?→ c.
that is, x̊ is identified with x and c̊ is identified with c. In this case,

c
if
prob. det. is unknown

pκD (c) if
prob. det. is known
p̊κD (c, c) =
(16.45)

pκD
if prob. det. is known and constant
κ
fk+1
(z|c, c)
=
κ
fk+1
(z|c).
(16.46)
Introduction to Part III
The integral
¨ is
space X
∫
513
f¨(ẍ)dẍ of a function f¨(ẍ) on the joint target-clutter state
∫
f¨(ẍ)dẍ =
∫
∫
f¨(x̊)dx̊ +
˚
X
f¨(c̊)dc̊
˚
C
with the following special cases (assuming that the target probability of detection
pD (x) is known):
• Poisson clutter:
∫
f¨(ẍ)dẍ =
∫
f¨(x)dx +
X
∫ ∫ ∞
f¨(c, c)dcdc.
0
C
• Bernoulli clutter, unknown clutter probability of detection:
∫
f¨(ẍ)dẍ =
∫
f¨(x)dx +
X
∫ ∫ 1
C
f¨(c, c)dcdc.
(16.47)
0
• Bernoulli clutter, known clutter probability of detection:
∫
f¨(ẍ)dẍ =
∫
f¨(x)dx +
∫
f¨(c)dc.
(16.48)
f ({ẍ1 , ..., ẍn })dẍ1 · · · dẍn .
(16.49)
X
C
The corresponding set integrals have the form
∫
¨ X
¨ =
f (X)δ
∑ 1 ∫
n!
n≥0
¨ (k) ) is the distribution of the RFS Ξ
¨ k||k then
If fk|k (X|Z
˚
¨ k|k ∩ ˚
Ξk|k = Ξ
X
(16.50)
˚k|k = Ξ
¨ k||k ∩ ˚
Ψ
C
(16.51)
is the RFS of targets and
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is the RFS of clutter generators. According to the discussion in Section 3.5.3, the
˚k|k are the marginals
distributions of ˚
Ξk|k and Ψ
˚ (k) )
fk|k (X|Z
˚ (k) )
fk|k (C|Z
=
∫
˚ ⊎ C|Z
˚ (k) )δ C
˚
f¨k|k (X
=
∫
˚ C|Z
˚ (k) )δ C
˚
fk|k (X,
=
∫
˚ ⊎ C|Z
˚ (k) )δ X
˚
f¨k|k (X
=
∫
˚ C|Z
˚ (k) )δ X.
˚
fk|k (X,
(16.52)
(16.53)
Given all of this, the p.g.fl. of the measurement model of (16.31) is:
˚
˚
¨ = (1 − p̊D + p̊D ˚
Gk+1 [g|X]
Lg )X · (1 − p̊κD + p̊κD ˚
Lκg )C .
(16.54)
¨ factors as
This is because, given the independence assumptions, Gk+1 [g|X]
¨
Gk+1 [g|X]
=
=
=
16.3.3
˚ ⊎ C]
˚
Gk+1 [g|X
Gk+1 [g|x̊1 ] · · · Gk+1 [g|x̊n ]
·Gk+1 [g|c̊1 ] · · · Gk+1 [g|c̊ν ]
˚
˚
(1 − p̊D + p̊D ˚
Lg )X · (1 − p̊κD + p̊κD ˚
Lκg )C .
(16.55)
(16.56)
(16.57)
Unknown-Clutter Models: Poisson-Mixture
This is the measurement model for the Poisson-mixture clutter-agnostic PHD filter
to be described in Section 18.10. It was introduced by Mahler in [179], Section
12.11, and is a generalization of a Bayesian static data-clustering approach due to
Cheeseman [33], [34]. In this case the clutter RFS in (16.8) has the form
˚ = Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (cν , cν )
Ck+1 (C)
(16.58)
˚ = {(c1 , c1 ), ..., (cν , cν )}. Since the Poisson RFSs Ck+1 (c1 , c1 ),...,
where C
˚ is itself a Poisson RFS. AccordCk+1 (cν , cν ) are independent, Ck+1 (C)
ing to (16.33), the respective PHDs (intensity functions) of the clutter generaκ
κ
tors are c1 · fk+1
(z|c1 ), ..., cν · fk+1
(z|cν ) and their respective p.g.fl.’s are
κ
κ
˚
˚
Lg−1 (c1 ,c1 )
Lg−1 (cν ,cν )
κ
˚
e
, ..., e
, where L (c, c) was defined in (16.34). Then the
g−1
Introduction to Part III
515
PHD (intensity function) of the total clutter process is
˚
κk+1 (z|C)
κ
κ
c1 · fk+1
(z|c1 ) + ... + cν · fk+1
(z|cν )
κ
κ
˚
˚
= Lz (c1 , c1 ) + ... + Lz (cν , cν )
(16.59)
=
(16.60)
where ˚
Lκz (c, c) was defined in (16.37). Thus the p.g.fl. of the total measurement
model is
˚ ⊎ C]
˚
Gk+1 [g|X
˚ · Gk+1 [g|C]
˚
Gk+1 [g|X]
(
)X
˚
˚κ
˚
1 − p̊D + p̊D ˚
Lg
· (eLg−1 )C
=
=
(16.61)
(16.62)
where
˚
Gk+1 [g|C]
˚κ
˚
(eLg−1 )C =
=
∏
˚κ
eLg−1 (c,c)
(16.63)
˚
(c,c)∈C
˚
Lκg−1 (c, c)
c·
=
∫
κ
(g(z) − 1) · fk+1
(z|c)dz.
(16.64)
Remark 69 It should also be noted (see Section 18.10) that the Poisson-mixture
model—as well as the Poisson-mixture CPHD filter—can be easily generalized to
clutter generators whose p.g.fl.’s have the form
Gκk+1 [g|c, c] = Gκk+1
(
1−c+c
∫
κ
g(z) · fk+1
(z|c)dz
)
,
where Gκk+1 (z) is some p.g.f.
16.3.4
Unknown-Clutter Models: General Bernoulli
This is the case considered in (16.35). It is the measurement model for the “κagnostic” CPHD filter of Section 18.5.
By analogy with the unknown-pD model in (16.8), the state representation
of a clutter generator has the form c̊ = (c, c) where 0 ≤ c ≤ 1 is the unknown
probability of detection of the clutter generator with kinematic state c. Thus
p̊κD (c̊)
˚
Lκz (c̊)
=
=
c
κ
Lκz (c) = fk+1
(z|c).
(16.65)
(16.66)
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The entire clutter intensity function κk+1 (z) is unknown and must be determined.
According to (16.54), its p.g.fl. is
(
)X
)C˚
˚ (
˚ ⊎ C]
˚ = 1 − p̊D + p̊D ˚
Gk+1 [g|X
Lg
· 1 − p̊κD + p̊κD ˚
Lκg
(16.67)
˚ = {x̊1 , ...,x̊n } = {(a1 , x1 ), ..., (an , xn )} and C
˚ = {(c1 , c1 ), ..., (cν , cν )}
where X
and
˚
˚ = (1 − p̊κD + p̊κD ˚
Gk+1 [g|C]
Lκg )C
(16.68)
is the p.g.fl. of the multi-Bernoulli clutter process.
16.3.5
Unknown-Clutter Models: Simplified Bernoulli
This was the case considered in (16.40). It is used for the “λ-agnostic” CPHD filter
of Section 18.4.
By assumption, the clutter RFS in (16.8) has the form
Ck+1 = Ck+1 (c1 ) ∪ ... ∪ Ck+1 (cν )
(16.69)
where the RFSs Ck+1 (c1 ), ..., Ck+1 (cν ) are Bernoulli and independent, with stateindependent clutter probability of detection and state-independent clutter likelihood
function:
pκD (c)
Lκz (c)
=
=
pκD
ck+1 (z).
(16.70)
(16.71)
The p.g.fl. of Ck+1 (c) is therefore independent of the state c:
Gk+1 [g|c] = 1 − pκD + pκD · ck+1 [g],
where
ck+1 [g] =
∫
g(z) · ck+1 (z)dz.
(16.72)
(16.73)
The p.g.fl. of the total measurement process, (16.54), becomes
(
)X
˚
|C|
¨ = 1 − p̊D + p̊D ˚
Gk+1 [g|X]
Lg
· (1 − pκD + pκD ck+1 [g])
(16.74)
Introduction to Part III
517
where X = {x̊1 , ...,x̊n } = {(a1 , x1 ), ..., (an , xn )} and C = {c1 , ..., cν } and
¨ =X
˚ ⊎ C and where the p.g.fl. of the clutter RFS is
X
Gk+1 [g|C] = (1 − pκD + pκD · ck+1 [g])|C| .
(16.75)
Note that the exponent on the right side of this equation is a scalar |C| rather than
a finite set C.
16.4
ORGANIZATION OF PART III
Part III is organized as follows:
1. Chapter 17: PHD filters, CPHD filters, and multi-Bernoulli filters that can
operate with unknown probability of detection.
2. Chapter 18: PHD filters, CPHD filters, and multi-Bernoulli filters that can
operate in unknown clutter.
In regard to the RFS filters described in these chapters, the following disclaimers should be emphasized:
• The detection profile must not change too rapidly in comparison to the
measurement-update rate.
• The clutter statistics must not change too rapidly in comparison to the
measurement-update rate.
Furthermore, all of these filters must simultaneously accomplish at least two
or more of the following difficult tasks, using the same measurement-stream:
1. Detect targets.
2. Track targets.
3. Implicitly or explicitly estimate pD .
4. Implicitly or explicitly estimate clutter rate λ or the clutter intensity function
κ.
Despite these difficulties, simulations have shown that these filters do seem
to perform reasonably well (although with decreasing performance as they are
required to simultaneously accomplish more and more tasks).
Chapter 17
RFS Filters for Unknown pD
17.1
INTRODUCTION
The model for an unknown target-detection profile was described in Section 16.3.1.
As noted there, if an RFS-based multitarget detection and tracking filter employs
the standard multitarget measurement model, then the unknown detection profile
model can be used to convert it into a filter that does not require a priori knowledge
of the probability of detection (see Section 17.1.1).
The purpose of this chapter is to describe the following filters and their
practical implementations:
• pD -agnostic PHD filter (“pD -PHD filter” for short): the classical PHD filter,
generalized to address unknown probability of detection.
• pD -agnostic CPHD filter (“pD -CPHD filter” for short): the classical CPHD
filter, generalized to address unknown probability of detection.
• pD -agnostic CBMeMBer filter (“pD -CBMeMBer filter” for short): the CBMeMBer filter, generalized to address unknown probability of detection.
The remainder of this Introduction is organized as follows:
1. Section 17.1.1: Overview of the approach—converting RFS tracking filters
into filters that do not require priori knowledge of the probability of detection.
2. Section 17.1.2: Defining motion models for probability of detection—that is,
the Markov transition density for the probability of detection.
3. Section 17.1.3: A summary of the major Lessons Learned in the chapter.
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4. Section 17.1.4: Organization of the chapter.
17.1.1
Converting RFS Filters into pD -Agnostic Filters
The conversion requires only a simple substitution of variables. Specifically:
• Whenever x occurs in a formula, substitute (a, x).
• Whenever pD (x) occurs, substitute a.
• Whenever Lz (x) occurs, do not change it.
∫
∫ ∫1
• Whenever an integral ·dx occurs, substitute the integral
·dadx.
0
Thus, for example, the measurement-update and time-update formulas for the
classical PHD filter are (see (8.50) through (8.52), and (8.15) through (8.16)):
Dk+1|k+1 (x)
Dk+1|k (x)
=
τk+1 (z)
=
Dk+1|k (x)
=
1 − pD (x) +
∑
pD (x) · Lz (x)
κk+1 (z) + τk+1 (z)
(17.1)
z∈Zk+1
∫
pD (x) · Lz (x) · Dk+1|k (x)dx
(17.2)
bk+1|k (x)
(17.3)
)
∫ (
pS (x′ ) · fk+1|k (x|x′ )
+
· Dk|k (x′ )dx′ .
+bk+1|k (x|x′ )
Making the indicated substitutions renders the PHD filter pD -agnostic:
˚k+1|k+1 (a, x)
D
˚k+1|k (a, x)
D
τk+1 (z)
=
∑
a · Lz (x)
κk+1 (z) + τk+1 (z)
(17.4)
˚k+1|k (a, x)dadx
a · Lz (x) · D
(17.5)
1−a+
z∈Zk+1
=
∫ ∫ 1
0
˚k+1|k (a, x)
D
= ˚
bk+1|k (a, x)
(17.6)
)
∫ ∫ 1(
p̊S (a′ , x′ ) · f˚k+1|k (a, x|a′ , x′ )
+
+˚
bk+1|k (a, x|a′ , x′ )
0
˚k|k (a,′ x′ )da′ dx′ .
·D
Some thought is, of course, required to suitably interpret items such as
˚
bk+1|k (a, x), p̊S (a′ , x′ ), and ˚
bk+1|k (a, x|a′ , x′ ). Since probability of detection
RFS Filters for Unknown pD
521
should not influence the appearance of new targets, one must have
˚
bk+1|k (a, x)
′
′
˚
bk+1|k (a, x|a , x )
=
=
(17.7)
bk+1|k (x)
′
(17.8)
bk+1|k (x|x ).
Similarly,
p̊S (a′ , x′ ) = pS (x′ )
(17.9)
since probability of detection should have no bearing on whether or not an existing
target disappears. For the Markov transition density, we assume that:
f˚k+1|k (a, x|a′ , x′ ) = fk+1|k (a|a′ ) · fk+1|k (x|x′ ).
(17.10)
This is an approximation, because the value of the probability of detection a will
in general be correlated with the kinematic target-state x.
With these changes, the time-update equation for the pD -PHD filter becomes
˚k+1|k (a, x)
D
17.1.2
=
bk+1|k (x)
(17.11)
)
∫ ∫ 1(
′
′
′
pS (x ) · fk+1|k (a|a ) · fk+1|k (x|x )
+
+bk+1|k (x|x′ )
0
˚k|k (a,′ x′ )da′ dx′ .
·D
A Motion Model for Probability of Detection
In (17.11), the Markov transition fk+1|k (a|a′ ) must be defined. In Appendix F it
is shown that it can be defined implicitly as
βuk+1|k ,vk+1|k (a) =
∫ 1
fk+1|k (a|a′ ) · βuk|k ,vk|k (a′ )da′
(17.12)
0
where βu,v (a) denotes a beta distribution with parameters u, v; and where
uk+1|k
v
k+1|k
θk|k
=
=
=
uk|k · θk|k
v
k|k
· θk|k
1
·
uk|k + v k|k
(17.13)
(17.14)
(
k|k
k|k
u ·v
1
· 2
−1
(uk|k + v k|k )2 σk+1|k
)
(17.15)
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and where the desired variance of the time-updated probability of detection,
(
)
1
uk|k · v k|k
2
(
)(
),
σk+1|k =
+
ε
·
(17.16)
uk|k + v k|k
uk|k + v k|k uk|k + v k|k + 1
is set by choosing a value of ε with 0 ≤ ε ≤ 1.
2
2
It is always the case that σk+1|k
≥ σk|k
—that is, that the uncertainty in a
never decreases during a time-update. The following two values of ε describe the
2
extremes of σk+1|k
:
ε
ε
17.1.3
=
=
0:
1:
2
2
σk+1|k
= σk|k
2
σk+1|k
= (
u
(no increase)
k|k
·v
(17.17)
k|k
)2 (largest increase).
uk|k + v k|k
(17.18)
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• A simple approach for addressing unknown probability of detection is to
replace the single-target state x with an augmented state x̊ = (a, x) where
0 ≤ a ≤ 1 is the unknown probability of detection of the unknown track x
(Section 17.1.1).
• A suitable Markov motion model for a allows us to model the increase
in uncertainty in the time interval between measurement-updates (Section
17.1.2).
• Any computationally tractable RFS multitarget detection and tracking filter
can be converted to a tractable filter that operates with unknown probability
of detection (Section 17.1.1).
• PHD and CPHD filters with unknown probability of detection can be implemented in exact closed form using beta-Gaussian mixture (BGM) techniques,
in which the statistical behavior of a is modeled by a beta distribution
βu,v (a) (Sections 17.4 and 17.4):
νk|k
˚k|k (a, x) =
D
∑
i=1
k|k
wi
k|k
· βuk|k ,vk|k (a) · NP k|k (x − xi ).
i
i
i
(17.19)
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523
• Garbage in, garbage out: For good performance to be possible in an RFS
pD -agnostic multitarget detection and tracking filter, the detection profile
must be slowly-varying in comparison to the measurement-update rate.
17.1.4
Organization of the Chapter
The chapter is organized as follows:
1. Section 17.2: The pD -CPHD filter—a version of the CPHD filter that does
not require a priori knowledge of the detection profile.
2. Section 17.3: The beta-Gaussian mixture (BGM) approximation of a PHD
Dk|k (a, x).
3. Section 17.4: BGM implementation of the pD -PHD filter.
4. Section 17.5: BGM implementation of the pD -CPHD filter.
5. Section 17.6: The pD -CBMeMBer filter—a version of the CBMeMBer filter
that does not require a priori knowledge of the detection profile.
6. Section 17.7: Implementations of pD -agnostic RFS filters.
17.2
THE PD -CPHD FILTER
The pD -CPHD filter is derived from the usual CPHD filter equations using the
procedure outlined in Section 17.1.1. The section is organized as follows:
1. Section 17.2.1: Modeling assumptions for the pD -CPHD filter.
2. Section 17.2.2: Time update equations for the pD -CPHD filter.
3. Section 17.2.3: Measurement update equations for the pD -CPHD filter.
4. Section 17.2.4: Multitarget state estimation for the pD -CPHD filter.
17.2.1
pD -CPHD Filter Models
The following models are used in the pD -CPHD filter:
• Probability of target survival: pS (x′ ).
• Target Markov density: fk+1|k (x|x′ ).
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• Markov density for probability of detection: fk+1|k (a|a′ )—as defined in
(17.12) through (17.16).
∫
def.
B
• PHD of the target appearance RFS: bk+1|k (x) with Nk+1|k
= bk+1|k (x)dx.
• Cardinality distribution and p.g.f. of the target appearance RFS: pB
k+1|k (n),
∑
∑
B
B
B
B
with Nk+1|k = n≥0 n · pk+1|k (n); and Gk+1|k (x) = n≥0 pk+1|k (n) ·
xn .
• Probability of detection at a track-state x: an unknown state variable
0 ≤ a ≤ 1, the explicit estimation of which is optional (as desired or not).
abbr,
• Sensor likelihood function: Lz (x) = fk+1 (z|x).
• Clutter cardinality distribution: pκk+1 (m).
• Clutter spatial distribution: ck+1 (z).
17.2.2
pD -CPHD Filter Time Update
The time-update equations for the pD -CPHD filter are:
• Predicted spatial distribution:

s̊k+1|k (a, x)
=
ψk
=

∫ ∫1
bk+1|k (x) + Nk|k
p (x′ )
0 S


·fk+1|k (a|a′ )
′
′
′
′
′
·fk+1|k (x|x ) · s̊k|k (a , x )da dx
(17.20)
B
Nk+1|k
+ Nk|k · ψk
∫ ∫ 1
s̊k|k [p̊S ] =
pS (x) · s̊k|k (a, x)dadx (17.21)
0
∫
B
where Nk+1|k
= bk+1|k (x)dx. Expressed in terms of PHDs rather than
spatial distributions, this becomes:
˚k+1|k (a, x)
D
=
bk+1|k (x)
∫ ∫ 1
+
pS (x′ ) · fk+1|k (a|a′ )
(17.22)
0
ψk
=
˚k|k (a′ , x′ )da′ dx′
·fk+1|k (x|x′ ) · D
∫ ∫ 1
1
˚k|k (a, x)dadx. (17.23)
pS (x) · D
Nk|k
0
RFS Filters for Unknown pD
525
• Predicted cardinality distribution and p.g.f.:
Gk+1|k (x)
=
pk+1|k (n)
=
pk+1|k (n|n′ )
GB
k+1 (x) · Gk|k (1 − ψk + ψk · x)
∑
pk+1|k (n|n′ ) · pk|k (n′ )
(17.24)
(17.25)
n′ ≥0
n
∑
i
n′ −i
=
pB
. (17.26)
k+1|k (n − i) · Cn′ ,i · ψk (1 − ψk )
i=0
• Predicted expected number of targets:
B
Nk+1|k = Nk+1|k
+ Nk|k · ψk .
17.2.3
(17.27)
pD -CPHD Filter Measurement Update
Let a new measurement set Zk+1 with |Zk+1 | = m be collected. Then the
measurement-update equations for the pD -CPHD filter are:
• Measurement updated cardinality distribution and p.g.f.:
pk+1|k+1 (n)
=
Gk+1|k+1 (x)
=
ℓZ (n) · pk+1|k (n)
∑ k+1
(17.28)
l≥0 ℓZk+1 (l) · pk+1|k (l)
( ∑m j
)
κ
j=0 x · (m − j)! · pk+1 (m − j)
·G(j) (x · ϕk ) · σj (Zk+1 )
( ∑m
) (17.29)
κ
i=0 (m − i)! · pk+1 (m − i)
·G(i) (ϕk ) · σi (Zk+1 )
where
( ∑
ℓZk+1 (n) =
min{m,n}
(m − j)! · pκk+1 (m − j)
j=0
·j! · Cn,j · ϕn−j
· σj (Zk+1 )
k
( ∑m
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
)
)
.
(17.30)
• Measurement updated spatial distribution or PHD:
ˆ Z (a, x) · s̊k+1|k (a, x)
s̊k+1|k+1 (a, x) = L
k+1
(17.31)
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or
˚k+1|k+1 (a, x) = ˚
˚k+1|k (a, x)
D
LZk+1 (a, x) · D
(17.32)
where
ˆ Z (a, x)
L
k+1

1
=
˚
LZk+1 (a, x)
=

ND
(1 − a) · L Zk+1
(17.33)


∑m a·Lz (x) D
Nk+1|k+1
+ j=1 ck+1j(zj ) · LZk+1 (zj )


ND
− a) · L Zk+1
1
 ∑ (1 a·L
 (17.34)
D
zj (x)
m
Nk+1|k
+
· LZk+1 (zj )
j=1 ck+1 (zj )
( ∑m
)
( ∑m
)
κ
j=0 (m − j)! · pk+1 (m − j)
(j+1)
·σj (Zk+1 ) · Gk+1|k (ϕk )
ND
L Zk+1
=
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
(17.35)
D
( ∑m−1
(17.36)
)
∑
(17.37)
LZk+1 (zj )
(m − i − 1)! · pκk+1 (m − i − 1)
(i+1)
·σi (Zk+1 − {zj }) · Gk+1|k (ϕk )
( ∑m
)
κ
l=0 (m − l)! · pk+1 (m − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
i=0
=
(l)
Gk+1|k (ϕk )
=
pk+1|k (n) · l! · Cn,l · ϕn−l
k
n≥l
(j+1)
(17.38)
Gk+1|k (ϕk )
=
∑
n≥j+1
pk+1|k (n) · (j + 1)! · Cn,j+1 · ϕn−j−1
k
RFS Filters for Unknown pD
527
where
ϕk
∫ ∫ 1
(1 − a) · s̊k+1|k (a, x)dadx (17.39)
(
)
τ̂k+1 (z1 )
τ̂k+1 (zm )
σm,i
, ...,
(17.40)
ck+1 (z1 )
ck+1 (zm )


τ̂?
τ̂k+1 (z1 )
k+1 (zj )
, ..., ck+1
(zj )  (17.41)
σm−1,i  ck+1 (z1 ) τ̂ (z
m)
k+1
, ..., ck+1
(zm )
=
0
σi (Zk+1 )
=
σi (Zk+1 − {zj })
=
τ̂k+1 (z)
=
s̊k+1|k [p̊D ˚
Lz ]
∫
a · Lz (x) · s̊k+1|k (a, x)dadx.
=
(17.42)
(17.43)
• Measurement updated expected number of targets:
ND
Nk+1|k+1 = ϕk · L Zk+1 +
m
∑
τ̂k+1 (zi )
ck+1 (zi )
D
· LZk+1 (z).
(17.44)
i=1
17.2.4
pD -CPHD Filter Multitarget State Estimation
At time tk+1 we are given a measurement-updated cardinality distribution
pk+1|k+1 (n) and a measurement-updated PHD Dk+1|k+1 (a, x) or spatial distribution sk+1|k+1 (a, x). We are to estimate the number and states of the targets.
Two approaches are considered here: target-only estimation and joint target-pD
estimation.
17.2.4.1
Method 1: Target-Only Estimation
Assume that the target states are to be estimated, but not their probabilities of
detection. We can then integrate a out as a nuisance variable:
Dk+1|k+1 (x) =
∫ 1
˚k+1|k+1 (a, x)da.
D
(17.45)
0
Then we can apply the usual state-estimation procedure for the CPHD filter as in
Section 8.5.5. That is, determine the MAP estimate of target number:
n̂ = arg sup pk+1|k+1 (n)
n≥0
(17.46)
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and determine the n̂ largest suprema of Dk+1|k+1 (x). The state estimates are the
states x̂1 , ..., x̂n̂ that correspond to those suprema.
The same approach applies to the pD -PHD filter of (17.4) and (17.11), except
that instead of n̂ we substitute the nearest integer to the measurement-updated
expected number of targets, Nk+1|k+1 .
17.2.4.2
Method 2: Joint Target-pD Estimation
Assume that we want to know not only the states of the targets, but also the
probabilities of detection at those states. In this case, determine the MAP estimate
of target number:
n̂ = arg sup pk+1|k+1 (n).
(17.47)
n≥0
˚k+1|k+1 (a, x) and let (â1 , x̂1 ), ...,
Then determine the n̂ largest suprema of D
(ân̂ , x̂n̂ ) be the augmented states that correspond to them. The estimated target
states are x̂1 , ..., x̂n̂ , and their respective probabilities of detection are â1 , ..., ân̂ .
The same approach applies to the pD -PHD filter of (17.4) and (17.11), except
that the nearest integer to Nk+1|k+1 is used in place of n̂.
17.3
BETA-GAUSSIAN MIXTURE (BGM) APPROXIMATION
Gaussian mixture (GM) implementation of the pD -PHD filter and pD -CPHD filter
˚k|k (a, x)
is not possible because the PHDs or spatial distributions have the form D
or s̊k|k (a, x), rather than the usual Dk|k (x) or sk|k (x) where x is a Euclidean
column vector. Consequently, some generalization must be devised. The purpose
of this section is to describe such a generalization: beta-Gaussian mixture (BGM)
approximation. The section is organized as follows:
1. Section 17.3.1: Overview of the approach.
2. Section 17.3.2: Beta-Gaussian mixtures (BGMs).
3. Section 17.3.3: Pruning BGMs.
4. Section 17.3.4: Merging BGMs.
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17.3.1
529
Overview of the BGM Approach
The two following items pose a challenge to the problem of generalizing the concept
of a Gaussian mixture to pD -agnostic CPHD filter:
• The factors a and 1 − a that occur in (17.4) and (17.34).
For the sake of conceptual simplicity, suppose for the moment that targets
are static (that is, their states do not change with time). Then time-updates are
unnecessary and the pD -PHD and pD -CPHD filters reduce to repeated application
of the measurement-update equations. As time progresses and because of (17.4)
and (17.34), the formulas for the PHD or the spatial distribution will contain factors
of the form
ai (1 − a)j
where i, j are arbitrarily large integers.
A simple way to address this challenge was proposed by Mahler, Vo, and Vo
in [194]. They noted that beta distributions have a similar form:
βu,v (a) =
where
β(u, v) =
au−1 (1 − a)v−1
β(u, v)
∫ 1
au−1 (1 − a)v−1 da
(17.48)
(17.49)
0
is the beta function. (See Appendix E for a discussion of beta distributions and their
properties.) The Gaussian mixture approximation is made possible by the fact that
Gaussian distributions are algebraically closed under multiplication:
NP1 (x − x1 ) · NP2 (x − x2 )
E −1
E −1 e
=
=
=
NP1 +P2 (x2 − x1 ) · NE (x − e) (17.50)
P1−1 + P2−1 ,
(17.51)
P1−1 x1 + P2−1 x2 .
(17.52)
Beta distributions are also algebraically closed under multiplication:
βu1 ,v1 (a) · βu2 ,v2 (a)
=
β(u1 + u2 − 1, v1 + v2 − 1)
β(u1 , v1 ) · β(u2 , v2 )
·βu1 +u2 −1,v1 +v2 −1 (a).
(17.53)
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In particular, by (E.6) and (E.7) in Appendix E, we have the following two special
cases:
a · βu,v (a)
=
=
β(u + 1, v)
· βu+1,v (a)
β(u, v)
u
· βu+1,v (a)
u+v
(17.54)
(17.55)
and
(1 − a) · βu,v (a)
=
=
17.3.2
β(u, v + 1)
· βu,v+1 (a)
β(u, v)
v
· βu,v+1 (a).
u+v
(17.56)
(17.57)
Beta-Gaussian Mixtures (BGMs)
˚ x) be a PHD on the space
These considerations suggest the following. Let D(a,
N
˚ x) is
[0, 1] × R . Then a beta-Gaussian mixture (BGM) approximation of D(a,
an approximation of the form
˚ x) ∼
D(a,
=
ν
∑
wi · βui ,vi (a) · NPi (x − xi )
(17.58)
i=1
where wi ≥ 0 for all i = 1, ..., ν.
Remark 70 (Implicit assumptions) Implicit in this approximation is the assumption that pD (xi ) is statistically independent of xi , which is not true in general.
However, the pD -PHD and pD -CPHD filters themselves depend on the implicit
assumption that the data rate is rapid enough that pD (x) changes relatively slowly
in both time and space.
It follows from (17.50) that the product of two BGMs is also a BGM. From
(17.55) and (17.57) it follows that the integrals
∫ 1
0
˚ x)da,
a · D(a,
∫ 1
˚ x)da
(1 − a) · D(a,
0
˚ x) is a BGM.
can be evaluated in exact closed form if D(a,
It follows that if the PHD is approximated as a BGM, then:
(17.59)
RFS Filters for Unknown pD
531
• The time-update and measurement-update equations for the pD -PHD and
pD -CPHD filters can be evaluated in exact closed form, provided that we
impose a few relatively minor restrictions (see Sections 17.4.1 and 17.5.1).
Thus we write
νk|k
˚k|k (a, x)
D
=
∑
k|k
k|k
· βuk|k vk|k (a) · NP k|k (x − xi )
wi
i
i
(17.60)
i
i=1
νk+1|k
˚k+1|k (a, x)
D
=
∑
k+1|k
· βuk+1|k vk+1|k (a)
wi
i
(17.61)
i
i=1
k+1|k
·NP k+1|k (x − xi
)
i
for all k ≥ 0. It follows that the Time propagation of the PHDs is equivalent to the
Time propagation of families of the form
k|k
k|k
k|k
k|k
k|k
(ℓi , wi , ui , vi , Pi
k|k ν
k|k
, xi )i=1
k|k
where, in addition to the other items in the family, ℓi is the track label of the ith
BGM component.
This is the basis for the BGM implementations of the pD -PHD filter and
pD -CPHD filter equations to be described shortly. Briefly stated, the measurementupdate equations for these filters are constructed by making the substitutions
k|k
pD,k (xi )
ui
?→
k|k
ui
(17.62)
k|k
+ vi
k|k
1 − pD,k (xi )
vi
?→
k|k
ui
17.3.3
.
(17.63)
k|k
+ vi
Pruning BGM Components
As time progresses, the number of components in a BGM approximation of a PHD
tends to increase without bound. As with Gaussian mixtures, various techniques
must be employed to merge similar components and prune insignificant components. For computational reasons, it is better to prune before merging. Doing so
avoids the computational cost associated with merging BGM components that will
end up being pruned anyway.
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Pruning BGM components is similar to pruning GM components. Suppose
that we are to prune components from the measurement-updated BGM system
k+1|k+1
(wi
k+1|k+1
, ui
k+1|k+1
, vi
k+1|k+1
, Pi
k+1|k+1 νk+1|k+1
)i=1
, xi
∑νk+1|k+1 k+1|k+1
with Nk+1|k+1 = i=1
wi
. Set a pruning threshold τprune , identify
those components for which
k+1|k+1
wi
(17.64)
< τprune ,
and then eliminate them. This results in a pruned system
k+1|k+1
(w̌i
k+1|k+1
, ǔi
k+1|k+1
, v̌i
k+1|k+1
, Pˇi
k+1|k+1 ν̌k+1|k+1
)i=1
, x̌i
with ν̌k+1|k+1 components. Let
ν̌k+1|k+1
w̌ k+1|k+1 =
∑
k+1|k+1
w̌i
(17.65)
i=1
be the combined weight of all components that remain. Define
k+1|k+1
k+1|k+1
= Nk+1|k+1 ·
ŵi
w̌i
w̌ k+1|k+1
(17.66)
for all i = 1, ..., ν̌k+1|k+1 . Then
k+1|k+1
(ŵi
k+1|k+1
, ǔi
k+1|k+1
, v̌i
k+1|k+1
, Pˇi
k+1|k+1 ν̌k+1|k+1
)i=1
, x̌i
is the final pruned BGM system.
17.3.4
Merging BGM Components
To merge BGM components, we must first specify a merging criterion. Suppose
that
f˚1 (a, x)
f˚2 (a, x)
=
w1 · βu1 ,v1 (a) · NP1 (x − x1 )
(17.67)
=
w2 · βu2 ,v2 (a) · NP2 (x − x2 )
(17.68)
RFS Filters for Unknown pD
533
are two components. From a purely mathematical point of view, it might seem
desirable to define a metric that measures the distance between the density functions
f˚1 (a, x) and f˚2 (a, x). However, the goal in tracking applications is to detect and
localize targets—not to determine their respective probabilities of detection. So,
instead, one should determine the distance between the marginal densities
f1 (x) = w1 · NP1 (x − x1 ),
f2 (x) = w2 · NP1 (x − x1 ).
(17.69)
Consequently, the merging criterion for BGMs is the same as for Gaussian mixtures
(as described in Section 9.5.3).
Now suppose that we are given n BGM components
f˚(a, x) = w1 · βu1 ,v1 (a) · NP1 (x − x1 ) + ... + wn · βun ,vn (a) · NPn (x − xn )
that are to be merged. In Section K.22 it is shown that the merged component that
has the same mean and covariance as f˚(a, x) is
w0 · βu0 ,v0 (a) · NP0 (x − x0 )
where:
w0
=
n
∑
wi
(17.70)
i=1
ŵi
=
x0
=
wi
w0
n
∑
(17.71)
ŵi · xi
(17.72)
(
)
ŵi · Pi + xi xTi
(17.73)
∑
(17.74)
i=1
and where
P0
=
−x0 xT0 +
n
∑
i=1
=
n
∑
ŵi · Pi +
i=1
u0
v0
=
=
θ0 µ0
θ0 (1 − µ0 )
ŵi · ŵj · (xi − xj )(xi − xj )T
1≤i<j≤n
(17.75)
(17.76)
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Advances in Statistical Multisource-Multitarget Information Fusion
and where
θ0
=
µ0
=
σ02
=
µ0 (1 − µ0 )
−1
σ02
n
1 ∑
wi · µi
w0 i=1
n
(
)
1 ∑
−µ +
wi · σi2 + µ2i
w0 i=1
2
(17.77)
(17.78)
(17.79)
and where, for i = 1, ..., n,
µi
=
σi2
=
ui
ui + vi
µi (1 − µi )
.
ui + vi + 1
(17.80)
(17.81)
(17.74) follows from (9.56).
17.4
BGM IMPLEMENTATION OF THE PD -PHD FILTER
The purpose of this section is to describe the BGM implementation of the pD -PHD
filter. It is organized as follows:
1. Section 17.4.1: Modeling assumptions for the BGM-pD -PHD filter.
2. Section 17.4.2: Time update equations for the BGM-pD -PHD filter.
3. Section 17.4.3: Measurement update equations for the BGM-pD -PHD filter.
4. Section 17.4.4: Multitarget state estimation for the BGM-pD -PHD filter.
17.4.1
BGM pD -PHD Filter Modeling Assumptions
The BGM implementation of the pD -PHD filter requires the following:
• The probability of target survival pS (x) = pS is constant.
RFS Filters for Unknown pD
535
• The target Markov density fk+1|k (x|x′ ) is linear-Gaussian:1
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(17.82)
• The Markov density fk+1|k (a|a′ ) for the detection profile is defined as in
(17.12) through (17.16).
• The birth-target PHD bk+1|k (x) is a Gaussian mixture:
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
k+1|k
· NB k+1|k (x − bi
bi
)
(17.83)
i
i=1
and so the expected number of birth targets is
B
νk+1|k
B
Nk+1|k
=
∑
k+1|k
bi
(17.84)
.
i=1
• The spawned-target PHD bk+1|k (x|x′ ) is a Gaussian mixture:
S
νk+1|k
bk+1|k (x) =
∑
k+1|k
ej
k+1|k+1 ′
· NGk+1|k (x − Ej
x)
(17.85)
j
j=1
and so the expected number of targets spawned by a target with state x′ is
S
νk+1|k
S
Nk+1|k
=
∑
k+1|k
ej
.
(17.86)
j=1
• The probability of detection is an unknown a, the explicit estimation of which
is optional.
• The sensor likelihood function is linear-Gaussian:2
Lz (x) = fk+1 (z|x) = NRk+1 (z − Hk+1 x).
1
2
(17.87)
This assumption can be relaxed to allow fk+1|k (x|x′ ) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
This assumption can be relaxed to allow fk+1 (z|x) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
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17.4.2
BGM pD -PHD Filter Time Update
We are given:
k|k
k|k
k|k
k|k
k|k
(ℓi , wi , ui , vi , Pi
k|k ν
k|k
, xi )i=1
with
νk|k
Nk|k =
∑
k|k
(17.88)
wi
i=1
and we are to determine formulas for
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, ui
k+1|k
, vi
k+1|k
, Pi
k+1|k νk+1|k
)i=1 .
, xi
These are:
• Time updated number of BGM components:
B
S
νk+1|k = νk|k + νk+1|k
+ νk|k · vk+1|k
.
(17.89)
B
There are νk|k components corresponding to persisting targets, νk+1|k
S
components corresponding to newly appearing targets, and νk|k · vk+1|k
components corresponding to spawned targets. The time-update components
are indexed as follows:
=
1, ..., νk|k
(persisting)
(17.90)
i
=
B
νk|k + 1, ..., νk|k + νk+1|k
(appearing)
(17.91)
i
=
S
1, ..., νk|k ; j = 1, ..., vk+1|k
(spawned).
(17.92)
i
Persisting-target BGM components, for i = 1, ..., νk|k :
k+1|k
ℓi
k+1|k
wi
k+1|k
xi
k+1|k
Pi
k+1|k
ui
k+1|k
vi
k|k
=
=
=
=
=
=
ℓi
(17.93)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk
k|k
k|k
ui · θi
k|k
k|k
vi · θi
(17.94)
(17.95)
(17.96)
(17.97)
(17.98)
RFS Filters for Unknown pD
537
where
1
k|k
θi =
k|k
ui
·
k|k
(
+ vi
k|k k|k
u i vi
1
·
−1
k|k
k|k 2 σ 2
(u + v )
i
i
)
(17.99)
i
and where, as in Section 17.1.2, the predicted variance σi2 is chosen subject
to
(
)
k|k k|k
1
ui vi
k+1|k
σi2 =
+
ε
·
(17.100)
i
k|k
k|k
k|k
k|k
k|k
ui + vi
(ui + v k|k )(ui + vi + 1)
for some choice of
k+1|k
0 ≤ εi
≤ 1.
• Appearing-target BGM components, for
k+1|k
(17.101)
B
i = νk|k + 1, ..., νk|k + νk+1|k
:
=
new label
(17.102)
=
k+1|k
bi−νk|k
(17.103)
k+1|k
=
bi−νk|k
(17.104)
k+1|k
=
Bi−νk|k
(17.105)
ui
=
1
(17.106)
k+1|k
vi
=
1.
(17.107)
ℓi
k+1|k
wi
k+1|k
xi
k+1|k
Pi
k+1|k
B
• Spawned-target GM components, for i = 1, ..., νk|k and j = 1, ..., vk+1|k
:
k+1|k
ℓi,j
k+1|k
wi,j
=
new label
(17.108)
=
k+1|k
k|k
ej
· wi
(17.109)
=
Ej
k+1|k k|k
xi
(17.110)
=
k+1|k k|k
k+1|k T
k+1|k
Ej
Pi (Ej
) + Gj
(17.111)
=
1
(17.112)
=
1.
(17.113)
k+1|k
xi
k+1|k
Pi,j
k+1|k
ui,J
k+1|k
vi,J
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Advances in Statistical Multisource-Multitarget Information Fusion
17.4.3
BGM pD -PHD Filter Measurement Update
We are given:
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, ui
k+1|k
, vi
k+1|k
, Pi
k+1|k νk+1|k
)i=1
, xi
with
νk+1|k
Nk+1|k =
∑
k+1|k
wi
(17.114)
.
i=1
We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | =
mk+1 . We are to determine formulas for
k+1|k=1
(ℓi
k+1|k+1
, wi
k+1|k+1
, ui
k+1|k+1
, vi
k+1|k+1 νk+1|k+1
)i=1
.
k+1|k+1
, Pi
, xi
These are:
• Measurement updated number of BGM components for the PHD:
νk+1|k+1 = νk+1|k + mk+1 · νk+1|k
(17.115)
where there are νk+1|k components for undetected tracks and mk+1 · νk+1|k
components for detected tracks. The measurement-update components are
indexed as follows:
i
i
=
=
1, ..., νk+1|k
1, ..., νk+1|k ; j = 1, ..., mk+1
(undetected) (17.116)
(detected).
(17.117)
• Measurement updated nondetection components: for i = 1, ..., νk+1|k ,
k+1|k=1
ℓi
k+1|k
=
k+1|k+1
wi
=
k+1|k+1
xi
k+1|k+1
Pi
k+1|k+1
ui
k+1|k+1
vi
=
=
=
=
ℓi
k+1|k
k+1|k
wi
· vi
k+1|k
k+1|k
ui
+ vi
k+1|k
xi
k+1|k
Pi
k+1|k
ui
k+1|k
vi
+ 1.
(17.118)
(17.119)
(17.120)
(17.121)
(17.122)
(17.123)
RFS Filters for Unknown pD
539
• Measurement updated nondetection components: for i = 1, ..., νk+1|k and
j = 1, ..., mk+1 ,
k+1|k=1
ℓi,j
k+1|k
=
(17.124)
ℓi
νk+1|k
τk+1 (zj )
∑ w k+1|k · uk+1|k
i
i
=
k+1|k
ui
i=1
(17.125)
k+1|k
+ vi
k+1|k
·NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
k+1|k
=
)
k+1|k
· ui
wi
k+1|k+1
wi,j
(zj − Hk+1 xi
k+1|k
(17.126)
k+1|k
ui
+ vi
k+1|k
NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
·
k+1|k+1
xi,j
k+1|k+1
Pi,j
Kik+1
k+1|k+1
vi,j
17.4.4
)
κk+1 (zj ) + τk+1 (zj )
=
k+1|k
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
)
(17.127)
=
(
(17.128)
=
k+1|k T
Pi
Hk+1
·
(
=
ui
=
k+1|k
vi
.
k+1|k+1
ui,j
(zj − Hk+1 xi
)
k+1|k
I − Kik+1 Hk+1 Pi
k+1|k
Hk+1 Pi
(17.129)
T
Hk+1
+ Rk+1
)−1
k+1|k
(17.130)
+1
(17.131)
BGM pD -PHD Filter Multitarget State Estimation
The target-only estimation or joint target-clutter estimation approaches in Section
17.2.4 can both be used.
For the target-only approach, note that the mixture distribution of (17.45) is a
Gaussian mixture with components
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1 νk+1|k+1
)i=1
.
k+1|k+1
, Pi
, xi
So round off
νk+1|k+1
Nk+1|k+1 =
∑
k+1|k+1
wi
(17.132)
i=1
to the nearest integer n and determine those Gaussian components that have the
k+1|k+1
n largest values of wi
.
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For the joint target-clutter approach, one instead determines those betak+1|k+1
Gaussian components that have the n largest values of wi
. Let
(u1 , v1 , x1 ), ..., (un , vn , xn ) be the parameters for those components. Then
x1 , ..., xn is the multitarget state estimate. From (E.10) in Appendix E, the modes
of the corresponding beta distributions are the respective estimates of the probabilities of detection:
a1 =
17.5
u1 − 1
un − 1
,..., an =
.
u1 + v1 − 2
un + vn − 2
(17.133)
BGM IMPLEMENTATION OF THE PD -CPHD FILTER
The purpose of this section is to describe the BGM implementation of the pD CPHD filter. It is organized as follows:
1. Section 17.5.1: Modeling assumptions for the BGM-pD -CPHD filter.
2. Section 17.5.2: Time update equations for the BGM-pD -CPHD filter.
3. Section 17.5.3: Measurement update equations for the BGM-pD -CPHD
filter.
4. Section 17.5.4: Multitarget state estimation for the BGM-pD -CPHD filter.
17.5.1
BGM pD -CPHD Filter Modeling Assumptions
The BGM implementation of the pD -CPHD filter requires the following:
• pk|k (n) = 0 for sufficiently large n.
• The probability of target survival pS (x) = pS is constant.
• The target Markov density fk+1|k (x|x′ ) is linear-Gaussian:3
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(17.134)
• The Markov density fk+1|k (a|a′ ) of the detection profile is defined as in
Eqs. (17.12) through (17.16).
3
This assumption can be relaxed to allow fk+1|k (x|x′ ) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
RFS Filters for Unknown pD
541
• The birth-target PHD bk+1|k (x) is a Gaussian mixture:
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
k+1|k
· NB k+1|k (x − bi
bi
)
(17.135)
i
i=1
and so the expected number of birth targets is
B
νk+1|k
B
Nk+1|k
=
∑
k+1|k
bi
(17.136)
.
i=1
• The cardinality distribution pB
k+1|k (n) of the birth-target RFS vanishes for
sufficiently large n.
• The probability of detection is an unknown a, the explicit estimation of which
is optional.
• The sensor likelihood function is linear-Gaussian:4
(17.137)
Lz (x) = fk+1 (z|x) = NRk+1 (z − Hk+1 x).
• The clutter cardinality distribution pκk+1 (m) is arbitrary.
• The clutter spatial distribution ck+1 (z) is arbitrary.
17.5.2
BGM pD -CPHD Filter Time Update
We are given:
k|k
pk|k (n),
k|k
k|k
k|k
k|k
(ℓi , wi , ui , vi , Pi
k|k ν
k|k
, xi )i=1
with
νk|k
Nk|k =
∑
k|k
wi
=
n∑
max
(17.138)
n · pk|k (n).
n=0
i=1
and we are to determine formulas for
k+1|k
pk+1|k (n),
(ℓi
k+1|k
, wi
k+1|k
, ui
k+1|k
, vi
k+1|k
, Pi
k+1|k νk+1|k
)i=1 .
, xi
These are:
4
This assumption can be relaxed to allow fk+1 (z|x) to be a Gaussian mixture, but at the expense
of increasing the computational burden.
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Advances in Statistical Multisource-Multitarget Information Fusion
• Time updated cardinality distribution:
pk+1|k (n)
=
n∑
max
pk+1|k (n|n′ ) · pk|k (n′ )
(17.139)
n′ =0
min{n,n′ }
′
pk+1|k (n|n )
∑
=
pB
k+1|k (n − i)
(17.140)
i=0
′
·Cn′ ,i · piS,k (1 − pS,k )n −i .
• Time updated number of BGM components for the PHD:
B
νk+1|k = νk|k + νk+1|k
.
(17.141)
Here, there are νk|k components corresponding to persisting targets and
B
νk+1|k
components corresponding to newly appearing targets. The timeupdate components are indexed as follows:
i
i
=
1, ..., νk|k
(persisting)
(17.142)
=
B
νk+1 + 1, ..., νk+1 + νk+1|k
(appearing).
(17.143)
• Persisting-target BGM components, for i = 1, ..., νk|k :
k+1|k
ℓi
k|k
=
k+1|k
wi
k+1|k
xi
k+1|k
Pi
k+1|k
ui
k+1|k
vi
=
=
=
=
=
(17.144)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk
k|k
k|k
ui · θi
k|k
k|k
vi · θi
(17.145)
1
k|k
where θi
ℓi
=
k|k
k|k
ui + vi
·
(
k|k k|k
ui vi
1
·
−1
k|k
k|k 2 σ 2
(ui + vi )
i
(17.146)
(17.147)
(17.148)
(17.149)
)
(17.150)
and where, as in Section 17.1.2, the predicted variance σi2 is chosen subject
to
)
(
k|k k|k
1
ui vi
k+1|k
2
+
ε
·
(17.151)
σi =
i
k|k
k|k
k|k
k|k
k|k
ui + vi
(ui + v k|k )(ui + vi + 1)
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543
for some choice of
k+1|k
0 ≤ εi
(17.152)
≤ 1.
• Appearing-target BGM components, for
k+1|k
B
i = νk|k + 1, ..., νk|k + νk+1|k
:
=
new label
(17.153)
=
k+1|k
bi−νk|k
(17.154)
k+1|k
=
bi−νk|k
(17.155)
k+1|k
=
Bi−νk|k
(17.156)
ui
=
1
(17.157)
k+1|k
vi
=
1.
(17.158)
ℓi
k+1|k
wi
k+1|k
xi
k+1|k
Pi
k+1|k
17.5.3
BGM pD -CPHD Filter Measurement Update
We are given:
k+1|k
pk+1|k (n),
(wi
k+1|k
, ui
k+1|k
, vi
k+1|k
, Pi
k+1|k νk+1|k
)i=1
, xi
with
νk+1|k
Nk+1|k =
∑
k+1|k
wi
=
n∑
max
(17.159)
n · pk+1|k (n).
n=0
i=1
A new measurement set Zk+1 with |Zk+1 | = mk+1 is collected, and we are to
determine formulas for
k+1|k+1
pk+1|k+1 (n),
(wi
k+1|k+1
, ui
k+1|k+1
, vi
k+1|k+1
, Pi
k+1|k+1 νk+1|k+1
)i=1
.
, xi
These are:
• Measurement updated number of BGM components for the PHD:
νk+1|k+1 = νk+1|k + mk+1 · νk+1|k
(17.160)
where, as with the pD -PHD filter, there are νk+1|k components for undetected tracks and mk+1 · νk+1|k components for detected tracks. The
measurement-update components are indexed as follows:
(undetected)
i
=
1, ..., νk+1|k
i
=
1, ..., νk+1|k ; j = 1, ..., mk+1 (detected).
(17.161)
(17.162)
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• Measurement updated cardinality distribution :
ℓZ (n) · pk+1|k (n)
pk+1|k+1 (n) = ∑ k+1
l≥0 ℓZk+1 (l) · pk+1|k (l)
(17.163)
where
( ∑
min{mk+1 ,n}
(mk+1 − j)! · pκk+1 (mk+1 − j)
j=0
·j! · Cn,j · ϕn−j
· σj (Zk+1 )
k
( ∑mk+1
)
κ
l=0 (mk+1 − l)! · pk+1 (mk+1 − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
ℓZk+1 (n) =
)
(17.164)
and where
νk+1|k
1
ϕk
=
Nk+1|k
(l)
Gk+1|k (ϕk )
=
nmax
∑
∑ w k+1|k · v k+1|k
i
i
k+1|k
i=1
ui
(17.165)
k+1|k
+ vi
pk+1|k (n) · l!Cn,l · ϕn−l
k
(17.166)
n=l
σi (Zk+1 )
=
σmk+1 ,i
(
τ̂k+1 (zmk+1 )
τ̂k+1 (z1 )
, ...,
ck+1 (z1 )
ck+1 (zmk+1 )
)
(17.167)
νk+1|k
1
τ̂k+1 (zj )
=
Nk+1|k
·NR
∑ w k+1|k · uk+1|k
i
l
k+1|k
l=1
ui
(17.168)
k+1|k
+ vl
k+1|k
T
Hk+1
k+1 +Hk+1 Pl
(zj − Hk+1 xk+1
).
l
• Measurement updated undetected-target components for the PHD: for i =
1, ..., vk+1|k ,
k+1|k+1
ℓi
k+1|k
=
k+1|k+1
wi
=
k+1|k+1
xi
k+1|k+1
Pi
k+1|k+1
ui
k+1|k+1
vi
=
=
=
=
ℓi
k+1|k
k+1|k
ND
wi
· vi
1
· k+1|k
· L Zk+1
k+1|k
Nk+1|k u
+ vi
i
k+1|k
xi
k+1|k
Pi
k+1|k
ui
k+1|k
vi
+1
(17.169)
(17.170)
(17.171)
(17.172)
(17.173)
(17.174)
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545
where
ND
L Zk+1
( ∑mk+1
j=0
=
(17.175)
(mk+1 − j)! · pκk+1 (mk+1 − j)
(j+1)
·σj (Zk+1 ) · Gk+1|k (ϕk )
( ∑mk+1
l=0
)
(mk+1 − l)! · pκk+1 (mk+1 − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
)
and
(j+1)
Gk+1|k (ϕk ) =
n∑
max
pk+1|k (n) · (j + 1)! · Cn,j+1 · ϕn−j−1
.
k
(17.176)
n=j+1
• Measurement updated detected-target components for the PHD: for i =
1, ..., vk+1|k and j = 1, ..., mk+1 ,
k+1|k+1
ℓi,j
k+1|k
=
(17.177)
ℓi
D
k+1|k
1
k+1|k+1
wi,j
=
Nk+1|k
·
k+1|k
· ui
wi
k+1|k
k+1|k
ui
+ vi
·
LZk+1 (zj )
ck+1 (zj )
(17.178)
k+1|k
·NR
k+1|k
xi,j
k+1|k+1
ui,j
(zj − Hk+1 xi
k+1|k
=
k+1|k
Pi,j
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
=
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
(
) k+1|k
I − Kik+1 Hk+1 Pi
=
k+1|k
ui
+1
vi,j
=
vi
Kik+1
=
k+1|k
)
k+1|k
(17.179)
(17.180)
(17.181)
k+1|k
Pi
)
(17.182)
(
k+1|k
T
Hk+1
Hk+1 Pi
T
Hk+1
+ Rk+1
(17.183)
)−1
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Advances in Statistical Multisource-Multitarget Information Fusion
where
D
LZk+1 (zj )
( ∑m −1
(mk+1 − i − 1)! · pκk+1 (mk+1 − i − 1)
(i+1)
·σi (Zk+1 − {zj }) · Gk+1|k (ϕk )
( ∑mk+1
)
κ
l=0 (mk+1 − l)! · pk+1 (mk+1 − l)
(l)
·σl (Zk+1 ) · Gk+1|k (ϕk )
k+1
i=0
=
(17.184)
)
and where
σi (Zk+1 − {zj })

=
(17.185)
τ̂?
k+1 (zj )

τ̂k+1 (z1 )
, ..., ck+1 (zj )

σmk+1 −1,i  ck+1 (z1τ̂)k+1 (zm
)
, ..., ck+1 (zmk+1 )
k+1
and where as usual x1 , ..., x?j , ..., xm indicates that the jth item xj is to
be removed from the list x1 , ..., xm .
17.5.4
BGM pD -CPHD Filter Multitarget State Estimation
There are two approaches, those described in Section 17.4.4, with the following
exception. Instead of the nearest integer n to Nk+1|k+1 , one instead uses the
MAP estimate of the measurement-updated cardinality distribution:
n̂ = arg sup pk+1|k+1 (n).
(17.186)
n≥0
17.6
THE PD -CBMEMBER FILTER
The CBMeMBER multi-Bernoulli filter was described in Chapter 13. Its timeupdate and measurement-update formulas were given in (13.36) through (13.44)
and (13.47) through (13.58), respectively. Its generalization to the pD -agnostic
case was described in 2011 by Vo, Vo, Hoseinnezhad, and Mahler [312], [313]. It
is accomplished using the methodology described in Section 17.1.1.
The filtering equations for the pD -CBMeMBer filter are as follows.
RFS Filters for Unknown pD
547
• pD -CBMeMBer filter time-update equations: Given the prior track table
ν
k|k
i
Tk|k = {(ℓik|k , qk|k
,s̊ik|k (a, x))}i=1
(17.187)
we are to determine the time-updated track table
Tk+1|k
persist
Tk+1|k
birth
Tk+1|k
=
persist
birth
Tk+1|k
∪ Tk+1|k
(17.188)
=
νk|k
{(ℓi , qi ,s̊i (a, x))}i=1
(17.189)
=
bk
B B
{(ℓB
i , qi , si (x))}i=1 .
(17.190)
The persisting tracks are given by, for i = 1, ..., νk|k ,
ℓi
=
ℓik|k
(17.191)
qi
=
i
qk|k
· s̊ik|k [p̊S ]
(17.192)
s̊i (a, x)
=
˚a,x ]
s̊ik|k [p̊S M
i
s̊k|k [p̊S ]
(17.193)
s̊ik|k [p̊S ]
=
∫ ∫ 1
pS (x′ ) · sik|k (a′ , x′ )da′ dx′
(17.194)
pS (x′ ) · fk+1|k (a|a′ )
(17.195)
0
˚a,x ]
s̊ik|k [p̊S M
∫ ∫ 1
=
0
·fk+1|k (x|x′ ) · s̊ik|k (a′ , x′ )da′ dx′ .
• pD -CBMeMBer filter measurement-update equations: Given the predicted
track table
ν
k+1|k
i
Tk+1|k = {(ℓik+1|k , qk+1|k
,s̊ik+1|k (a, x))}i=1
(17.196)
and a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | = mk+1 ,
we are to determine the measurement-updated track table
Tk+1|k+1
=
legacy
Tk+1|k+1
=
meas
Tk+1|k+1
=
legacy
meas
Tk+1|k+1
∪ Tk+1|k+1
νk+1|k
L L
{(ℓL
i , qi ,s̊i (a, x))}i=1
mk+1
U U
{(ℓU
j , qj ,s̊j (a, x))}j=1 .
(17.197)
(17.198)
(17.199)
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The legacy tracks are given by, for for i = 1, ..., νk+1|k :
ℓL
i
=
ℓik+1|k
qiL
=
i
qk+1|k
·
s̊L
i (a, x)
=
s̊ik+1|k (a, x) ·
=
∫ ∫ 1
s̊ik+1|k [p̊D ]
(17.200)
1 − s̊ik+1|k [p̊D ]
(17.201)
i
1 − qk+1|k
· s̊ik+1|k [p̊D ]
1−a
1 − s̊ik+1|k [p̊D ]
(17.202)
a · s̊ik+1|k (a, x)dadx.
(17.203)
0
The measurement-updated tracks are given by, for j = 1, ..., mk+1 :
ℓU
j
=
ℓ∗k+1|k
(17.204)
qjU
(17.205)
∑
i
i
i
˚z ]
νk+1|k qk+1|k (1−qk+1|k )·s̊k+1|k [p̊D L
j
i
i
i=1
(1−qk+1|k ·s̊k+1|k [p̊D ])2
=
κk+1 (zj ) +
( ∑
i
∑νk+1|k qk+1|k
·s̊ik+1|k [p̊D ˚
Lzj ]
i
1−qk+1|k
·s̊ik+1|k [p̊D ]
i=1
i
νk+1|k qk+1|k
i
i=1
1−qk+1|k
s̊U
j (a, x)
=
=
)
·a · Lzj (x)
∑νk+1|k
i=1
s̊ik+1|k [p̊D ˚
L zj ]
· s̊ik+1|k (a, x)
∫ ∫ 1
i
qk+1|k
i
1−qk+1|k
(17.206)
· s̊ik+1|k [p̊D ˚
L zj ]
a · Lzj (x) · s̊ik+1|k (a, x)dadx
(17.207)
0
and where ℓ∗k+1|k is the label of the predicted track that has the largest
contribution to the current measurement-updated probability of existence in
(17.206).
The filtering equations for the pD -CBMeMBer filter can be implemented in
exact closed form using the BGM approximation technique described in Section
17.3. This implementation approach will not be further discussed here. For more
detail, see [312], [313].
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17.7
549
IMPLEMENTATIONS OF PD -AGNOSTIC RFS FILTERS
Most implementations of pD -agnostic CPHD and CBMeMBer filters have been
accomplished jointly with the clutter-agnostic methods to be described in Chapter
18. As of this writing, there was only one implementation of uniquely pD -agnostic
RFS filters—namely, a pD -CPHD filter and pD -PHD filter.
BGM implementations of these filters were described in 2011 by Mahler, Vo,
and Vo [195], [194] (although employing a Markov transition for a different than
that described in Section 17.1.2). The results reported here are from [194], pp.
3510-3511.
In this implementation, a linear-Gaussian sensor observes up to 10 appearing
and disappearing targets following linear trajectories. The Poisson clutter process,
which is known, is spatially uniform with clutter rate λ = 20. The probability
of detection, pD = 0.98, is constant but unknown, and thus must be implicitly
estimated by the pD -CPHD filter and pD -PHD filter.
In a simulation involving 100 Monte Carlo runs, it was observed that the pD CPHD filter converged to the correct number of targets in general, whereas the
pD -PHD filter exhibited an upward bias that slowly self-corrected. In terms of miss
distance (as measured by the OSPA metric of Section 6.2.2), both filters performed
reasonably, but with the pD -CPHD filter performing better than the pD -PHD filter.
It was observed that both filters tended to have difficulty resolving closely-spaced
targets.
The filters were also compared with conventional CPHD and PHD filters.
As expected, the tracking performances of the pD -CPHD and pD -PHD filter were
worse than that of the conventional filters. This is because the conventional CPHD
and PHD filters are provided with a priori information that the pD -CPHD and pD PHD filters are not given: the actual detection profile pD (x).
Chapter 18
RFS Filters for Unknown Clutter
18.1
INTRODUCTION
The purpose of this chapter is to address RFS multitarget detection and tracking
when the detection profile pD (x) is known but the clutter is partially or completely
unknown.
The conventional approach is to try to estimate the clutter process separately
prior to tracking. Examples include the approaches of X. Rong Li and Ning Li
[263], and of Teak Lyul Song and D. Musicki [287].
The general RFS approach described in this chapter was proposed by Mahler
in [189]. It is unusual in that clutter estimation and multitarget detection and
tracking are unified within a single, statistically unified algorithm. Special cases
of the CPHD filter version of the approach have been implemented by Mahler, Vo,
and Vo [195], [194]. A CBMeMBer filter version has been addressed by Vo, Vo,
Hosseinezhad, and Mahler [312], [313].
Measurement models for unknown clutter were described in Section 16.3.2
and will be extensively applied in this chapter. Three types of unknown clutter will
be addressed:
1. Poisson clutter generators, as discussed in Section 16.3.3. This will lead to
the clutter-agnostic Poisson-clutter PHD filter of Section 18.10, which has
combinatorial complexity.
2. General Bernoulli clutter generators, as discussed in Section 16.3.4. The
clutter cardinality distribution pκk+1 (m) and the entire clutter intensity
function (clutter PHD) κk+1 (z) = λk+1 · ck+1 (z) are unknown. This model
will lead to the general CPHD filter for the Bernoulli clutter-generator model
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(Section 18.2), which includes a very general motion Markov motion model.
Restricting this general motion model to a “phenomenology-nonintermixing”
model will, in turn, lead to the “κ-CPHD filter” of Section 18.5. This filter
can estimate both pκk+1 (m) and κk+1 (z), and has the same combinatorial
complexity as the classical PHD filter.
3. Simplified Bernoulli clutter generators, as discussed in Section 16.3.5. The
clutter cardinality distribution pκk+1 (m) (and thus also the clutter rate λk+1 )
are unknown, but the clutter spatial distribution ck+1 (z) is known. This
model will lead to the “λ-CPHD filter” of Section 18.4. This filter can
estimate pκk+1 (m) (and thus also λk+1 ), and has the same combinatorial
complexity as the classical PHD filter.
The focus of the chapter will be on the κ-CPHD and λ-CPHD filters and
their implementation. The λ-CPHD filter can be implemented in exact closed form
using Gaussian mixture (GM) techniques (Section 18.4.7). The κ-CPHD filter
can also be implemented in exact closed form, using either beta-Gaussian mixture
(BGM) techniques (Section 18.5.7) or normal-Wishart mixture (NWM) techniques
(Section 18.5.8). Multisensor versions of both filters are also addressed (Section
18.6).
A secondary focus will be on the κ-CBMeMBer filter, which is a CBMeMBer
filter that employs the general Bernoulli clutter-generator model (Section 18.7).
A ternary and cautionary focus will be on “pseudofilters” (Section 18.9).
These are CPHD and PHD filters that use the Bernoulli clutter generator model, but
which also employ a problematic phenomenology-intermixing motion model. This
model allows clutter generators to transition to targets, and/or targets to transition
to clutter generators. It is problematic because it blends target statistics with
clutter statistics, thereby making it more difficult to distinguish targets from clutter.
Moreover, and as will be shown, such an approach leads to pathological algorithmic
behavior (Section 18.9.2).
18.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• Unknown clutter can be modeled by extending the single-target state space
¨ = X ⊎ C, where C is the space of
X to a state space of the general form X
the states c of “clutter generators” and ‘⊎’ denotes disjoint union (Section
18.2.1).
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553
• A more general such approach is to assume that any clutter generator’s
measurement generation process is Poisson, as in Section 16.3.3. However,
this leads to PHD and CPHD filters that have combinatorial computational
complexity (Section 18.10).
• Any computationally tractable RFS multitarget detection and tracking filter
can be converted to one that can operate in unknown clutter but is still
computationally tractable. This is possible if the clutter generators in C are
assumed to be Bernoulli—that is, if their p.g.fl.’s have the form
∫
κ
Gk+1 [g|c] = 1 − pκD (c) + pκD (c) g(z) · fk+1
(z|c)dz
(18.1)
where pκD (c) is the probability that the generator c will be detected, and
κ
fk+1
(z|c) is the likelihood that measurement z will be produced if a clutter
generator with state c is present (Section 16.3.4).
• The simplest such RFS filter, the λ-CPHD filter, can estimate the cardinality
distribution pκk+1 (m) of the clutter RFS (and therefore also the cutter rate
λ), given that the clutter spatial distribution ck+1 (z) is known a priori. This
filter consists of three coupled filters:
– A filter for target PHDs Dk|k (x).
˚k|k of clutter generators.
– A filter for the expected number N
– A filter on the probability distribution p̈k|k (n̈) on n̈ = n + n̊, where
n is the number of targets and n̊ is the number of clutter generators
(Section 18.4). In addition, the respective cardinality distributions
pk|k (n) and p̊k|k (n̊) for n and for n̊ can be computed from p̈k|k (n̈)
(Section 18.3.4.2).
• The next simplest RFS filter, the κ-CPHD filter (Section 18.5), can estimate
the clutter cardinality distribution pκk+1 (m) and the entire clutter intensity
function κk+1 (z) = λk+1 · ck+1 (z). It consists of three coupled filters:
– A filter for target PHDs Dk|k (x).
˚k|k (c, c).
– A filter for clutter PHDs D
– A filter on the probability distribution p̈k|k (n̈). Once again, the cardinality distributions pk|k (n) and p̊k|k (n̊) for n and for n̊ can be
computed from p̈k|k (n̈).
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Advances in Statistical Multisource-Multitarget Information Fusion
• Both the λ-CPHD filter and κ-CPHD filter have the same computational
complexity as the classical PHD filter: O(mn), where m is the current
number of measurements and n is the current number of tracks.
• The λ-CPHD filter can be implemented using Gaussian mixture (GM)
approximation (Section 18.4.7).
• The κ-CPHD filter can be implemented using beta-Gaussian mixture (BGM)
approximation techniques (Section 18.5.7); or using normal-Wishart mixture
(NWM) approximation (Section 18.5.8).
• The next more complex RFS filter, the κ-CBMeMBer filter, can address
more nonlinear motion and measurement models than can the κ-CPHD filter
(Section 18.7).
• Garbage in, garbage out: For all of these filters, the clutter background must
be slowly varying in comparison to the measurement-update rate. Otherwise,
there is insufficient information to be able to estimate the clutter process while
also detecting and tracking targets.
• It is possible to claim—mistakenly—that similar techniques lead to multitarget detection and tracking filters that can estimate not only the clutter rate but
also new-target birth-rate. This misconception results if one adopts a problematic “phenomoneology-intermixing” motion model, which allows targets
to transition to clutter generators and/or clutter generators to transition to targets (Section 18.2.3).
• Even the simplest such intermixing CPHD “pseudofilter”—the λ-PHD
pseudofilter—exhibits pathological behavior that is directly attributable to the
intermixing motion model (Section 18.9.2).
18.1.2
Organization of the Chapter
The chapter is organized as follows:
1. Section 18.2: A general model for unknown Bernoulli clutter generators,
including two multitarget motion models: a phenomenology nonintermixing
model and a problematic phenomenology intermixing model.
2. Section 18.3: The CPHD filter corresponding to this general Bernoulli model.
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3. Section 18.4: The λ-CPHD filter—a CPHD filter that is applicable when
the clutter spatial density ck+1 (z) is known but the clutter cardinality
distribution pκk+1 (m) (and therefore also the clutter rate λk+1 ) is unknown.
4. Section 18.5: The κ-CPHD filter—a CPHD filter that is applicable when the
clutter cardinality distribution pκk+1 (m) and the clutter intensity function
κk+1 (z) = λk+1 · ck+1 (z) are unknown.
5. Section 18.6: Multisensor generalizations of the λ-CPHD and κ-CPHD
filters.
6. Section 18.7: The κ-CBMeMBer filter—a version of the CBMeMBer filter
that is applicable when the entire clutter intensity function is unknown (as
well as when the target probability of detection pD (x) is unknown).
7. Section 18.8: Implementations of the λ-CPHD filter, κ-CPHD filter, and
κ-CBMeMBer filter.
8. Section 18.9: Erroneous clutter-agnostic “pseudofilters,” based on the phenomenology intermixing motion model.
9. Section 18.10: A clutter-agnostic PHD filter based on the Poisson-mixture
clutter model of Section 16.3.3.
10. Section 18.11: Related work.
18.2
A GENERAL MODEL FOR UNKNOWN BERNOULLI CLUTTER
Consider the general Bernoulli clutter-generator model described in Section 16.3.4.
The state space of the dynamical system is assumed to have the form
¨ = X ⊎˚
X
C
(18.2)
where X is the target state space and ˚
C is the state space for Bernoulli clutter
generators. The probability of detection and the likelihood function for ˚
C will be
κ
denoted by, respectively, p̊κD (c̊) and ˚
Lκz (c̊) = f˚k+1
(z|c̊). It will be assumed that
˚
C can take two forms:
˚
C =
˚
C =
C
(18.3)
[0, 1] × C.
(18.4)
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That is, in the second case the states of the clutter generators have the augmented
form c̊ = (c, c), where c is the probability of detection of c; or in the first case
¨ 0 is
c̊ = c is not augmented. Thus the integral on X
∫
∫
∫
∫
∫
abbr.
¨
¨
¨
¨
f (ẍ)dẍ =
f (x)dx +
f (c̊)dc̊ =
f (x)dx + f¨(c̊)dc̊
(18.5)
˚
C
X
where, respectively,
or
∫
∫
f˚(c̊)dc̊ =
f˚(c̊)dc̊ =
∫
∫ ∫ 1
f˚(c)dc
(18.6)
f˚(c, c)dcdc.
(18.7)
0
In these two cases, the probability of detection and likelihood functions will
then be assumed to have the form
{
p̊D (c) if
c̊ = c
κ
p̊D (c̊) =
(18.8)
c
if c̊ = (c, c)
˚
Lκz (c̊) =
{
˚
Lz (c)
˚
Lz (c)
if
if
c̊ = c
c̊ = (c, c)
(18.9)
where p̊D (c) is the probability of detection on C0 and ˚
Lz (c) is the likelihood
function on C.
¨ is assumed to be endowed
Given this, the joint target-clutter state space X
with the joint model described in the following section.
18.2.1
The General Joint Target-Clutter Model
This consists of the following items:
• Joint probability of detection:
p̈D (ẍ) =
{
pD (x)
p̊D (c̊)
if
if
ẍ = x
.
ẍ = c̊
(18.10)
Lz (x)
˚
Lz (c̊)
if
if
ẍ = x
.
ẍ = c̊
(18.11)
• Joint likelihood function:
¨ z (ẍ) =
L
{
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557
• Joint target-survival probability:
p̈S (ẍ) =
{
if
if
pS (x)
p̊S (c̊)
ẍ = x
.
ẍ = c̊
(18.12)
if
if
(18.13)
• Joint target appearance PHD:
¨bk+1|k (ẍ) =
{
bk+1|k (x)
˚
bk+1|k (c̊)
ẍ = x
.
ẍ = c̊
Joint Markov transition density: The specification of this density is more
complicated, because it must be the case that, for all ẍ′ ,
1
=
∫
=
∫
f¨k+1|k (ẍ|ẍ′ )dẍ
f¨k+1|k (x|ẍ′ )dx +
(18.14)
∫
f¨k+1|k (c̊|ẍ′ )dc̊.
(18.15)
Thus there are four possibilities

pT (x′ ) · fk+1|k (x|x′ )


 (1 − p (x′ )) · f ⇐ (c̊|x′ )
ẍ = x, ẍ′ = x′
ẍ = c̊, ẍ′ = x′
T
k+1|k
f¨k+1|k (ẍ|ẍ′ ) =
.
⇒
(1 − p̊T (c̊′ )) · fk+1|k
(x|c̊′ )
ẍ = x, ẍ′ = c̊′



p̊T (c̊′ ) · f˙k+1|k (c̊|c̊′ )
ẍ = c̊, ẍ′ = c̊′
(18.16)
In (18.16), pT (x′ ) abbr.
= pT,k+1|k (x′ ) and p̊T (c̊′ ) abbr.
= p̊T,k+1|k (c̊′ ) and:
if
if
if
if
• pT (x′ ) = probability that a target x′ will transition to a target.
• 1−pT (x′ ) = probability that a target x′ will transition to a clutter generator.
• p̊T (c̊′ ) = probability that a clutter generator c̊′ will transition to a clutter
generator.
• 1−p̊T (c̊′ ) = probability that a clutter generator c̊′ will transition to a target.
• fk+1|k (x|x′ ) = probability (density) that target x′ will transition to target
x.
• f˚k+1|k (c̊|c̊′ ) = probability (density) that generator c̊′ will transition to
generator c̊.
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⇐
• fk+1|k
(c̊|x′ ) = probability (density) that target x′
generator c̊.
will transition to
⇒
• fk+1|k
(x|c̊′ ) = probability (density) that generator c̊′ will transition to
target x.
As in [153], two different motion models for targets and clutter generators
will be highlighted.
18.2.2
Phenomenology-Nonintermixing Motion Model
This model is the dynamic basis for the λ-agnostic CPHD filter of Section 18.4
and the κ-agnostic PHD filter of Section 18.5. It is characterized by the following
assumptions:
1. Phenomenology-nonintermixing model, assumption 1: Clutter generators
can transition only to clutter generators:
p̊T (c̊′ ) = 1.
(18.17)
2. Phenomenology-nonintermixing model, assumption 2: Targets can transition
only to targets:
pT (x′ ) = 1.
(18.18)
These assumptions can be expressed more formally as:

fk+1|k (x|x′ ) if ẍ = x, ẍ′ = x′



0
if ẍ = c̊, ẍ′ = x′
f¨k+1|k (ẍ|ẍ′ ) =
0
if ẍ = x, ẍ′ = c̊′


 ˚
′
fk+1|k (c̊|c̊ ) if ẍ = c̊, ẍ′ = c̊′
(18.19)
where fk+1|k (x|x′ ) is the Markov transition density for targets and f˚k+1|k (c̊|c̊′ )
is the Markov transition density for clutter generators.
18.2.3
Phenomenology-Intermixing Motion Model
This model is characterized by one or both of the following assumptions:
1. Phenomenology-intermixing model, assumption 1: Clutter generators can
transition to targets:
p̊T (c̊′ ) < 1.
(18.20)
RFS Filters for Unknown Clutter
559
Furthermore, as part of this assumption the generator-to-target transition
c̊ ?→ x is interpreted as being, simultaneously, a birth model for targets (targets appear when clutter generators transition to them) and a death model for
generators (clutter generators disappear by transitioning to targets). Mathematically, this compels the following:
(a) bk+1|k (x) = 0 (since target births are already modeled, as transitions
from generators).
(b) p̊S (c̊) = 1 (since generator deaths are already modeled, as transitions
to targets).
2. Phenomenology-intermixing model, assumption 2: Targets can transition to
clutter generators:
pT (x′ ) < 1
(18.21)
Furthermore, as part of this assumption the target-to-generator transition
x ?→ c̊ is interpreted as being, simultaneously, a birth model for generators (clutter generators appear when targets transition to them) and a death
model for targets (targets disappear by transitioning to clutter generators).
Mathematically, this compels the following:
(a) ˚
bk+1|k (c̊) = 0 (since generator births are already modeled—as transitions from targets).
(b) pS (x) = 1 (since target deaths are already modeled—as transitions to
generators).
In short, the single space ˚
C of parameters is being pressed into service not
only to model clutter, but also to model target appearance and target disappearance.
The following point cannot be emphasized too strongly:
• The phenomenology-intermixing motion model is highly questionable from
both a statistical and a phenomenological point of view.
Suppose, for example, that the measurement-statistics of targets are fundamentally different from the measurement-statistics of clutter; or that clutter motion
is fundamentally different than target motion. Then one should exploit the differences between them, in order to more efficiently distinguish targets from clutter.
If targets are allowed to become generators and generators are allowed to
become targets, the statistical differences between the two will be obscured, thereby
making target detection, localization, and discrimination more difficult. To express
matters in more forthright language:
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• Tanks do not transition to treelines and treelines do not transition to tanks.
• Assuming the contrary means that it will become very difficult to detect a tank
that is hiding in a treeline.
Furthermore, suppose that the intermixing motion model is used in the clutteragnostic CPHD filter of Section 18.3 rather than the nonintermixing motion model.
Then as we shall see in Section 18.9.2, simple analytical examples demonstrate that
the resulting CPHD and PHD filters will exhibit pathological behavior—and that
these pathologies are direct consequences of the intermixing motion model.
18.3
CPHD FILTER FOR GENERAL BERNOULLI CLUTTER
The CPHD filter corresponding to the general multi-Bernoulli model is just a
¨ = X ⊎˚
conventional CPHD filter defined on the state space X
C, assuming the
models described in Section 18.2.1. That is, it has the form
... →
pk|k (n̈)
... →
¨ k|k (ẍ)
D
→
↑
→
pk+1|k (n̈)
¨ k+1|k (ẍ)
D
→
↑↓
→
pk+1|k+1 (n̈)
→ ...
¨ k+1|k+1 (ẍ)
D
→ ...
¨ and pk|k (n̈)
¨ k|k (ẍ) is a PHD defined on the joint state-variable ẍ ∈ X
where D
is a probability distribution on the number n̈ = n + n̊, where n is the number of
targets and n̊ is the number of clutter generators.
¨ is the disjoint union of X and ˚
¨ k|k (ẍ) can
Because X
C, the joint PHD D
equivalently be represented as two separate PHDs, one defined on the target state
space and one on the clutter state space:
Dk|k (x)
˚k|k (c̊)
D
=
=
¨ k|k (x)
D
¨ k|k (c̊).
D
(18.22)
(18.23)
¨ can be expressed as three coupled filters:
Thus the CPHD filter on X
... →
p̈k|k (n̈)
... →
Dk|k (x)
... →
˚k|k (c̊)
D
→
↑
→
↑
→
p̈k+1|k (n̈)
Dk+1|k (x)
˚k+1|k (c̊)
D
→
↑↓
→
↑↓
→
p̈k+1|k+1 (n̈)
→ ...
Dk+1|k+1 (x)
→ ...
˚k+1|k+1 (c̊)
D
→ ...
RFS Filters for Unknown Clutter
561
Furthermore, the cardinality distribution pk|k (n) of n (the target cardinality
distribution) and the cardinality distribution p̊k|k (n̊) of n̊ (the clutter-generator
cardinality distribution) can both be computed from the joint cardinality distribution
p̈k|k (n̈) (see Section 18.3.4.2).
Since clutter measurements are due to clutter generators, this means that
κ̈k+1 (z) = 0, where κ̈k+1 (z) is the intensity function for the a priori clutter
¨ is actually a ZFA-CPHD filter, as
RFS. Consequently, the CPHD filter on X
described in Section 8.6. Its filtering equations are, therefore,
• Time update (see (8.86), (8.89), and (8.90)):
¨ k+1|k (ẍ)
D
p̈k+1|k (n̈)
= ¨bk+1|k (ẍ)
(18.24)
∫
¨ k|k (ẍ′ )dẍ′
+ p̈S (ẍ′ ) · f¨k+1|k (ẍ|ẍ′ ) · D
∑
=
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.25)
n̈′ ≥0
and
p̈k+1|k (n̈|n̈′ )
(18.26)
=
n̈
∑
¨i
¨ n̈′ −i
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i · ψk (1 − ψk )
i=0
ψ¨k
=
¨k|k
N
=
∫
1
¨ k|k (ẍ′ )dẍ′
p̈S (ẍ′ ) · D
¨k|k
N
∫
¨ k|k (ẍ′ )dẍ′ .
D
(18.27)
(18.28)
In this case, the cardinality distribution corresponding to both target appearances and clutter–generator appearances has the form
p̈B
k+1|k (n̈) =
∑
B
pB
k+1|k (n) · p̊k+1|k (n̊)
(18.29)
n+n̊=n̈
where pB
k+1|k (n) is the cardinality distribution of the target appearance RFS,
and p̊B
k+1|k (n̊) is the cardinality distribution of the target appearance RFS
for clutter generators.
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• Measurement update (see (8.134) through (8.141)): Let the new measurement set be Zk+1 = {z1 , ..., zm } with |Zk+1 | = m. Then:


¨k )
¨ (m+1) (ϕ
G
k+1|k
¨
Dk+1|k+1 (ẍ)
1
 (1 − p̈D (ẍ)) · G¨ (m) (ϕ¨k ) 
k+1|k
=

 (18.30)
∑
¨
¨
¨ z (ẍ)
L
Dk+1|k (ẍ)
Nk+1|k
+ z∈Zk+1 p̈Dτ̈(ẍ)·
k+1 (z)
∫
¨k+1|k =
¨ k+1|k (ẍ)dẍ
N
D
(18.31)
and
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈)
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
Cn̈,m · ϕ¨n̈−m
k
(18.32)
(18.33)
where
1
ϕ¨k
=
τ̈k+1 (z)
=
∫
¨ k+1|k (ẍ)dẍ
(1 − p̈D (ẍ)) · D
¨k+1|k
N
∫
1
¨ z (ẍ) · D
¨ k+1|k (ẍ)dẍ
p̈D (ẍ) · L
¨k+1|k
N
(18.34)
(18.35)
and where Cn̈,m was defined in (2.1).
The filtering equations for the general-model CPHD filter are just these equations, after the model formulas described in Section 18.2.1 have been substituted
into them. The remainder of the section is organized as follows:
1. Section 18.3.1: Time update equations for the general Bernoulli model
CPHD filter.
2. Section 18.3.2: Measurement update equations for the general Bernoulli
model CPHD filter.
3. Section 18.3.3: The PHD filter special case of the general Bernoulli model
CPHD filter.
4. Section 18.3.4: Multitarget state estimation for the general Bernoulli model
CPHD filter.
5. Section 18.3.5: Clutter estimation for the general Bernoulli model CPHD
filter.
RFS Filters for Unknown Clutter
18.3.1
563
General Bernoulli Clutter-Generator Model: CPHD Filter Time Update
We are given the joint cardinality distribution p̈k|k (n̈), the target PHD Dk|k (x), and
˚k|k (c̊). We are to determine the time-updated joint cardinality
the clutter PHD D
distribution p̈k+1|k (n̈), the time-updated target PHD Dk+1|k (x), and the time˚k+1|k (c̊). These are:
updated clutter PHD D
• Time updated cardinality distribution: Let
Nk|k
˚k|k
N
=
∫
Dk|k (x)dx.
(18.36)
=
∫
˚k|k (c̊′ )dc̊′ .
D
(18.37)
Then using (18.5) for the joint integral and making the proper substitutions
into (18.25)-(18.28), we get
p̈k+1|k (n̈)
=
∑
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.38)
n̈′ ≥0
p̈k+1|k (n̈|n̈′ )
(18.39)
=
n̈
∑
¨i
¨ n̈′ −i
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i · ψk (1 − ψk )
i=0
( ∫
ψ¨k
=
p (x′ ) · Dk|k (x′ )dx′
∫S
˚k|k (c̊′ )dc̊′
+ p̊S (c̊′ ) · D
)
(18.40)
˚k|k
Nk|k + N
where as in (18.29),
p̈B
k+1|k (n̈) =
∑
n+n̊=n̈
B
pB
k+1|k (n) · p̊k+1|k (n̊).
(18.41)
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• Time updated target PHD: Substituting (18.16) into (18.24), and applying
(18.5), results in
Dk+1|k (x)
=
bk+1|k (x)
(18.42)
∫
+ pS (x′ ) · pT (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
∫
⇒
˚k|k (c̊′ )dc̊′ .
+ p̊κS (c̊′ ) · (1 − p̊T (c̊′ )) · fk+1|k
(x|c̊′ ) · D
• Time updated clutter PHD: Substituting (18.16) into (18.24), and applying
(18.5), results in
˚k+1|k (c̊)
D
18.3.2
= ˚
bk+1|k (c̊)
(18.43)
∫
⇐
+ pS (x′ ) · (1 − pT (x′ )) · fk+1|k
(c̊|x′ ) · Dk|k (x′ )dx′
∫
˚k|k (c̊′ )dc̊′ .
+ p̊S (c̊′ ) · p̊T (c̊′ ) · f˙k+1|k (c̊|c̊′ ) · D
General Bernoulli Clutter Model: CPHD Filter Measurement Update
We are given the time-updated joint cardinality distribution p̈k+1|k (n̈′ ), and the
˚k+1|k (c̊). Let a new
time-updated target and clutter PHDs Dk+1|k (x) and D
measurement set Zk+1 with |Zk+1 | = m be collected. We are to determine the measurement-updated joint cardinality distribution p̈k+1|k (n̈′ ), the
measurement-updated target PHD Dk+1|k (x), and the measurement-updated clut˚k+1|k (c̊). Let
ter PHD D
¨ k+1|k (ẍ)
G
=
∑
p̈k+1|k (n̈) · ẍn̈
(18.44)
n̈≥0
Nk+1|k
˚k+1|k
N
=
∫
Dk+1|k (x)dx
(18.45)
=
∫
˚k+1|k (c̊)dc̊
D
(18.46)
RFS Filters for Unknown Clutter
=
∫
κ
τ̊k+1
(z)
=
∫
ϕ¨k
=
τk+1 (z)
565
pD (x) · Lz (x) · Dk+1|k (x)dx
(18.47)
˚k+1|k (c̊)dc̊
p̊κD (c̊) · ˚
Lκz (c̊) · D
( ∫
)
(1 − pD (x)) · Dk+1|k (x)dx
∫
˚k+1|k (c̊)dc̊
+ (1 − p̊κD (c̊)) · D
.
˚k+1|k
Nk+1|k + N
(18.48)
(18.49)
Then:
• Measurement updated joint cardinality distribution:
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈)
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
Cn̈,m · ϕ¨n̈−m .
(18.50)
(18.51)
k
• Measurement updated target PHD:
Dk+1|k+1 (x)
Dk+1|k (x)
=
¨ (m+1) (ϕ¨k )
G
1 − pD (x)
k+1|k
· (m)
˚
¨
Nk+1|k + Nk+1|k G
(ϕ¨k )
(18.52)
k+1|k
∑
+
z∈Zk+1
pD (x) · Lz (x)
κ (z) .
τk+1 (z) + τ̊k+1
• Measurement updated clutter PHD:
˚k+1|k+1 (c̊)
D
˚k+1|k (c̊)
D
=
¨ (m+1) (ϕ¨k )
G
1 − p̊κD (c̊)
k+1|k
· (m)
˚k+1|k G
¨
Nk+1|k + N
(ϕ¨ )
k+1|k
+
∑
z∈Zk+1
(18.53)
k
p̊κD (c̊) · ˚
Lκz (c̊)
κ (z) .
τk+1 (z) + τ̊k+1
Remark 71 (Computability of clutter-agnostic CPHD filters) The computational
complexity of the CPHD filter just described is the same as that for the classical
PHD filter: O(mn), where m is the current number of measurements and n is
the current number of targets.
Remark 72 From (18.52) and (18.53) it is clear that, when pD is constant and
nearly 1, clutter-agnostic CPHD filters will behave like the classical PHD filter.
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18.3.3
General Bernoulli Clutter-Generator Model: PHD Filter Special Case
The PHD filter special case occurs when we assume, for the measurement-update
step, that the predicted-target RFS is Poisson:
¨ = e−N¨k+1|k · D
¨ X¨
fk+1|k (X)
k+1|k .
(18.54)
In this case the previous filtering equations reduce to the following time-update and
measurement-update equations:
bk+1|k (x)
(18.55)
∫
+ pS (x′ ) · pT (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
∫
⇒
˚k|k (c̊′ )dc̊′
+ p̊κS (c̊′ ) · (1 − p̊T (c̊′ )) · fk+1|k
(x|c̊′ ) · D
Dk+1|k (x)
=
˚k+1|k (c̊)
D
= ˚
bk+1|k (c̊)
(18.56)
∫
⇐
+ pS (x′ ) · (1 − pT (x′ )) · fk+1|k
(c̊|x′ ) · Dk|k (x′ )dx′
∫
˚k|k (c̊′ )dc̊′
+ p̊S (c̊′ ) · p̊T (c̊′ ) · f˙k+1|k (c̊|c̊′ ) · D
and
18.3.4
Dk+1|k+1 (x)
Dk+1|k (x)
=
˚k+1|k+1 (c̊)
D
˚k+1|k (c̊)
D
=
1 − pD (x) +
∑
z∈Zk+1
1 − p̊κD (c̊) +
∑
z∈Zk+1
pD (x) · Lz (x)
κ (z)
τk+1 (z) + τ̊k+1
(18.57)
p̊κD (c̊) · ˚
Lκz (c̊)
κ (z) .
τk+1 (z) + τ̊k+1
(18.58)
General Bernoulli Clutter Model: Multitarget State Estimation
For the classical CPHD filter, state estimation consists of two steps (Section 8.5.5).
First, estimate target number from the cardinality distribution using the MAP
estimator:
n̈k+1|k+1 = arg sup p̈k+1|k+1 (n̈).
(18.59)
n̈
Second, determine the states corresponding to the n̈k+1|k+1 largest suprema of the
¨ k+1|k+1 (ẍ).
PHD D
RFS Filters for Unknown Clutter
567
In the case of the CPHD filters considered in this chapter, this approach is not
appropriate. This is because the cardinality distribution p̈k+1|k+1 (n̈) is now a
distribution on the number n̈ = n + n̊, where n is the number of targets and n̊
is the number of clutter generators. What we actually need to estimate is n rather
than n̈. The purpose of this section is to explain how this can be accomplished.
18.3.4.1
Original Approach to State Estimation
The state-estimation procedure originally proposed in [189],[195], and [194] was
as follows. First, determine the expected number of targets by integrating the target
PHD:
∫
∫
¨ k+1|k+1 (x)dx.
Nk+1|k+1 = Dk+1|k+1 (x)dx = D
(18.60)
Second, round Nk+1|k+1 off to the nearest integer n̂. Third, determine the states
corresponding to the n̂ largest suprema of Dk+1|k+1 (x).
The disadvantage of this procedure is that Nk+1|k+1 does not provide an
accurate instantaneous estimate. Thus one of the primary advantages of the CPHD
filter, as compared to the PHD filter, is lost.
18.3.4.2
A Better Approach to State Estimation
Because of an insight due to Chen Xin, McDonald, and Kirubarajan [36], a procedure that estimates the target cardinality distribution (that is, the distribution of the
number of actual targets) can be used instead. Let
rk+1 =
Nk+1|k+1
.
˚k+1|k+1
Nk+1|k+1 + N
(18.61)
Then it can be shown that the cardinality distribution for the actual targets is
pk+1|k+1 (n) =
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 ).
n!
(18.62)
Consequently, we can determine the MAP estimate for the current number of
targets as
nk+1|k+1 = arg sup pk+1|k+1 (n)
(18.63)
n
and use this instead of Nk+1|k+1 .
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Simulations have shown that this approach results in improved estimation of
target number (compared to the original approach in Section 18.3.4.1). Specifically,
the original approach tends to have an upward bias in the cardinality estimate,
whereas this approach avoids this bias. The improvement is not as great as one
might expect, however. The reason is probably the fact that the clutter-agnostic
CPHD filters strongly resemble the classical PHD filter.
The cardinality distribution for the clutter generators may be determined
similarly as:
r̊ n̊
¨ (n̊)
p̊k+1|k+1 (n̊) = k+1 · G
(18.64)
k+1|k+1 (1 − r̊k+1 )
n̊!
where
˚k+1|k+1
N
r̊k+1 =
= 1 − rk+1 .
(18.65)
˚k+1|k+1
Nk+1|k+1 + N
Remark 73 It can further be shown that the multitarget probability distribution for
actual targets is i.i.d.c. and is given by the following formulas:
fk+1|k+1 (X|Z (k+1) )
=
¨ (|X|) (1 − rk+1 ) · s̈X
G
k+1|k+1
k+1|k+1
(18.66)
=
¨ (|X|) (1 − rk+1 )
G
k+1|k+1
(18.67)
|X|
·rk+1 · sX
k+1|k+1
with corresponding p.g.fl.
¨ k+1|k (sk+1|k+1 [1 − rk+1 + rk+1 · h]).
Gk+1|k+1 [h] = G
(18.68)
Equation (18.66) is proved in the same way that (K.441) is proved in Section K.21.
¨ k+1|k+1 be the measurement-updated joint
To see why (18.62) is true, let Ξ
¨ k+1|k+1 that are
target/clutter RFS. Then the (random) number of elements of Ξ
actual targets is
¨ k+1|k+1 ∩ X|.
|Ξ
(18.69)
¨ k+1|k+1 ∩X is also the cardinality distribution
Thus the cardinality distribution of Ξ
¨ k+1|k+1 ∩ X
of the number of actual targets. According to (4.137), the p.g.f. of Ξ
is
GΞ¨ k+1|k+1 ∩X (x) = GΞ¨ k+1|k+1 [1 − 1X + x · 1X ].
(18.70)
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569
¨ k+1|k+1 is i.i.d.c., this becomes
Since Ξ
GΞ¨ k+1|k+1 ∩X (x)
(18.71)
=
¨ k+1|k+1 (s̈k+1|k+1 [1 − 1X + x · 1X ])
G
¨ k+1|k+1 (1 − s̈k+1|k+1 [1X ] + x · s̈k+1|k+1 [1X ])
G
(18.72)
=
¨ k+1|k+1 (1 − rk+1 + x · rk+1 ).
G
(18.73)
=
The last equation follows because
s̈k+1|k+1 [1X ]
=
=
=
∫
¨ k+1|k+1 [1X ]
¨
D
D
(ẍ)dẍ
X k+1|k+1
=
¨
˚
Nk+1|k+1
Nk+1|k+1 + Nk+1|k+1
∫
¨
Dk+1|k+1 (x)dx
˚k+1|k+1
Nk+1|k+1 + N
Nk+1|k+1
= rk+1 .
˚k+1|k+1
Nk+1|k+1 + N
(18.74)
(18.75)
(18.76)
Thus the cardinality distribution for the actual number of targets is, as claimed,
=
=
=
18.3.5
pk+1|k+1 (n)
[
]
1 dn ¨
Gk+1|k+1 (1 − rk+1 + x · rk+1 )
n! dxn
[ n
] x=0
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 + x · rk+1 )
n!
x=0
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 ).
n!
(18.77)
(18.78)
(18.79)
General Bernoulli Clutter-Generator Model: Clutter Estimation
Inspection of (18.52) and (18.53) would lead one to infer that the term
κ
τk+1
(z) =
∫
˚k+1|k (c̊)dc̊
p̊κD (c̊) · ˚
Lκz (c̊) · D
(18.80)
must be an estimate κ̂k+1 (z) of the intensity function of the unknown clutter RFS,
but which has been determined prior to the collection of the new measurement set
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Zk+1 . However, Remark 21 of Section 8.3.3 warns us that such an inference cannot
be taken for granted. Rather, it must be demonstrated.
κ
That is the purpose of this section. Specifically, it will be shown that τk+1
(z)
is the predicted average clutter intensity function. Moreover, it will be shown that
the entire multiobject probability distribution of the clutter process can be estimated
in terms of the predicted average p.g.fl. of the clutter RFS. Specifically, in Section
K.21 the following formulas will be derived:
• Estimated intensity function (PHD) of the clutter RFS:
∫
(k)
˚k+1|k (c̊|Z (k) )dc̊.
κ̂k+1 (z|Z ) = p̊κD (c̊) · ˚
Lκz (c̊) · D
(18.81)
• Estimated clutter rate of the clutter RFS:
∫
ˆ k+1|k (Z (k) ) = p̊κ (c̊) · D
˚k+1|k (c̊|Z (k) )dc̊.
λ
D
(18.82)
• Estimated spatial distribution of the clutter RFS:
ĉk+1 (z|Z (k) ) =
κ̂k+1 (z|Z (k) )
.
ˆ k+1 (Z (k) )
λ
(18.83)
• Estimated p.g.fl. of the clutter RFS:
ˆ κk+1 [g|Z (k) ] = G
¨ k+1|k (s̊k+1 [1 + (1 − rk+1 )p̊κD ˚
G
Lκg−1 ])
(18.84)
where
rk+1
=
˚
Lκg−1 (c̊)
=
Nk+1|k
˚k+1|k
Nk+1|k + N
∫
κ
(g(z) − 1) · fk+1
(z|c̊)dz
(18.85)
(18.86)
and where rk+1 is the proportion of objects that are targets (rather than
clutter generators).
• Estimated multiobject probability distribution of the clutter RFS:
κ
˜ k+1 ) · λ
˜ |Z| · ĉZ
¨ (|Z|) (1 − λ
fˆk+1
(Z|Z (k) ) = G
k+1
k+1
k+1|k
(18.87)
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571
where
˜ k+1
λ
=
ĉZ
k+1
=
ˆ k+1 (Z (k) )
λ
˚k+1|k
Nk+1|k + N
∏
ĉk+1 (z|Z (k) )
(18.88)
(18.89)
z∈Z
˜ k+1 can be intuitively interpreted as the proportion of objects
and where λ
that generate clutter measurements (see (18.128)).
• Estimated p.g.f. of the clutter RFS:
˜ k+1 + z · λ
˜ k+1 ).
ˆ k+1 (z|Z (k) ) = G
¨ k+1|k (1 − λ
G
(18.90)
ˆ k+1 (z|Z (k) ) is, as it should be, the
Note that the expected value of G
estimated clutter rate:
[
]
d ˆ
Gk+1 (z|Z (k) )
(18.91)
dz
z=1
[
]
˜ k+1 + z · λ
˜ k+1 ) · λ
˜ k+1
¨ (1) (1 − λ
= G
k+1|k
z=1
=
˜ k+1 = N
˜ k+1 = λ
ˆ k+1 .
¨ (1) (1) · λ
¨k+1|k · λ
G
k+1|k
(18.92)
• Estimated cardinality distribution of the clutter RFS:
p̂k+1 (m|Z (k) ) =
˜m
λ
k+1
˜ k+1 ).
¨ (m) (1 − λ
·G
k+1|k
m!
(18.93)
In particular, note that because of (18.87) and (18.93), the estimated clutter
process is i.i.d.c. with spatial distribution ĉk+1 (z|Z (k) ) and cardinality distribution
p̂k+1 (m|Z (k) ).
18.4
THE λ-CPHD FILTER
The λ-agnostic CPHD filter, or λ-CPHD filter for short, was proposed by Mahler
et al. in 2010 [189]. It is a CPHD filter that recursively estimates the clutter rate
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λk+1 and, more generally, the clutter cardinality distribution pκk+1 (m)—but not
the clutter spatial distribution ck+1 (z), which is assumed to be known a priori.
The λ-CPHD filter has the form
... →
p̈k|k (n̈)
... →
Dk|k (x)
... →
˚k|k
N
→
↑
→
↑
→
p̈k+1|k (n̈)
¨ k+1|k (x)
D
˚k+1|k
N
→
↑↓
→
↑↓
→
p̈k+1|k+1 (n̈)
→ ...
Dk+1|k+1 (x)
→ ...
˚k+1|k+1
N
→ ...
and consists of three mutually coupled filters. The top filter propagates the joint
probability distribution p̈k|k (n̈) on the number n̈ = n+n̊, where n is the number
of targets and n̊ is the number of clutter generators. The middle filter propagates
the PHD Dk|k (x) on targets. The bottom filter propagates the expected number
˚k|k of clutter generators.
N
In Section 18.3.4.2 it was shown that, for the general clutter-agnostic CPHD
filter, it is possible to derive from p̈k|k (n̈) the cardinality distribution pk|k (n) for
targets and the cardinality distribution p̊k|k (n̊) for clutter generators. The same,
of course, is true for the λ-CPHD filter—see Section 18.4.4.
The section is organized as follows:
1. Section 18.4.1: Models for the λ-CPHD filter.
2. Section 18.4.2: Time update equations for the λ-CPHD filter.
3. Section 18.4.3: Measurement update equations for the λ-CPHD filter.
4. Section 18.4.4: Multitarget state estimation for the λ-CPHD filter.
5. Section 18.4.5: Clutter estimation for the λ-CPHD filter.
6. Section 18.4.6: The λ-PHD filter (the PHD filter special case of the λ-CPHD
filter).
7. Section 18.4.7: Gaussian mixture (GM) implementation of the λ-CPHD
filter.
18.4.1
λ-CPHD Filter: Models
The λ-CPHD filter is based on the model of (18.3). That is, the clutter generator
state space is ˚
C = C (that is, the clutter probability of detection is known) and
clutter generators have the form c̊ = c. It also presumes the phenomenologynonintermixing motion model of Section 18.2.2. That is, targets can transition only
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573
to targets and clutter generators can transition only to clutter generators:
pT (x′ )
=
1
(18.94)
′
=
1.
(18.95)
p̊T (c )
It thereby requires the following models:
• Target probability of survival: pS (x) abbr.
= pS,k+1 (x).
• Target Markov density: fk+1|k (x|x′ ).
• PHD for target appearance: bk+1|k (x).
• Cardinality distribution for target appearance: pB
k+1|k (n), with
B
Nk+1|k
=
∫
bk+1|k (x)dx =
∑
n · pB
k+1|k (n).
(18.96)
n≥0
• Clutter-generator probability of survival is constant: p̊S abbr.
= pS,k+1 .
• Clutter-generator Markov density: f˚k+1|k (c|c′ ).
• PHD for appearance of clutter generators: ˚
bk+1|k (c).
• Cardinality distribution for clutter-generator appearance: p̊k+1|k (n̊) with
˚B
N
k+1|k =
∫
˚
bk+1|k (c)dc =
∑
n̊ · p̊B
k+1|k (n̊).
(18.97)
n̊≥0
• Target probability of detection: pD (x) abbr.
= pD,k+1 (x).
• Target likelihood function: Lz (x) abbr.
= fk+1 (z|x).
• Clutter spatial distribution: ck+1 (z).
• Clutter-generator probability of detection is known and constant: p̊D abbr.
=
p̊D,k+1 .
• Clutter-generator likelihood function is state-independent:
f˚k+1 (z|c) = ck+1 (z) where ck+1 (z) is known a priori.
Lκz (c) abbr.
=
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For computational reasons, it may be convenient to assume that clutter generators cannot appear (though this is not assumed in general):
˚
bk+1|k (x)
=
0
(18.98)
p̊B
k+1|k (n̊)
=
δ0,n̊ .
(18.99)
In this case, the cardinality distribution for the total appearance RFS, (18.41),
reduces to the target appearance RFS:
p̈B
k+1|k (n̈)
∑
=
B
pB
k+1|k (n) · p̊k+1|k (n̊)
(18.100)
n+n̊=n̈
pB
k+1|k (n̈).
=
18.4.2
(18.101)
λ-CPHD Filter: Time Update
The time-update formulas for the λ-CPHD filter result from substituting the models
in Section 18.4.1 into (18.36) through (18.43). We are given the joint cardinality
˚k|k
distribution p̈k|k (n̈), the target PHD Dk|k (x), and the expected number N
of clutter generators. We are to find the time-updates p̈k+1|k (n̈), Dk+1|k (x), and
˚k+1|k . These are:
N
• Time updated joint target/clutter cardinality distribution:
p̈k+1|k (n̈)
=
∑
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.102)
n̈′ ≥0
p̈k+1|k (n̈|n̈′ )
(18.103)
=
n̈
∑
¨i
¨ n̈′ −i
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i · ψk (1 − ψk )
i=0
ψ¨k
=
∫
˚k|k
pS (x′ ) · Dk|k (x′ )dx′ + p̊S · N
˚k|k
Nk|k + N
(18.104)
where, as in (18.29),
p̈B
k+1|k (n̈) =
∑
n+n̊=n̈
B
pB
k+1|k (n) · p̊k+1|k (n̊).
(18.105)
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575
• Time updated target PHD: Substituting (18.16) into (18.24), and applying
(18.5), results in
∫
Dk+1|k (x) = bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ . (18.106)
• Time updated expected number of clutter generators:
˚k+1|k = N
˚B
˚
N
k+1|k + p̊S · Nk|k .
For, substituting (18.16) into (18.24) and applying (18.5), we get:
∫
˚k+1|k (c) = ˚
¨ k|k (c′ )dc′ .
D
bk+1|k (c) + p̊S f˚k+1|k (c|c′ ) · D
(18.107)
(18.108)
Integrating both sides of (18.108) then leads to the claimed result.
18.4.3
λ-CPHD Filter: Measurement Update
We are given the joint cardinality distribution
p̈k+1|k (n̈), the target PHD
˚k+1|k of clutter generators, and a new meaDk+1|k (x), the expected number N
surement set Zk+1 with |Zk+1 | = m. We are to find the measurement-updates
˚k+1|k+1 . We are also to arrive at an estimate
p̈k+1|k+1 (n̈), Dk+1|k+1 (x), and N
ˆ
λk+1 of the clutter rate and, more generally, an estimate p̂κk+1 (m) of the clutter
cardinality distribution. Let
∫
Nk+1|k = Dk+1|k (x)dx.
(18.109)
Then:
• Measurement updated joint cardinality distribution and p.g.f.:
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈)
=
ϕ¨k
=
¨ k+1|k (ẍ)
G
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
Cn̈,m · ϕ¨n̈−m
k
( ∫
)
(1 − pD (x)) · Dk+1|k (x)dx
˚k+1|k
+(1 − p̊D ) · N
˚k+1|k
Nk+1|k + N
∑
p̈k+1|k (n̈) · ẍn̈ .
n̈≥0
(18.110)
(18.111)
(18.112)
(18.113)
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• Measurement updated target PHD:
Dk+1|k+1 (x)
Dk+1|k (x)
¨ (m+1) (ϕ¨k )
G
1 − pD (x)
k+1|k
· (m)
˚
¨
Nk+1|k + Nk+1|k G
(ϕ¨k )
=
(18.114)
k+1|k
+
∑
pD (x) · Lz (x)
ˆ
z∈Zk+1 λk+1 · ck+1 (z) + τk+1 (z)
where
ˆ k+1
λ
=
τk+1 (z)
=
˚k+1|k
p̊D · N
∫
pD (x) · Lz (x) · Dk+1|k (x)dx.
(18.115)
(18.116)
• Measurement updated expected number of clutter generators:
˚k+1|k+1
N
˚k+1|k
N
=
¨ (m+1) (ϕ¨k )
G
1 − p̊D
k+1|k
· (m)
˚k+1|k G
¨
Nk+1|k + N
(ϕ¨k )
(18.117)
k+1|k
+
∑
p̊D · ck+1 (z)
.
ˆ
z∈Zk+1 λk+1 · ck+1 (z) + τk+1 (z)
For, from (18.53) the clutter PHD is
˚k+1|k+1 (c)
D
˚k+1|k (c)
D
=
¨ (m+1) (ϕ¨k )
G
1 − p̊D
k+1|k
· (m)
˚
¨
Nk+1|k + Nk+1|k G
(ϕ¨k )
(18.118)
k+1|k
+
∑
p̊D · ck+1 (z)
.
ˆ
z∈Zk+1 λk+1 · ck+1 (z) + τk+1 (z)
Since the right side does not depend on c, we can integrate both sides to get
the claimed result.
18.4.4
λ-CPHD Filter: Multitarget State Estimation
State estimation is accomplished as described in Section 18.3.4. In particular, recall
that in Section 18.3.4.2 it was shown that the cardinality distributions for targets and
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577
for clutter were given by (18.62) and (18.64):
pk+1|k+1 (n)
=
p̊k+1|k+1 (n̊)
=
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 )
n!
(1 − rk+1 )n̊ ¨ (n̊)
· Gk+1|k+1 (rk+1 )
n̊!
(18.119)
(18.120)
where
rk+1 =
Nk+1|k+1
.
˚k+1|k+1
Nk+1|k+1 + N
Thus, the multitarget state can be estimated as in Section 18.3.4.2.
determine the MAP estimate
n̂k+1|k+1 = arg sup pk+1|k+1 (n)
(18.121)
That is,
(18.122)
n
and then determine the target states corresponding to the n̂k+1|k+1 largest peaks
of Dk+1|k+1 (x).
18.4.5
λ-CPHD Filter: Clutter Estimation
The clutter rate is given by
• Estimated clutter rate:
ˆ k+1 = p̊D · N
˚k+1|k .
λ
(18.123)
Equation (18.123) follows from (18.81),
κ̂k+1 (z|Z
(k)
)=
∫
˚k+1|k (c̊)dc̊.
p̊κD (c̊) · ˚
Lκz (c̊) · D
(18.124)
Set p̊κD (c̊) = p̊κD and ˚
Lκz (c̊) = ck+1 (z), in which case
˚k+1|k ,
κ̂k+1 (z|Z (k) ) = p̊κD · ck+1 (z) · N
(18.125)
from which (18.123) follows.
However, the λ-CPHD filter can estimate not only λ but also the clutter
cardinality distribution. From (18.93), this is given by
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Advances in Statistical Multisource-Multitarget Information Fusion
• Estimated clutter cardinality distribution:
p̂k+1 (m|Z (k) ) =
1 ¨ (m)
˜ k+1|k ) · λ
˜m
G
(1 − λ
k+1|k
m! k+1|k
where, since p̊D (c̊) = p̊D is constant,
˜ k+1
λ
=
=
=
ˆ k+1 (Z (k) )
λ
˚k+1|k
Nk+1|k + N
∫
˚k+1|k (c̊|Z (k) )dc̊
p̊D (c̊) · D
˚k+1|k
Nk+1|k + N
˚k+1|k
p̊D · N
.
˚k+1|k
Nk+1|k + N
(18.126)
(18.127)
(18.128)
ˆ k+1 = p̊D · N
˚k+1|k
Remark 74 Recall from the discussion in Section 18.3.5 that λ
has not been chosen to be the estimated clutter rate simply because it appears in the
“right place” in (18.114). Rather, it has been chosen because it can be provably
shown to be the predicted average clutter rate—see Section 18.3.5.
18.4.6
Special Case: The λ-PHD Filter
The λ-PHD filter is not of much practical interest, since the λ-CPHD filter
will perform better while also having roughly the same computational complexity.
Rather, it will be of interest later in Section 18.9 as part of the discussion of
“pseudofilters”.
Assume that the predicted-target RFS is Poisson, so that
˚k+1|k )·(x−1)
¨ k+1|k (ẍ) = eN¨k+1|k ·(x−1) = e(Nk+1|k +N
G
.
(18.129)
Then the λ-CPHD filter reduces to its PHD filter special case, the λ-PHD filter.
The filtering equations for this filter are:
• λ-PHD filter time-update:
Dk+1|k (x)
=
bk+1|k (x)
(18.130)
∫
+ pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
˚k+1|k
N
=
˚B
˚
N
k+1|k + p̊S · Nk|k .
(18.131)
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579
• λ-PHD filter measurement-update:
Dk+1|k+1 (x)
Dk+1|k (x)
+
˚k+1|k+1
N
˚k+1|k
N
(18.132)
1 − pD (x)
=
∑
pD (x) · Lz (x)
ˆ
z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)
(18.133)
1 − p̊D
=
+
∑
p̊D · ck+1 (z)
ˆ
z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)
where
18.4.7
ˆ k+1
λ
=
τk+1 (z)
=
p̊D · Nk+1|k
∫
pD (x) · Lz (x) · Dk+1|k (x)dx.
(18.134)
(18.135)
λ-CPHD Filter Implementation: Gaussian Mixtures
Equation (18.114) has the same general form as the measurement-update equation
for the classical PHD filter. Consequently, the λ-CPHD filter can be implemented
using Gaussian mixtures, in essentially the same way as the classical PHD filter.
That is, the PHDs are approximated as
νk|k
Dk|k (x)
=
∑
k|k
k|k
(18.136)
· NP k|k (x − xi )
wi
i
i=1
νk+1|k
Dk+1|k (x)
=
∑
k+1|k
wi
k+1|k
· NP k+1|k (x − xi
).
(18.137)
i
i=1
18.4.7.1
GM-λ-CPHD Filter: Models
The GM implementation of the λ-CPHD filter requires the following:
• The probability of target survival pS (x) = pS is constant.
• The target Markov density fk+1|k (x|x′ ) is linear-Gaussian:
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(18.138)
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• The birth-target PHD bk+1|k (x) is a Gaussian mixture:
B
νk+1|k
bk+1|k (x)
=
∑
k+1|k
k+1|k
· NB k+1|k (x − bi
bi
)
(18.139)
i
i=1
B
νk+1|k
B
Nk+1|k
=
∑
k+1|k
bi
(18.140)
.
i=1
• Cardinality distribution for target appearance: pB
k+1|k (n), with
B
Nk+1|k
=
∑
n · pB
k+1|k (n).
(18.141)
n≥0
• Clutter-generator probability of survival is constant: p̊S abbr.
= pS,k+1 .
• Cardinality distribution for clutter-generator appearance: p̊k+1|k (n̊) with
˚B
N
k+1|k =
∑
n · p̊B
k+1|k (n̊).
(18.142)
n̊≥0
• Target probability of detection is constant: pD (x) = pD (as usual, this
assumption can be removed using the approximation described in Section
9.5.6).
• Sensor likelihood function is linear-Gaussian:
Lz (x) = fk+1 (z|x) = NRk+1 (z − Hk+1 x).
(18.143)
• Clutter-generator probability of detection is constant: p̊D abbr.
= p̊D,k+1 .
• Clutter spatial distribution: ck+1 (z).
18.4.7.2
GM-λ-CPHD Filter: Time Update
We are given:
˚k|k ,
p̈k|k (n̈), N
k|k
k|k
k|k
(ℓi , wi , Pi
k|k ν
k|k
, xi )i=1
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581
with
νk|k
Nk|k =
∑
k|k
(18.144)
wi
i=1
and we are to determine formulas for
˚k+1|k ,
N
p̈k+1|k (n̈),
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
k+1|k νk+1|k
)i=1 .
, xi
These are:
• Time updated cardinality distribution:
∑
p̈k+1|k (n̈) =
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.145)
n̈′ ≥0
p̈k+1|k (n̈|n̈′ )
=
n̈
∑
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i
(18.146)
i=0
ψ¨k
=
′
·ψ¨ki (1 − ψ¨k )n̈ −i
∫
˚k|k
pS (x′ ) · Dk|k (x′ )dx′ + p̊S · N
(18.147)
˚k|k
Nk|k + N
where as in (18.29),
p̈B
k+1|k (n̈) =
∑
B
pB
k+1|k (n) · p̊k+1|k (n̊).
(18.148)
n+n̊=n̈
• Time updated expected number of clutter generators:
˚k+1|k = N
˚B
˚
N
k+1|k + p̊S · Nk|k .
(18.149)
• Time updated number of GM components in the PHD:
B
νk+1|k = νk|k + νk+1|k
.
(18.150)
Here, there are νk|k components corresponding to persisting targets and
B
νk+1|k
components corresponding to newly appearing targets. The timeupdate components are indexed as follows:
i
i
=
1, ..., νk|k
(persisting)
(18.151)
=
B
νk|k + 1, ..., νk|k + νk+1|k
(appearing).
(18.152)
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• Persisting-target GM components, for i = 1, ..., νk|k :
k+1|k
ℓi
k|k
=
k+1|k
wi
k+1|k
xi
k+1|k
Pi
=
=
=
ℓi
(18.153)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk .
(18.154)
(18.155)
(18.156)
B
• Appearing-target GM components, for i = νk|k + 1, ..., νk|k + νk+1|k
:
k+1|k
ℓi
k+1|k
wi
=
new label
(18.157)
=
k+1|k
bi−νk|k
(18.158)
k+1|k
=
bi−νk|k
(18.159)
=
k+1|k
Bi−νk|k .
(18.160)
k+1|k
xi
k+1|k
Pi
18.4.7.3
GM-λ-CPHD Filter: Measurement Update
We are given:
˚k+1|k ,
p̈k+1|k (n̈), N
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
k+1|k νk+1|k
)i=1
, xi
with
νk+1|k
Nk+1|k =
∑
k+1|k
wi
(18.161)
.
i=1
We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | =
mk+1 . We are to determine formulas for
p̈k+1|k+1 (n̈),
˚k+1|k+1 ,
N
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1
, Pi
k+1|k+1 νk+1|k+1
)i=1
.
, xi
RFS Filters for Unknown Clutter
583
• Measurement updated joint cardinality distribution and p.g.f.:
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈)
=
ϕ¨k
=
¨ k+1|k (ẍ)
G
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
(18.162)
n̈−m
Cn̈,m · ϕ¨k k+1
(18.163)
( ∫
)
(1 − pD (x)) · Dk+1|k (x)dx
˚k+1|k
+(1 − p̊D ) · N
(18.164)
˚k+1|k
Nk+1|k + N
∑
p̈k+1|k (n̈) · ẍn̈ .
(18.165)
n̈≥0
• Measurement updated expected number of clutter generators:
˚k+1|k+1
N
˚k+1|k
N
=
¨ (mk+1 +1) (ϕ¨k )
G
1 − p̊D
k+1|k
·
(m
)
˚
¨
Nk+1|k + Nk+1|k
G k+1 (ϕ¨ )
k+1|k
+
(18.166)
k
∑
p̊D · ck+1 (z)
.
ˆ
z∈Zk+1 λk+1 · ck+1 (z) + τk+1 (z)
• Estimated clutter rate:
ˆ k+1 = p̊D · N
˚k+1|k .
λ
(18.167)
• Estimated clutter cardinality distribution:
p̂κk+1 (m) =
˜m
λ
k+1
˜ k+1 )
¨ (m) (1 − λ
·G
k+1|k
m!
(18.168)
ˆ k+1
λ
.
˚k+1|k
Nk+1|k + N
(18.169)
where
˜ k+1 =
λ
• Measurement updated number of GM components for the PHD:
νk+1|k+1 = νk+1|k + mk+1 · νk+1|k
(18.170)
where there are νk+1|k components for undetected tracks and mk+1 · νk+1|k
components for detected tracks. The measurement-update components are
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Advances in Statistical Multisource-Multitarget Information Fusion
indexed as follows:
(undetected)
i
=
1, ..., νk+1|k
i
=
1, ..., νk+1|k ; j = 1, ..., mk+1 (detected).
(18.171)
(18.172)
• Measurement updated nondetection components: for i = 1, ..., νk+1|k ,
k+1|k=1
ℓi
k+1|k
=
k+1|k+1
wi
¨ (mk+1 +1) (ϕ¨k )
G
1 − pD
k+1|k
k+1|k
=
·
· wi
(18.174)
˚k+1|k
¨ (mk+1 ) (ϕ¨k )
Nk+1|k + N
G
k+1|k
k+1|k+1
xi
k+1|k+1
Pi
(18.173)
ℓi
k+1|k
=
xi
(18.175)
=
k+1|k
Pi
.
(18.176)
• Measurement updated nondetection components: for i = 1, ..., νk+1|k and
j = 1, ..., mk+1 ,
k+1|k=1
ℓi,j
k+1|k
=
(18.177)
ℓi
νk+1|k
τk+1 (zj )
=
∑
pD
k+1|k
(18.178)
wi
i=1
k+1|k
·NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
(zj − Hk+1 xi
)
k+1|k
k+1|k+1
wi,j
NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
(zj − Hk+1 xi
)
(18.179)
=
ˆ k+1 · ck+1 (zj ) + τk+1 (zj )
λ
k+1|k
pD · w i
k+1|k+1
xi,j
k+1|k
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
)
(18.180)
(
)
k+1|k
= I − Kik+1 Hk+1 Pi
(18.181)
(
−1
)
k+1|k T
T
= Pik+1|k Hk+1
Hk+1 Pi
Hk+1 + Rk+1
(18.182)
.
=
k+1|k+1
Pi,j
Kik+1
• Measurement updated target cardinality distribution:
pk+1|k+1 (n) =
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 )
n!
(18.183)
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585
Nk+1|k+1
˚k+1|k+1
Nk+1|k+1 + N
(18.184)
where
rk+1
=
νk+1|k
Nk+1|k+1
=
∑
νk+1|k mk+1
k+1|k+1
wi
+
i=1
18.5
∑ ∑
i=1
k+1|k+1
wi,j
. (18.185)
j=1
THE κ-CPHD FILTER
The κ-agnostic CPHD filter, or κ-CPHD filter for short, was proposed by Mahler et
al. in 2010 [189]. Whereas the λ-CPHD filter recursively estimates only the clutter
cardinality distribution pκk+1 (m) (and thus also the clutter rate λk+1 ), the κCPHD filter also recursively estimates the entire clutter intensity function κk+1 (z).
It consists of three coupled filters:
... →
p̈k|k (n̈)
... →
Dk|k (x)
... →
˚k|k (c, c)
D
→
↑
→
↑
→
p̈k+1|k (n̈)
¨ k+1|k (x)
D
˚k+1|k (c, c)
D
→
↑↓
→
↑↓
→
p̈k+1|k+1 (n̈)
→ ...
Dk+1|k+1 (x)
→ ...
˚k+1|k+1 (c, c)
D
→ ...
The top filter propagates the joint probability distribution p̈k|k (n̈) on the number
n̈ = n + n̊, where n is the number of targets and n̊ is the number of clutter
generators. The middle filter propagates the PHD Dk|k (x) on targets. The bottom
˚k|k (c, c) on augmented clutter generators, where c
filter propagates the PHD D
denotes the (unknown) probability of detection of the clutter generator with state
c.
As with the λ-CPHD filter, it is possible to derive from p̈k|k (n̈) the
cardinality distribution pk|k (n) for targets and the cardinality distribution p̊k|k (n̊)
for clutter generators—see Section 18.5.4.
The section is organized as follows:
1. Section 18.5.1: Models for the κ-CPHD filter.
2. Section 18.5.2: Time update equations for the λ-CPHD filter.
3. Section 18.5.3: Measurement update equations for the κ-CPHD filter.
4. Section 18.5.4: Multitarget state estimation for the κ-CPHD filter.
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Advances in Statistical Multisource-Multitarget Information Fusion
5. Section 18.5.5: Clutter estimation for the κ-CPHD filter.
6. Section 18.5.6: The κ-PHD filter special case of the κ-CPHD filter.
7. Section 18.5.7: Beta-Gaussian mixture (BGM) implementation of the κCPHD filter.
8. Section 18.5.8: Normal-Wishart mixture (NWM) implementation of the κCPHD filter.
18.5.1
κ-CPHD Filter: Models
The clutter state space is ˚
C = [0, 1] × C, and a clutter state has the form c̊ = (c, c).
As with the λ-CPHD filter, we presume the nonintermixing motion model of
Section 18.2.2:
pT (x′ )
p̊T (c̊′ )
=
=
(18.186)
(18.187)
1
1.
That is, targets transition only to targets, and clutter generators transition only to
clutter generators. The κ-CPHD filter thereby requires the following models:
• Target probability of survival: pS (x) abbr.
= pS,k+1 (x).
• Target Markov density: fk+1|k (x|x′ ).
• PHD for target appearance: bk+1|k (x).
• Cardinality distribution for target appearance: pB
k+1|k (n), with
B
Nk+1|k
=
∫
bk+1|k (x)dx =
∑
n · pB
k+1|k (n).
(18.188)
n≥0
• Clutter-generator probability of survival does not depend on the clutter
probability of detection: p̊S (c, c) = p̊S (c) abbr.
= pS,k+1 (c).
• Clutter-generator Markov density: f˚k+1|k (c|c′ ).
• Markov density for clutter probability of detection: f˚k+1|k (c|c′ ), defined as
in (17.12) through (17.16).
• PHD for appearance of clutter generators does not depend on the clutter
probability of detection: ˚
bk+1|k (c, c) = ˚
bk+1|k (c).
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587
• Cardinality distribution for clutter-generator appearance: p̊k+1|k (n̊) with
˚B
N
k+1|k =
∫
˚
bk+1|k (c)dc =
∑
n · p̊B
k+1|k (n̊).
(18.189)
n̊≥0
• Target probability of detection: pD (x) abbr.
= pD,k+1 (x).
• Target likelihood function: Lz (x) abbr.
= fk+1 (z|x).
κ
• Clutter-generator likelihood function: ˚
Lκz (c, c) = Lκz (c) abbr.
= fk+1
(z|c)—
likelihood that measurement z will be generated if a clutter generator with
state c is present and detected.
Remark 75 (Interpretation of clutter likelihood function) Lκz (c) is a family of
“elemental” clutter models parametrized by c. Selection of c specifies which
elemental model governs the generation of a measurement. For example, c could
define a rectangular solid at a particular location—a simple model of a building.
In this case ˚
Lz (c) would be the spatial distribution of any measurement generated
by the building. Or, c could be a point in three-dimensional space—a simple
model of wind-driven point clutter. Then ˚
Lz (c) would in this case be the spatial
distribution of any measurement generated by the point clutter generator.
As with the λ-CPHD filter, for computational reasons it may sometimes
prove convenient to assume that clutter generators cannot appear (though this is
not assumed in general):
˚
bk+1|k (x)
=
0
(18.190)
p̊B
k+1|k (n̊)
=
δ0,n̊
(18.191)
in which case
p̈B
k+1|k (n̈) =
∑
B
B
pB
k+1|k (n) · p̊k+1|k (n̊) = pk+1|k (n̈).
(18.192)
n+n̊=n̈
18.5.2
κ-CPHD Filter: Time Update
The time-update formulas for the κ-CPHD filter result from substituting the models
in Section 18.5.1 into (18.36) through (18.43). We are given the joint cardinality
˚k|k (c, c).
distribution p̈k|k (n̈), the target PHD Dk|k (x), and the clutter PHD D
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˚k+1|k (c, c). Let
We are to find the time-updates p̈k+1|k (n̈), Dk+1|k (x), and D
Nk|k
=
˚k|k
N
=
∫
Dk|k (x)dx
∫ ∫ 1
˚k|k (c, c)dcdc.
D
(18.193)
(18.194)
0
Then:
• Time updated cardinality distribution: As with the λ-CPHD filter,
p̈k+1|k (n̈)
∑
=
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.195)
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i
(18.196)
n̈′ ≥0
p̈k+1|k (n̈|n̈′ )
n̈
∑
=
i=0
ψ¨k
′
·ψ¨ki (1 − ψ¨k )n̈ −i
∫
(
)
p (x) · Dk|k (x)dx
∫ ∫1 S
˚k|k (c, c)dcdc
+ 0 p̊S (c) · D
(18.197)
˚k|k
Nk|k + N
=
whereas in (18.29),
p̈B
k+1|k (n̈) =
∑
B
pB
k+1|k (n) · p̊k+1|k (n̊).
(18.198)
n+n̊=n̈
• Time updated target PHD: As with the λ-CPHD filter,
Dk+1|k (x) = bk+1|k (x) +
∫
pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ . (18.199)
• Time updated clutter PHD: Substituting (18.16) into (18.24), and applying
(18.5), results in
˚k+1|k (c, c) = ˚
D
bk+1|k (c)
(18.200)
∫ ∫ 1
¨ k|k (c′ , c′ )dc′ dc′ .
+
p̊S (c′ ) · f˚k+1|k (c|c′ ) · f˚k+1|k (c|c′ ) · D
0
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18.5.3
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κ-CPHD Filter: Measurement Update
Substituting the models in Section 18.5.1 into (18.44) through (18.53), we get the
measurement-update equations for the κ-CPHD filter. We are given the joint
cardinality distribution p̈k+1|k (n̈), the target PHD Dk+1|k (x), the clutter PHD
˚k+1|k (c, c), and a new measurement set Zk+1 with |Zk+1 | = mk+1 . We are to
D
˚k+1|k+1 (c, c).
find the measurement-updates p̈k+1|k+1 (n̈), Dk+1|k+1 (x), and D
We are also to arrive at an estimate κ̂k+1 (z) of the clutter intensity function and
an estimate p̂κk+1 (m) of the clutter cardinality distribution. Let
Nk+1|k
=
˚k+1|k
N
=
∫
(18.201)
Dk+1|k (x)dx
∫ ∫ 1
˚k+1|k (c, c)dcdc.
D
(18.202)
0
Then
• Measurement updated joint cardinality distribution and p.g.f.:
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈)
=
ϕ¨k+1
=
¨ k+1|k (ẍ)
G
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
(18.203)
n̈−m
Cn̈,mk+1 · ϕ¨k+1 k+1
(18.204)
 ∫

(1 − pD (x)) · Dk+1|k (x)dx
∫ ∫1


+ 0 (1 − c)
˚
·Dk+1|k (c, c)dcdc
(18.205)
˚k+1|k
Nk+1|k + N
∑
p̈k+1|k (n̈) · ẍn̈ .
(18.206)
n̈≥0
• Measurement updated target PHD:
Dk+1|k+1 (x)
Dk+1|k (x)
=
¨ (mk+1 +1) (ϕ¨k+1 )
G
1 − pD (x)
k+1|k
·
(18.207)
˚k+1|k
¨ (mk+1 ) (ϕ¨ )
Nk+1|k + N
G
k+1|k
+
∑
z∈Zk+1
pD (x) · Lz (x)
κ̂k+1 (z) + τk+1 (z)
k+1
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where Cn̈,m was defined in (2.1) and where
κ̂k+1 (z)
τk+1 (z)
=
=
∫ ∫ 1
∫
˚k+1|k (c, c)dcdc
c · Lκz (c) · D
(18.208)
0
(18.209)
pD (x) · Lz (x) · Dk+1|k (x)dx.
• Measurement updated expected clutter PHD:
˚k+1|k+1 (c, c)
D
˚k+1|k (c, c)
D
=
1−c
˚k+1|k
Nk+1|k + N
(18.210)
¨ (mk+1 +1) (ϕ¨k+1 )
G
k+1|k
·
+
¨ (mk+1 ) (ϕ¨k+1 )
G
k+1|k
∑
c · Lκ (c)
z
.
κ̂k+1 (z) + τk+1 (z)
z∈Zk+1
Remark 76 (Computational complexity) The computational complexity of the κCPHD filter is the same as that of the classical PHD filter: O(mn) where m is
the current number of measurements and n is the current number of tracks.
18.5.4
κ-CPHD Filter: Multitarget State Estimation
State estimation is accomplished as described in Section 18.3.4. In particular, the
measurement-updated cardinality distributions for targets and for clutter were given
by Eqs. (18.62) and (18.64):
pk+1|k+1 (n)
=
p̊k+1|k+1 (n̊)
=
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 )
n!
(1 − rk+1 )n̊ ¨ (n̊)
· Gk+1|k+1 (rk+1 )
n̊!
(18.211)
(18.212)
where
rk+1 =
Nk+1|k+1
.
˚k+1|k+1
Nk+1|k+1 + N
(18.213)
Thus, the multitarget state can be estimated as in Section 18.3.4.2. Determine the
MAP estimate
n̂k+1|k+1 = arg sup pk+1|k+1 (n)
(18.214)
n
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591
and then determine the target states corresponding to the n̂k+1|k+1 largest peaks
of Dk+1|k+1 (x).
18.5.5
κ-CPHD Filter: Clutter Estimation
The κ-CPHD filter can estimate both the clutter cardinality distribution and the
clutter intensity function. The estimated clutter process turns out to be the predicted clutter process. It is i.i.d.c., and is therefore completely determined by its
cardinality distribution and spatial distribution. The clutter intensity function and
its derived quantities are given by:
• Estimated clutter intensity function:
κ̂k+1 (z) =
∫ ∫ 1
˚k+1|k (c, c)dcdc.
c · Lκz (c) · D
(18.215)
0
(which is, note, the same as (18.208)).
• Estimated clutter rate:
ˆ k+1 =
λ
∫ ∫ 1
˚k+1|k (c, c)dcdc.
c·D
(18.216)
0
• Estimated clutter spatial distribution:
ĉk+1 (z) =
κ̂k+1 (z)
.
ˆ k+1
λ
(18.217)
Equation (18.215) follows from (18.81),
κ̂k+1 (z|Z
(k)
)=
∫
˚k+1|k (c̊)dc̊.
p̊κD (c̊) · ˚
Lκz (c̊) · D
(18.218)
To see why, notice that for c̊ = (c, c), we set p̊κD (c̊) = c and ˚
Lκz (c̊) = Lκz (c) in
which case
κ̂k+1 (z|Z
(k)
)=
∫ ∫ 1
˚k+1|k (c, c)dcdc.
c · Lκz (c) · D
0
As for the cardinality distribution, note that from (18.93),
(18.219)
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• Estimated clutter cardinality distribution:
p̂k+1 (m|Z (k) ) =
˜m
λ
k+1|k
m!
˜ k+1|k )
¨ (m) (1 − λ
·G
k+1|k
where, since p̊D (c̊) = c,
˜ k+1
λ
=
=
=
ˆ k+1 (Z (k) )
λ
˚k+1|k
Nk+1|k + N
∫
˚k+1|k (c̊|Z (k) )dc̊
p̊D (c̊) · D
˚k+1|k
Nk+1|k + N
∫ ∫1
˚k+1|k (c, c|Z (k) )dcdc
c·D
0
.
˚k+1|k
Nk+1|k + N
(18.220)
(18.221)
(18.222)
Remark 77 Recall from the discussion in Section 18.3.5 that κ̂k+1 (z) has not
been chosen to be the estimated clutter intensity function simply because it appears
in the “right place” in (18.207). Rather, it has been chosen because it can be
provably shown to be the predicted average clutter intensity function—see Section
18.3.5.
Remark 78 (Infinite mixture interpretation) Equation (18.215) can be regarded
as an infinite mixture of the parametrized elemental clutter models ˚
Lz (c). To see
why, first suppose that Dk+1|k (c, c) has the form
Dk+1|k (c, c) = δc1 (c) · δc1 (c) + ... + δcν (c) · δcν (c)
(18.223)
with 0 ≤ c1 , ..., cν ≤ 1. That is, the unknown clutter generators and their unknown
probabilities of detection are precisely known. Then the estimated clutter intensity
function is a finite mixture of the parametrized elemental clutter models ˚
Lz (c):
κ̂k+1 (z)
=
∫ ∫ 1
c·˚
Lz (c) · Dk+1|k (c, c)dcdc
(18.224)
0
=
c1 · ˚
Lz (c1 ) + ... + cν · ˚
Lz (cν ).
(18.225)
Thus, intuitively speaking, the clutter RFS is an infinite superposition of elemental
clutter contributors. It is this representation of κk+1 (z) as an infinite mixture
that potentially allows the κ-CPHD filter to estimate quite complicated clutter
processes.
RFS Filters for Unknown Clutter
18.5.6
593
Special Case: The κ-PHD Filter
The κ-PHD filter is unlikely to be of practical interest, since the κ-CPHD filter will
perform better while having approximately the same computational complexity. It
results when we assume that the predicted multitarget distribution is Poisson, in
which case the predicted p.g.f. becomes
˚
Gk+1|k (ẍ) = e(Nk+1|k +Nk+1|k )·(ẍ−1) .
(18.226)
Given this, the κ-CPHD filter time-update and measurement-update equations reduce to:
• Target-PHD time-update:
Dk+1|k (x) = bk+1|k (x) +
∫
pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ . (18.227)
• Clutter-PHD time-update:
˚k+1|k (c, c)
D
=˚
bk+1|k (c) +
∫ ∫ 1
p̊S (c′ ) · f˚k+1|k (c|c′ ) (18.228)
0
¨ k|k (c′ , c′ )dc′ dc′ .
·f˚k+1|k (c|c′ ) · D
• Target-PHD measurement-update:
∑
Dk+1|k+1 (x)
pD (x) · Lz (x)
= 1 − pD (x) +
Dk+1|k (x)
κ̂k+1 (z) + τk+1 (z)
z∈Zk+1
where
τk+1 (z)
=
κ̂k+1 (z)
=
∫
pD (x) · Lz (x) · Dk+1|k (x)dx
∫ ∫ 1
˚k+1|k (c, c)dadc.
c · Lκz (c) · D
(18.229)
0
• Clutter-PHD measurement-update:
˚k+1|k+1 (c, c)
∑
D
c · Lκz (c)
=1−c+
.
˚k+1|k (c, c)
κ̂k+1 (z) + τk+1 (z)
D
z∈Zk+1
(18.230)
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Advances in Statistical Multisource-Multitarget Information Fusion
18.5.7
κ-CPHD Filter: Beta-Gaussian Mixtures
The clutter PHDs in the κ-CPHD filter can be approximated as beta-Gaussian
mixtures in the same manner as in (17.60) and (17.61):
ν̊k|k
˚k|k (c, c)
D
∑
=
k|k
ẘi
k|k
· βrk|k ,sk|k (c) · NC k|k (c − ci ) (18.231)
i
i
i
i=1
ν̊k+1|k
˚k+1|k (c, c)
D
∑
=
k+1|k
(18.232)
· βrk+1|k ,sk+1|k (c)
ẘi
i
i
i=1
k+1|k
·NC k+1|k (c − ci
).
i
Likewise, target PHDs can be approximated as Gaussian mixtures in the usual way:
νk|k
Dk|k (x)
∑
=
k|k
k|k
(18.233)
· NP k|k (x − xi )
wi
i
i=1
νk+1|k
Dk+1|k (x)
∑
=
k+1|k
k+1|k
· NP k+1|k (x − xi
wi
).
(18.234)
i
i=1
Consequently, propagation of these two PHDs can be replaced by propagation of
systems of the form
p̈k|k (n̈),
k|k
k|k
k|k
(ℓi , wi , Pi
k|k
k|k
k|k
k|k ν
k|k
, xi )i=1
,
k|k
k|k ν̊
k|k
( ẘi , ri , si , Ci , ci )i=1
.
Furthermore, it turns out that:
• The estimate κ̂k+1 (z) of the clutter intensity function is a Gaussian mixture—see (18.279).
Remark 79 Notice that the BGM components for clutter generators are not labeled, because—unlike the case for the GM components for targets—there is no
need to do so.
In this section, the formulas for the BGM-κ-CPHD filter are summarized.
RFS Filters for Unknown Clutter
18.5.7.1
595
BGM-κ-CPHD Filter: Models
The BGM-κ-CPHD filter requires the following models:
• Target probability of survival is assumed constant: pS (x) abbr.
= pS .
• Target Markov density is linear-Gaussian:
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(18.235)
• PHD for target appearance is a Gaussian mixture:
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
bi
k+1|k
· NB k+1|k (x − bi
).
(18.236)
i
i=1
• Cardinality distribution for target appearance: pB
k+1|k (n), with
B
νk+1|k
B
Nk+1|k
=
∑
k+1|k
bi
=
i=1
∑
n · pB
k+1|k (n).
(18.237)
n≥0
• Clutter-generator probability of survival is constant: p̊S (c, c) = p̊S .
• Clutter-generator Markov density is linear-Gaussian:
˚ ′
f˚k+1|k (c|c′ ) = NQ
˚k (c − Fk c ).
(18.238)
• Markov density for clutter probability of detection: f˚k+1|k (c|c′ ), defined as
in (17.12) through (17.16).
• PHD for appearance of clutter generators is a BGM:
B
νk+1|k
˚
bk+1|k (c) =
∑
k+1|k
˚
˚k+1|k ).
bi
· NB
˚k+1|k (c − bi
(18.239)
i
i=1
• Cardinality distribution for clutter-generator appearance: p̊k+1|k (n̊) with
B
νk+1|k
˚B
N
k+1|k =
∑
i=1
k+1|k
˚
bi
=
∑
n̊≥0
n · p̊B
k+1|k (n̊).
(18.240)
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Advances in Statistical Multisource-Multitarget Information Fusion
• Target probability of detection is constant: pD (x) = pD .
• Target likelihood function is linear-Gaussian:
Lz (x) = NRk+1 (z − Hk+1 x).
(18.241)
• Elemental clutter-generator model (clutter likelihood function) is a Gaussian
mixture:
Lκz (c)
=
k+1
ν̊∑
˚ k+1 c)
ek+1
· NR
˚k+1 (z − Hi
i
(18.242)
ek+1
= 1.
i
(18.243)
i
i=1
k+1
ν̊∑
i=1
The approximation of Lκz (c) as a Gaussian mixture permits the BGM-κCPHD filter to estimate more general clutter processes.
18.5.7.2
BGM-κ-CPHD Filter: Time Update
We are given the BGM system
p̈k|k (n̈),
k|k
k|k
k|k
(ℓi , wi , Pi
k|k
k|k
k|k ν
k|k
, xi )i=1
,
k|k
k|k
k|k ν̊
k|k
(ẘi , ri , si , Ci , ci )i=1
and we are to determine the formulas for the BGM system
p̈k+1|k (n̈),
k+1|k νk+1|k
)i=1 ,
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k ν̊k+1|k
(ẘi
, ri
, si
, Ci
, ci
)i=1 .
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
, xi
Let
νk|k
Nk|k
=
∑
k|k
wi
(18.244)
i=1
ν̊k|k
˚k|k
N
=
∑
i=1
k|k
ẘi .
(18.245)
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597
Then
• Time updated cardinality distribution:
∑
p̈k+1|k (n̈) =
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.246)
n̈′ ≥0
p̈k+1|k (n̈|n̈′ )
n̈
∑
=
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i
(18.247)
i=0
′
·ψ¨ki (1 − ψ¨k )n̈ −i
=
˚k|k
pS · Nk|k + p̊S · N
˚k|k
Nk|k + N
(18.248)
∑
B
pB
k+1|k (n) · p̊k+1|k (n̊).
(18.249)
ψ¨k
whereas in (18.29),
p̈B
k+1|k (n̈) =
n+n̊=n̈
B
There are νk+1|k = νk|k +νk+1|k
Gaussian components for the time-updated
B
target PHD and ν̊k+1|k = ν̊k|k + ν̊k+1|k
BGM components for the time-updated
clutter PHD.
• Time updated persisting-target components—for i = 1, ..., νk|k :
k+1|k
ℓi
k|k
=
k+1|k
wi
k+1|k
xi
k+1|k
Pi
=
=
=
ℓi
(18.250)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk .
(18.251)
(18.252)
(18.253)
• Time updated appearing-target components—for i = νk|k + 1, ..., νk|k +
B
νk+1|k
:
k+1|k
ℓi
k+1|k
wi
k+1|k
xi
=
new labels
(18.254)
=
k+1|k
bi−νk|k
(18.255)
=
k+1|k
bi−νk|k
(18.256)
k+1|k
=
Bi−νk|k .
(18.257)
k+1|k
Pi
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Advances in Statistical Multisource-Multitarget Information Fusion
• Time updated expected number of targets:
B
νk|k +νk+1|k
∑
Nk+1|k =
k+1|k
wi
(18.258)
.
i=1
• Time updated persisting-generator components—for l = 1, ..., ν̊k|k :
k+1|k
k|k
ẘl
=
k+1|k
cl
k+1|k
Cl
k+1|k
rl
k+1|k
sl
=
p̊S · ẘl
˚k ck|k
F
(18.259)
˚k C k|k F
˚kT + Q
˚k
F
l
k|k ˚k|k
rl · θ l
k|k
k|k
sl · ˚
θl
(18.261)
(18.260)
l
=
=
=
(18.262)
(18.263)
where
k|k
˚
θl =
1
k|k
rl
k|k
·
(
+ sl
k|k k|k
rl s l
1
·
−1
k|k
k|k 2 σ̊ 2
(r + s )
l
l
)
(18.264)
l
and where the predicted variance σ̊l2 is chosen subject to
(
)
k|k k|k
rl s l
1
k+1|k
2
σ̊l =
+
ε̊
·
(18.265)
l
k|k
k|k
k|k
k|k
k|k
k|k
rl + s l
(rl + sl )(rl + sl + 1)
for some
k+1|k
0 ≤ ε̊l
≤ 1.
(18.266)
Time updated appearing-generator components—for l = ν̊k|k + 1, ..., ν̊k|k +
B
ν̊k+1|k
:
k+1|k
ẘl
k+1|k
cl
k+1|k
Cl
k+1|k
rl
k+1|k
sl
k+1|k
= ˚
bl−ν̊k|k
(18.267)
=
k+1|k
˚
bl−ν̊k|k
(18.268)
=
˚k+1|k
B
l−ν̊k|k
(18.269)
=
1
(18.270)
=
1.
(18.271)
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599
• Time updated expected number of clutter generators:
B
ν̊k|k +ν̊k+1|k
˚k+1|k =
N
∑
k+1|k
ẘi
.
(18.272)
i=1
18.5.7.3
BGM-κ-CPHD Filter: Measurement Update
We are given the BGM system
p̈k+1|k (n̈),
k+1|k νk|k
)i=1 ,
k+1|k
k+1|k
k+1|k
k+1|k
k+1|k ν̊k+1|k
(ẘi
, ri
, si
, Ci
, ci
)i=1
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
, xi
Given a new measurement set Zk+1 with |Zk+1 | = mk+1 , we are to determine
the formulas for the BGM system
p̈k+1|k+1 (n̈),
k+1|k+1 νk+1|k+1
)i=1
,
k+1|k+1
k+1|k+1
k+1|k+1
k+1|k+1
k+1|k+1 ν̊k+1|k+1
(ẘi
, ri
, si
, Ci
, ci
)i=1
.
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1
, Pi
, xi
We are also to determine the formulas for the measurement-updated target cardinality distribution pk+1|k+1 (n) and the estimated clutter intensity function κ̂k+1 (z)
and the estimated clutter cardinality distribution pκk+1 (m).
Let
νk+1|k
Nk+1|k
=
∑
k+1|k
(18.273)
wi
i=1
ν̊k+1|k
˚k+1|k
N
=
∑
i=1
Then:
k+1|k
ẘi
.
(18.274)
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Advances in Statistical Multisource-Multitarget Information Fusion
• Measurement updated joint target/clutter cardinality distribution and p.g.f.:
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈)
=
ϕ¨k
=
¨ k+1|k (ẍ)
G
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
n̈−m
Cn̈,m · ϕ¨k k+1
(
)
(1 − pD ) · Nk+1|k
∑ν̊k+1|k ẘik+1|k ·sk+1|k
i
+ i=1
k+1|k
k+1|k
ri
+si
(18.275)
(18.276)
(18.277)
˚k+1|k
Nk+1|k + N
∑
p̈k+1|k (n̈) · ẍn̈ .
(18.278)
n̈≥0
• Estimated clutter intensity function:
ν̊k+1|k ν̊ k+1
κ̂k+1 (z)
∑ ∑ ẘ k+1|k · ek+1 · r k+1|k
i
i
l
=
k+1|k
i=1
ri
l=1
(18.279)
k+1|k
+ si
˚ k+1 ck+1|k ).
·NR
˚k+1 +H
˚ k+1 C k+1|k (H
˚ k+1 )T (z − Hl
i
l
i
l
l
• Estimated clutter rate:
ν̊k+1|k ν̊ k+1
ˆ k+1 =
λ
∑ ∑ ẘ k+1|k · ek+1 · r k+1|k
i
i
l
k+1|k
i=1
ri
l=1
.
(18.280)
k+1|k
+ si
• Estimated clutter cardinality distribution:
p̂κk+1 (m) =
˜ k+1
λ
˜ k+1 )
¨ (m) (1 − λ
·G
k+1|k
m!
(18.281)
ˆ k+1
λ
.
˚k+1|k
Nk+1|k + N
(18.282)
where
˜ k+1 =
λ
There are νk+1|k+1 = νk+1|k + mk+1 · νk+1|k Gaussian components for
the measurement-updated target PHD and ν̊k+1|k+1 = ν̊k+1|k + mk+1 · ν̊k+1|k ·
ν̊ k+1 BGM components for the measurement-updated clutter PHD (where, recall,
ν̊ k+1 is the number of components in the Gaussian mixture representation of the
elemental clutter model).
RFS Filters for Unknown Clutter
601
• Measurement updated undetected-target components—for i = 1, ..., νk+1|k :
k+1|k+1
ℓi
k+1|k
(18.283)
=
ℓi
=
¨ (mk+1 +1) (ϕ¨k )
G
1 − pD
k+1|k
k+1|k
·
· wi
(mk+1 ) ¨
˚
¨
Nk+1|k + Nk+1|k
Gk+1|k (ϕk )
=
xi
(18.285)
=
k+1|k
Pi
.
(18.286)
k+1|k+1
wi
(18.284)
k+1|k+1
xi
k+1|k+1
Pi
k+1|k
• Measurement updated detected-target components—for i = 1, ..., νk+1|k
and j = 1, ..., mk+1 :
k+1|k+1
ℓi,j
k+1|k+1
wi,j
k+1|k
=
ℓi
(18.287)
=
k+1|k
wi
· pD
(18.288)
k+1|k
NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
·
k+1|k
xi,j
(zj − Hk+1 xi
)
κ̂k+1 (zj ) + τk+1 (zj )
=
k+1|k
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
)
k+1|k
Pi,j
=
(
Kik+1
=
T
Pi
Hk+1
(
)−1
k+1|k T
· Hk+1 Pi
Hk+1 + Rk+1
τk+1 (zj )
=
pD
I − Kik+1 Hk+1
)
k+1|k
Pi
k+1|k
(18.289)
(18.290)
(18.291)
νk+1|k
∑
k+1|k
(18.292)
wi
i=1
k+1|k
·NR
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
(zj − Hk+1 xi
).
• Measurement updated undetected-generator components— l = 1, ..., ν̊k+1|k :
¨ (mk+1 +1) (ϕk )
G
k+1|k
k+1|k
k+1|k+1
ẘl
=
ẘl
˚k+1|k
Nk+1|k + N
k+1|k
s
· k+1|kl
k+1|k
rl
+ sl
·
(18.293)
¨ (mk+1 ) (ϕk )
G
k+1|k
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Advances in Statistical Multisource-Multitarget Information Fusion
k+1|k+1
cl
k+1|k
=
k+1|k+1
Cl
k+1|k+1
rl
k+1|k+1
sl
=
=
=
cl
(18.294)
k+1|k
Cl
k+1|k
rl
k+1|k
sl
+ 1.
(18.295)
(18.296)
(18.297)
• Measurement updated detected-generator components— i = 1, ..., ν̊k+1|k ,
l = 1, ..., ν̊ k+1 , and j = 1, ..., mk+1 :
k+1|k
k+1|k+1
ẘi,l,j
=
k+1|k
ẘi
· ek+1
r
l
· k+1|ki
k+1|k
κ̂k+1 (zj ) + τk+1 (zj ) r
+s
(18.298)
˚ k+1 (zj − H
˚ k+1 ck+1|k )
+K
i
l
i,l
(18.299)
i
i
k+1 k+1|k
˚
·NR
ci
)
˚k+1 +HC
˚ k+1|k (H
˚ k+1 )T (zj − Hl
i
l
l
k+1|k
ci,l,j
k+1|k
Ci,l,j
k+1|k+1
ri,l,j
k+1|k
=
cl
=
˚ k+1 H
˚ k+1 )C k+1|k
(I − K
i
i,l
l
(18.300)
=
k+1|k
ri
+1
(18.301)
k+1|k
=
si
(18.302)
k+1|k
si,l,j
where
(
)−1
˚ k+1 )T H
˚ k+1 C k+1|k (H
˚ k+1 )T + R
˚k+1
˚ k+1 = C k+1|k (H
. (18.303)
K
i
i
l
l
i,l
l
l
• Measurement updated target cardinality distribution:
pk+1|k+1 (n) =
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 )
n!
(18.304)
where
rk+1
=
Nk+1|k+1
=
Nk+1|k+1
˚k+1|k+1
Nk+1|k+1 + N
νk+1|k
∑
(18.305)
νk+1|k mk+1
k+1|k+1
wi
+
i=1
∑ ∑
i=1
k+1|k+1
wi,j
(18.306)
j=1
˚k+1|k+1
N
(18.307)
ν̊k+1|k
=
∑
l=1
ν̊k+1|k ν̊
k+1|k+1
ẘl
+
k+1 m
k+1
∑ ∑ ∑
i=1
l=1
j=1
k+1|k+1
ẘi,l,j
.
RFS Filters for Unknown Clutter
18.5.8
603
κ-CPHD Filter Implementation: Normal-Wishart Mixtures
The approximation approach described in this section was proposed in 2009 by
Chen Xin, Kirubarajan, Tharmarasa, and Pelletier ([37], Section 3) and further
refined in 2012 ([39], Section IV). The multitarget detection and tracking filter
described by these authors is essentially the κ-PHD filter of Section 18.5.6, except
that clutter probability of detection is presumed to be unity (c = 1).
Their implementation approach is similar to that of the BGM-κ-PHD filter, except that the clutter PHDs are approximated by normal-Wishart mixtures
(NWMs) rather than beta-Gaussian mixtures. The NWM approximation is theoretically and computationally more complex than the BGM approximation, but has
potentially attractive performance characteristics.
The NWM approach has one limitation: clutter estimation is not possible,
if such is desired. Although the formula for the estimated clutter intensity function κ̂k+1 (z) can be computed in closed form, the formula is analytically quite
complicated. As a result, it is not possible to derive a closed-form formula for the
estimated clutter cardinality distribution pκk+1 (m).
The purpose of this section is to describe the NWM approximation approach
and generalize it to the κ-CPHD filter.
In the NWM approach, the unknown clutter state c is assumed to have the
specific parametrization
´
c = (ć, C)
(18.308)
where
• ć ∈ Z = RM is an element of the M -dimensional measurement space.
• C´ = R−1 is the inverse of some M × M measurement-covariance matrix
R.
Thus the clutter state space C is a Euclidean space of dimension M (M +
3)/2. Given this, Chen et al. make the following assumptions about the motion and
measurement models for clutter:
• Clutter probability of detection—clutter generators are always detected:
p̊D (c) = 1.
(18.309)
In what follows, this will be slightly generalized by allowing p̊D to be
constant but otherwise arbitrary:
p̊D (c) = p̊D .
(18.310)
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• Clutter likelihood function—the measurements produced by a clutter generator are Gaussian-distributed, but with unknown mean and covariance
´ = N ´ −1 (z − ć).
Lκz (c) = Lκz (ć, C)
C
(18.311)
In what follows, this will be slightly generalized by defining
´ =N
Lκz (ć, C)
´ −1 (z − ć)
(η̊k+1 ·C)
(18.312)
for some η̊k+1 > 0.
´
• Clutter Markov transition density—in a transition from (ć′ , C´ ′ ) to (ć, C),
′
´
the matrix C does not change whereas the uncertainty in the mean ć′
increases with covariance proportional to the inverse of C´ ′ :
′
´ ′ , C´ ′ ) = N
´
f˚k+1|k (ć, C|ć
´ ′ )−1 (ć − ć ) · δC
´ ′ (C)
(φ̊k ·C
(18.313)
´ denotes the Dirac delta function on
for some φ̊k > 0, where δC´ ′ (C)
´
information matrices C, concentrated at the information matrix C´ ′ .
The approach in [39] was used to implement a NWM-κ-PHD filter. The most
significant innovation of this section will be to generalize this to a NWM-κ-CPHD
filter.
18.5.8.1
Normal-Wishart Mixtures (NWMs)
When the clutter likelihood function has the form of (18.242),
Lκz (c) =
k+1
ν̊∑
˚ k+1 c),
ek+1
· NR
˚k+1 (z − Hi
i
(18.314)
i
i=1
it is possible to implement the κ-CPHD filter using BGM approximation. When
the likelihood function has the form of (18.312), however, this is no longer possible.
The NWM approximation was designed to address this generalized likelihood
function, and is as follows.
An introduction to normal-Wishart distributions can be found in Appendix G.
These have the form
´
N Wd,o,o,O (ć, C)
RFS Filters for Unknown Clutter
605
where d, o, o, O are parameters, with o > 0, d > M , o ∈ Z = RM , and O
an M × M positive-definite matrix that has the same units as a measurementcovariance matrix R.
Normal-Wishart distributions satisfy the following two identities. First, the
following slight generalization of [39], Eq. (27):
´ · N Wd,o,o,O (ć, C)
´ = qz,d,o,o,O · N Wd∗ ,o∗ ,o∗ ,O∗ (ć, C)
´
Lκz (ć, C)
z
z
(18.315)
where
d∗
o∗
=
=
o∗z
=
Oz∗
=
qz,d,o,o,O
=
d+1
η̊k+1 + o
η̊k+1 · z + o · o
η̊k+1 + o
η̊k+1 · o
· (z − o)(z − o)T
O+
η̊k+1 + o
( ∗)
(η̊k+1 · o)M/2 · Γ d2 · (det O)d/2
( ∗
)
.
(π · o∗ )M/2 · Γ d −M
· (det Oz∗ )d∗ /2
2
(18.316)
(18.317)
(18.318)
(18.319)
(18.320)
Second ([39], Eq. (37)),
∫
´ ′ , C´ ′ ) · N Wd,o,o,O (ć′ , C´ ′ )dć′ dC´ ′ = N Wd,õ,o,O (ć, C)
´ (18.321)
f˚k+1|k (ć, C|ć
where
φ̊k · a
.
(18.322)
φ̊k + o
For the sake of conceptual completeness, both identities are verified in Appendix
G.
Given this, a normal-Wishart mixture (NWM) approximation of the clutter
´ has the form:
PHD Dk|k (ć, C)
õ =
νk|k
˚k|k (ć, C)
´ =
D
∑
k|k
wi
´
· N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C).
i
i
i
(18.323)
i
i=1
Because of (18.315) and (18.321), the κ-CPHD filter equations can be solved in
exact closed form. It thus becomes possible to propagate the system
k|k
k|k
k|k
k|k
k|k ν
k|k
(ℓi , di , oi , oi , Oi )i=1
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Advances in Statistical Multisource-Multitarget Information Fusion
´
of normal-Wishart parameters rather than propagate the clutter PHD Dk|k (ć, C)
itself. This process is described in the following subsections. For notational clarity,
labels ℓ will be suppressed.
18.5.8.2
NWM-κ-CPHD Filter: Models
The NWM-κ-CPHD filter requires the following models:
• Target probability of survival is constant: pS (x) abbr.
= pS .
• Target Markov density is linear-Gaussian:
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(18.324)
• PHD for target appearance is a Gaussian mixture:
B
νk+1|k
bk+1|k (x) =
∑
k+1|k
bi
k+1|k
· NB k+1|k (x − bi
).
(18.325)
i
i=1
• Cardinality distribution for target appearance: pB
k+1|k (n), with
B
νk+1|k
B
Nk+1|k
=
∑
i=1
k+1|k
bi
=
∑
n · pB
k+1|k (n).
(18.326)
n≥0
´ = p̊S .
• Clutter-generator probability of survival is constant: p̊S (c, ć, C)
´ = p̊D .
• Clutter-generator probability of detection is constant: p̊D (c, ć, C)
• Clutter-generator Markov density: has the form
′
´ ′ , C´ ′ ) = N
´
f˚k+1|k (ć, C|ć
´ ′ )−1 (ć − ć ) · δC
´ ′ (C)
(φ̊k ·C
(18.327)
´ is the Dirac delta function concentrated
for some φ̊k > 0, where δC´ ′ (C)
′
´
´
at C . Because δC´ ′ (C) cannot model any increase in the uncertainty in
´ Chen et al. compensated for this fact by introducing a time-update for the
C,
parameter d of the form
δ · d′
(18.328)
d=
δ + d′
where δ is a fading factor that causes knowledge of d′ to diminish over
time (see [39], Eq. (38)).
RFS Filters for Unknown Clutter
607
• PHD for appearance of clutter generators is a NWM:1
B
νk+1|k
˚
´ =
bk+1|k (ć, C)
∑
k+1|k
˚
´
bi
· N Wδk+1|k ,uk+1|k ,uk+1|k ,U k+1|k (ć, C).
i
i
i
i
i=1
(18.329)
• Cardinality distribution for clutter-generator appearance: p̊k+1|k (n̊) with
B
νk+1|k
˚B
N
k+1|k =
∑
k+1|k
˚
bi
=
i=1
∑
n · p̊B
k+1|k (n̊).
(18.330)
n̊≥0
• Target probability of detection is constant: pD (x) = pD .
• Target likelihood function is linear-Gaussian:
Lz (x) = NRk+1 (z − Hk+1 x).
(18.331)
• Clutter likelihood function:
´ =N
Lκz (ć, C)
´ −1 (z − ć)
(η̊k+1 ·C)
(18.332)
for some η̊k+1 > 0.
18.5.8.3
NWM-κ-CPHD Filter: Time Update
Suppose that we are given the NWM system
p̈k|k (n̈),
k|k
k|k
k|k
(ℓi , wi , Pi
k|k
k|k
k|k
k|k ν
k|k
, xi )i=1
,
k|k
k|k ν̊
k|k
(ẘi , di , oi , oi , Oi )i=1
.
1
Chen et al. construct this PHD from the new measurements. Let zj be one of the measurements at
2
time-step k + 1 and let σk+1
be the variance of the measurement noise. Then they create mk+1
2
NWM components with dj = 1.5, oj = zj , oj = 0.5, and Oj = 8d−1
· σk+1
· IM ×M ,
j
where IM ×M is the M × M identity matrix (see [39], p. 1227, Section IV-C-1).
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Advances in Statistical Multisource-Multitarget Information Fusion
We are to update it to a NWM system
p̈k+1|k (n̈),
k+1|k νk+1|k
)i=1 ,
k+1|k k+1|k k+1|k
k+1|k
k+1|k ν̊k+1|k
(ẘi
, di
, oi
, oi
, Oi
)i=1 .
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
, xi
This is accomplished as follows. Let
νk|k
Nk|k
=
∑
k|k
(18.333)
wi
i=1
ν̊k|k
˚k|k
N
=
∑
k|k
(18.334)
ẘi .
i=1
Then
• Time update for cardinality distribution (same as for BGM-κ-CPHD filter):
p̈k+1|k (n̈)
=
∑
p̈k+1|k (n̈|n̈′ ) · p̈k|k (n̈′ )
(18.335)
p̈B
k+1|k (n̈ − i) · Cn̈′ ,i
(18.336)
n̈′ ≥0
p̈k+1|k (n̈|n̈′ )
=
n̈
∑
i=0
′
·ψ¨ki (1 − ψ¨k )n̈ −i
ψ¨k
=
˚k|k
pS · Nk|k + p̊S · N
˚k|k
Nk|k + N
(18.337)
∑
B
pB
k+1|k (n) · p̊k+1|k (n̊).
(18.338)
where as in (18.29),
p̈B
k+1|k (n̈) =
n+n̊=n̈
B
There are νk+1|k = νk|k + νk+1|k
GM components for the time-updated
B
target PHD and ν̊k+1|k = ν̊k|k + ν̊k+1|k NWM components for the time-updated
clutter PHD.
RFS Filters for Unknown Clutter
609
• Time update for persisting-target GM components (same as for BGM-κCPHD filter)—for i = 1, ..., νk|k :
k+1|k
ℓi
k|k
=
k+1|k
wi
k+1|k
xi
k+1|k
Pi
=
=
=
ℓi
(18.339)
k|k
pS · w i
k|k
Fk xi
k|k
Fk Pi FkT + Qk
(18.340)
(18.341)
(18.342)
• Time update for appearing-target GM components (same as for BGM-κB
CPHD filter)—for i = νk|k + 1, ..., νk|k + νk+1|k
:
k+1|k
ℓi
k+1|k
wi
=
new labels
(18.343)
=
k+1|k
bi−νk|k
(18.344)
k+1|k
=
bi−νk|k
(18.345)
=
k+1|k
Bi−νk|k .
(18.346)
k+1|k
xi
k+1|k
Pi
• Time update for persisting-generator NWM components—for l = 1, ..., ν̊k|k :
k+1|k
ẘl
k|k
=
k+1|k
dl
=
k+1|k
oi
=
k+1|k
oi
=
k+1|k
Oi
=
p̊S · ẘl
k|k
δk · dl
k|k
δk + dl
k|k
φ̊k · oi
k|k
φ̊k + oi
k|k
oi
Oik+k
where δk is the fading factor of (18.328).
(18.347)
(18.348)
(18.349)
(18.350)
(18.351)
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Advances in Statistical Multisource-Multitarget Information Fusion
• Time update for appearing-generator NWM components—for l = ν̊k|k +
B
1, ..., ν̊k|k + ν̊k+1|k
:
k+1|k
k+1|k
= ˚
bl−ν̊ B
ẘl
(18.352)
k+1|k
k+1|k
k+1|k
dl
k+1|k
oi
k+1|k
oi
k+1|k
Oi
18.5.8.4
=
δl−ν̊k|k
(18.353)
=
k+1|k
ul−ν̊k|k
(18.354)
=
k+1|k
ul−ν̊k|k
(18.355)
=
k+1|k
Ul−ν̊k|k .
(18.356)
NWM-κ-CPHD Filter: Measurement Update
Suppose that we are given the predicted NWM system
p̈k+1|k (n̈),
k+1|k νk+1|k
)i=1 ,
k+1|k k+1|k k+1|k
k+1|k
k+1|k ν̊k+1|k
(ẘi
, di
, oi
, oi
, Oi
)i=1
k+1|k
(ℓi
k+1|k
, wi
k+1|k
, Pi
, xi
Given a new measurement set Zk+1 with |Zk+1 | = mk+1 , we are to update it to
a NWM system
p̈k+1|k (n̈),
k+1|k+1 νk+1|k+1
)i=1
,
k+1|k+1 k+1|k+1 k+1|k+1
k+1|k+1
k+1|k+1 ν̊k+1|k+1
(ẘi
, di
, oi
, oi
, Oi
)i=1
.
k+1|k+1
(ℓi
k+1|k+1
, wi
k+1|k+1
, Pi
, xi
We are also to determine the measurement-updated target cardinality distribution
pk+1|k+1 (n).
This is accomplished as follows. Let
νk+1|k
Nk+1|k
=
∑
k+1|k
(18.357)
wi
i=1
ν̊k+1|k
˚k+1|k
N
=
∑
i=1
Then:
k+1|k
ẘi
.
(18.358)
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611
• Measurement update for joint target/clutter cardinality distribution:
p̈k+1|k+1 (n̈)
=
ℓ¨Zk+1 (n̈) · p̈k+1|k (n̈)
∑
¨
l≥0 ℓZk+1 (l) · p̈k+1|k (l)
ℓ¨Zk+1 (n̈)
=
Cn̈,m · ϕ¨k
ϕ¨k
=
n̈−mk+1
(18.359)
(18.360)
˚k+1|k
(1 − pD ) · Nk+1|k + (1 − p̊D ) · N
. (18.361)
˚k+1|k
Nk+1|k + N
There are νk+1|k+1 = νk+1|k + mk+1 · νk+1|k Gaussian components for the
measurement-updated target PHD and ν̊k+1|k+1 = ν̊k+1|k + mk+1 ·ν̊k+1|k NWM
components for the measurement-updated clutter PHD.
• Measurement update for undetected-target GM components (same as for the
BGM-κ-CPHD filter)—for i = 1, ..., ν̊k+1|k :
k+1|k+1
ℓi
k+1|k
=
ℓi
(18.362)
=
1 − pD
˚k+1|k
Nk+1|k + N
(18.363)
k+1|k+1
wi
¨ (mk+1 +1) (ϕ¨k )
G
k+1|k
·
k+1|k+1
xi
k+1|k+1
Pi
¨ (mk+1 ) (ϕ¨k )
G
k+1|k
k+1|k
· wi
k+1|k
=
xi
(18.364)
=
k+1|k
Pi
.
(18.365)
• Measurement update for detected-target GM components (same as for the
BGM-κ-CPHD filter)—for i = 1, ..., νk+1|k and j = 1, ..., mk+1 :
k+1|k+1
ℓi,j
k+1|k
=
(18.366)
ℓi
k+1|k+1
wi,j
pD · N R
(18.367)
k+1|k
T
Hk+1
k+1 +Hk+1 Pi
k+1|k
(zj − Hk+1 xi
)
=
κ̂k+1 (zj ) + τk+1 (zj )
k+1|k
k+1|k
xi,j
=
k+1|k
Pi,j
=
k+1|k
xi
+ Kik+1 (zj − Hk+1 xi
(
) k+1|k
I − Kik+1 Hk+1 Pi
)
k+1|k
· wi
(18.368)
(18.369)
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Advances in Statistical Multisource-Multitarget Information Fusion
where
k+1|k
Kik+1 = Pi
(
)−1
k+1|k T
T
Hk+1
Hk+1 Pi
Hk+1 + Rk+1
.
(18.370)
• Measurement update for undetected-generator NWM components—for l =
1, ..., ν̊k+1|k :
(m
+1)
¨ k+1 (ϕk )
· (1 − p̊D ) G
k+1|k
·
(m
)
˚k+1|k
¨
Nk+1|k + N
G k+1 (ϕ )
k+1|k
ẘl
k+1|k+1
ẘl
=
k+1|k
k+1|k+1
k+1|k
dl
=
k+1|k+1
oi
k+1|k+1
oi
k+1|k+1
Oi
=
=
=
(18.371)
k
dl
(18.372)
k+1|k
oi
k+1|k
oi
k+1|k
Oi
.
(18.373)
(18.374)
(18.375)
• Measurement update for detected-generator NWM components—for
1, ..., ν̊k+1|k and j = 1, ..., mk+1 :
i =
k+1|k
k+1|k+1
ẘi,j
p̊D · ẘi
k+1|k+1
oi,j
i
i
k+1|k
i
(18.376)
(18.377)
=
di
=
k+1|k
η̊k+1 + oi
=
η̊k+1 · zj + oi
+1
(18.378)
k+1|k
k+1|k+1
oi,j
i
κ̂k+1 (zj ) + τk+1 (zj )
k+1|k+1
di,j
· qzj ,dk+1|k ,ok+1|k ,ok+1|k ,Ok+1|k
=
k+1|k
· oi
(18.379)
k+1|k
η̊k+1 + oi
k+1|k
k+1|k+1
Oi,j
k+1|k
=
Oi
+
η̊k+1 · oi
(18.380)
k+1|k
η̊k+1 + oi
k+1|k
·(zj − oi
k+1|k T
)(zj − oi
)
RFS Filters for Unknown Clutter
613
and where
qzj ,dk+1|k ,ok+1|k ,ok+1|k ,Ok+1|k
i
i
i
i
(
) )
(
k+1|k M/2
k+1|k+1
(η̊k+1 · oi
)
· ΓM di,j
/2
(18.381)
k+1|k dk+1|k /2
=


·(det Oi
) i
(
) 
k+1|k
· ΓM di
/2
.
k+1|k+1 dk+1|k+1
/2
·(det Oi,j
) i,j
k+1|k+1 M/2
(π · oi,j
)
• Target cardinality distribution:
pk+1|k+1 (n) =
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 )
n!
(18.382)
where
rk+1
=
¨ k+1|k+1 (ẍ)
G
=
Nk+1|k+1
˚k+1|k+1
Nk+1|k+1 + N
∑
p̈k+1|k+1 (n̈) · ẍn̈
(18.383)
(18.384)
n̈≥0
νk+1|k
Nk+1|k+1
=
∑
νk+1|k mk+1
k+1|k+1
wi
+
i=1
∑ ∑
i=1
k+1|k+1
wi,j
(18.385)
j=1
and
ν̊k+1|k
˚k+1|k+1
N
=
∑
k+1|k+1
(18.386)
ẘl
l=1
ν̊k+1|k ν̊ k+1 mk+1
+
∑ ∑ ∑
i=1
18.5.8.5
l=1
k+1|k+1
ẘi,l,j
.
j=1
NWM-κ-CPHD Filter Merging and Pruning
Merging and pruning is more complicated for NWMs than it is for Gaussian mixtures. In what follows, two possible approaches are considered: a more theoretically justified (and more computationally demanding) one; and an approximate
approach along the lines of that proposed in Chen Xin et al. [39].
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Advances in Statistical Multisource-Multitarget Information Fusion
• Exact Merging of NWMs—Merging Criterion. Suppose that we are given two
NMW components
k|k
´
D1 (ć, C)
=
w1
´
· N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C)
(18.387)
´
· N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C)
(18.388)
1
k|k
´
D2 (ć, C)
=
w2
2
1
2
1
2
1
2
and that we want to determine if they should be merged. This can be
accomplished by determining the probability (density) of overlap,
∫
´
N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C)
1
=
1
1
(18.389)
1
´
C´
·N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C)dćd
(2 2 2 M/22
)
(o1 · o2 )
· (det O1 )d1 /2
·(det O2 )d2 /2 · ΓM (d/2)
( M (M +1)/2
)
2
· (π · o)M/2 · (det O)d/2
·ΓM (d1 /2) · ΓM (d2 /2)
(18.390)
where the last equation follows from (G.29) in Appendix G. Thresholding
this quantity determines which components should be merged.
• Exact Merging of NWMs—Merging Formulas: Suppose that it has been
determined that the following NWM
k|k
´ + ... + wνk|k · N W k|k k|k k|k k|k (ć, C)
´
w1 · N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C)
d
,o
,o
,O
1
1
1
ν
1
ν
ν
ν
should be merged into a single component
k|k
w0
´
· N Wdk|k ,ok|k ,ok|k ,Ok|k (ć, C).
0
0
0
0
Set these two PHDs equal to each other and solve for d0 , o0 , o0 , O0 . Let
w0
=
ν
∑
wi
(18.391)
i=1
ŵi
=
wi
w0
(18.392)
RFS Filters for Unknown Clutter
615
and apply (G.4) through (G.8). Then we get
o0
o0
=
=
( ν
∑
ŵi · o−1
i
)−1
(18.393)
i=1
ν
∑
(18.394)
ŵi · oi
i=1
d0
d0 − M2−1
O0
=
=
(
)
M −1 −1
· trOi
i=1 ŵi · dl −
2
((∑
)
)
ν
−1 −1
tr
l=1 ŵl · dl · Ol
∑ν
d0 ·
( ν
∑
ŵi · di · Oi−1
)−1
.
(18.395)
(18.396)
i=1
Equation (18.395) results from (G.8), which leads to
)−1
(
M −1
ŵi · dl −
· trOi
2
i=1
(
)−1
M −1
d0 −
· trO0
2
((
)−1 )
d0
· tr d0 O0−1
M −1
d0 − 2
ν
∑
=
=
(18.397)
(18.398)
and then to (G.5), which finally leads to (18.395).
• Approximate Merging of NWMs—Merging Criterion. Let
´ = w k|k · N
´
D1 (ć, C)
´ −1 (ć − o1 ) · Wd1 ,O1 (C)
1
(o1 ·C)
´ is (d1 − M − 1) · O −1 .
be an NWM component. The mode of Wd1 ,O1 (C)
1
´ is sufficiently tightly concentrated around its mode.
Assume that Wd1 ,O1 (C)
According to (G.9) in Appendix G, this occurs if
√
d1 − M − 1 · trO1−1
(18.399)
is sufficiently small. Then we can approximate
´ ∼
´
D1 (ć, C)
= w1 · N(o1 ·(d1 −M −1))−1 ·O1 (ć − o1 ) · Wd1 ,O1 (C).
(18.400)
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Advances in Statistical Multisource-Multitarget Information Fusion
Now let
´ = w k|k · N
´
D2 (ć, C)
´ −1 (ć − o2 ) · Wd2 ,O2 (C)
2
(o2 ·C)
(18.401)
´
be another tightly concentrated component. Then to determine if D1 (ć, C)
´
and D2 (ć, C) should be merged, it is enough to determine if
˜ 1 (ć) = N(o ·(d −M −1))−1 ·O (ć − o1 )
D
1
1
1
(18.402)
˜ 2 (ć) = N(o ·(d −M −1))−1 ·O (ć − o2 )
D
2
2
2
(18.403)
and
should be merged, using the usual merging criteria for Gaussian components
described in Section 9.5.3.
• Approximate Merging of NWMs—Merging Formulas. If it is determined that
these components should be merged, then the associated merged component
is
˜ 0 (ć) = w0 · NO (ć − o0 )
D
(18.404)
0
where
w0
ŵ1
=
=
w1 + w2
w1 /w0
(18.405)
(18.406)
ŵ2
o0
=
=
w2 /w0
ŵ1 · o1 + ŵ2 · o2
(18.407)
(18.408)
O0
=
ŵ1 · (o1 · (d1 − M − 1))−1 · O1
+ŵ2 · (o1 · (d2 − M − 1))−1 · O2
+ŵ1 · ŵ2 · (o1 − o2 )(o1 − o2 )T .
(18.409)
´ and D2 (ć, C)
´ are tightly
Finally, note that, by assumption, D1 (ć, C)
concentrated at their respective matrix modes (o1 · (d1 − M − 1))−1 · O1
and (o2 · (d2 − M − 1))−1 · O2 . But—since they are to be merged—they
must have nearly identical modes. Thus the merged component can be taken
to have the form
´ =D
˜ 0 (C)
´ · Wd ,O (C)
´ ∼
˜ 0 (C)
´ · Wd ,O (C).
´
D0 (ć, C)
=D
1
1
2
2
(18.410)
RFS Filters for Unknown Clutter
18.6
617
MULTISENSOR κ-CPHD FILTERS
The λ-CPHD and κ-CPHD filters are single-sensor filters. However, they can be
extended to the multisensor case using the techniques described in Chapter 10.
18.6.1
Iterated-Corrector κ-CPHD Filter
The easiest approach is the iterated-corrector method of Section 10.5. In this case,
one simply repeats the measurement-update for the λ-CPHD or κ-CPHD filter, once
for each sensor. Of course, this approach inherits the limitations of the iteratedcorrector approach.
18.6.2
Parallel-Combination κ-CPHD Filter
One can also apply the approximate parallel-combination approach of Section 10.6.
The purpose of this section is to briefly illustrate how this approach is applied to the
κ-CPHD filter.
Let the augmented target-clutter state space be
1
s
¨ = X ⊎˚
X
C ⊎ ... ⊎ ˚
C
(18.411)
j
j
j
j
j
where ˚
C = [0, 1] × C is the space of clutter generators c̊ = (c, c) for the jth
sensor. The integral on this space is defined as
∫
∫
∫
∫
1
1
s
s
f¨(ẍ)dẍ =
f¨(x)dx + 1 f¨(c̊)dc̊ + ... + s f¨(c̊)dc̊
(18.412)
˚
C
X
where
∫
j
j
j
f¨(c̊)dc̊ =
˚
C
∫ ∫ 1
˚
C
j j
j j
f¨(c, c)dcdc.
(18.413)
0
The PCAM-κ-CPHD filter has the form
... →
p̈k|k (n̈)
→
p̈k+1|k (n̈)
... →
sk|k (x)
→
sk+1|k (x)
... →
s̊k|k (c, c)
..
.
→
s̊k+1|k (c, c)
..
.
... →
s̊k|k (c, c)
→
s̊k+1|k (c, c)
1
→
↑↓
→
↑↓
p̈k+1|k+1 (n̈)
→ ...
sk+1|k+1 (x)
→ ...
→
..
.
s̊k+1|k+1 (c, c)
..
.
→
s̊k+1|k+1 (c, c)
1
1
1
1
1
s
1
1
s
s
s
1
→ ...
s
s
s
s
s
→ ...
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Advances in Statistical Multisource-Multitarget Information Fusion
Here, the top filter propagates the cardinality distribution p̈k|k (n̈) on the number
1
j
s
n̈ = n + n + ... + n, where n is the number of targets and n is the number
of clutter generators for the jth sensor. The second filter propagates the spatial
distribution sk|k (x) on targets; and the following rows are filters that propagate
j
j
j
the clutter-generator distributions s̊k|k (c, c) of the sensors.
Suppose that at time tk the s sensors collect the respective measurement
1
j
s
j
sets Z k+1 , ..., Z k+1 with |Z k+1 | = m. Then we are to construct the joint targetclutter spatial distribution s̈k+1|k+1 (ẍ) and joint target-clutter cardinality distri1
s
bution p̈k+1|k+1 (n̈), updated using all of the measurement sets Z k+1 , ..., Z k+1 .
Given this, the measurement-updated spatial distribution and cardinality distribution are direct analogs of (10.94) through (10.103). Let:
=
∫
(1 − pD (ẍ)) · s̈k+1|k (ẍ)dẍ
=
∫
pD (ẍ) · Lj (ẍ) · s̈k+1|k (ẍ)dẍ
j
ϕ¨k+1
j
j
τ̈ k+1 (z)
j
(18.414)
j
j
(18.415)
z
and
s̈k+1|k+1 (ẍ)
=
p̈k+1|k+1 (n̈)
=
1
¨ 1
·L
(ẍ) · s̈k+1|k (ẍ) (18.416)
s
¨k+1|k+1
Z k+1 ,...,Z k+1
N
n̈
p̃(n̈) · θ¨k+1
(18.417)
˜ θ¨k+1 )
G(
RFS Filters for Unknown Clutter
619
where
1
¨ 1
L
(ẍ)
=
¨ k+1|k+1
N
=
s
Z k+1 ,...,Z k+1
1...s
s
¨1
L
¨s
(ẍ) · · · L
(ẍ)
Z k+1
G (θ¨k+1 )
Z
· 1 k+1
(18.418)
s
˜ θ¨k+1 )
G(
¨ k+1|k+1 · · · N
¨ k+1|k+1
N
∫ 1
¨ 1 (ẍ)
L
(18.419)
˜ (1)
Z k+1
s
¨s
···L
Z k+1
(ẍ) · s̈k+1|k (ẍ)dẍ
1...s
θ¨k+1
¨ k+1|k+1
N
(18.420)
=
1
s
¨ k+1|k+1 · · · N
¨ k+1|k+1
N
¨k+1|k+1
N
=
p̃k+1|k+1 (n̈)
=
˜ k+1|k+1 (ẍ)
G
=
¨
˜ (1)
G
k+1|k+1 (θk+1 ) ¨
·θ
˜ k+1|k+1 (θ¨k+1 ) k+1
G
1
(18.421)
s
ℓ¨z1 (n) · · · ℓ¨zs (n̈) · p̈k+1|k (n̈)
∑
p̃k+1|k+1 (n̈) · ẍn̈
(18.422)
(18.423)
n̈≥0
and where
j
j
ℓ¨j
(n̈)
=
C
j
n̈,m
Z k+1
m
· ϕ¨n̈−
k+1
(18.424)
j
j
¨j
L
(ẍ)
=
Z k+1
j
¨ (m+1) (ϕ¨k+1 )
1 − p̈D (ẍ) G
k+1|k
·
j
¨
Nk+1|k
¨ (m) (ϕ¨k+1 )
G
k+1|k
(18.425)
j
j
∑ p̈D (ẍ) · Lj (ẍ)
z
+
j
j
j
j
τ̈ k+1 (z)
z∈Z k+1
j
j
¨ k+1|k+1
N
=
j
¨ (m+1) (ϕ¨k+1 )
ϕ¨k+1 G
j
k+1|k
·
+ m.
j
¨
Nk+1|k ¨ (m) ¨
Gk+1|k (ϕk+1 )
(18.426)
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Advances in Statistical Multisource-Multitarget Information Fusion
The multisensor PCAM-κ-CPHD filter results when we define PHDs and
spatial distributions for the targets and for each of the clutter spaces:
Dk+1|k (x)
=
Nk+1|k
=
sk+1|k (x)
=
Dk+1|k+1 (x)
=
Nk+1|k+1
=
sk+1|k+1 (x)
=
¨ k+1|k (x)
D
∫
Dk+1|k (x)dx
(18.427)
(18.428)
Dk+1|k (x)
Nk+1|k
¨
Dk+1|k+1 (x)
∫
Dk+1|k+1 (x)dx
(18.430)
Dk+1|k+1 (x)
Nk+1|k+1
(18.432)
(18.429)
(18.431)
and
j
j
˚k+1|k (c̊)
D
j
=
j
˚k+1|k
N
=
¨ k+1|k (c̊),
D
∫ j
j
j
˚k+1|k (c̊)dc̊
D
(18.433)
(18.434)
j
j
j
˚k+1|k (c̊)
D
j
s̊k+1|k (c̊)
(18.435)
=
j
˚k+1|k
N
j
j
˚k+1|k+1 (c̊)
D
j
=
j
˚k+1|k+1
N
=
¨ k+1|k+1 (c̊),
D
∫ j
j
j
˚k+1|k+1 (c̊)dc̊
D
(18.436)
(18.437)
j
j
j
˚k+1|k+1 (c̊)
D
j
s̊k+1|k+1 (c̊)
=
.
(18.438)
j
˚k+1|k+1
N
Given this, the conventional PCAM-CPHD formulas are rewritten so that they are
expressed in terms of these PHDs and/or spatial distributions. The details will not
be considered here.
RFS Filters for Unknown Clutter
621
For the purpose of multitarget state estimation, the cardinality distribution on
targets is given by the obvious analog of (18.62):
pk+1|k+1 (n)
=
rk+1
=
n
rk+1
¨ (n)
·G
k+1|k+1 (1 − rk+1 )
n!
Nk+1|k+1
(18.439)
1
. (18.440)
s
˚k+1|k+1 + ... + N
˚k+1|k+1
Nk+1|k+1 + N
The estimated clutter intensity function for the jth sensor is the analog of
(18.81):
j
j
κ̂k+1 (z) =
∫ ∫ 1
j
j
j j
j
j
j
j
j
˚k+1|k+1 (c, c)dcdc
c · f κk+1 (z|c) · D
(18.441)
∫ ∫ 1
(18.442)
0
with estimated clutter rate
j
ˆ k+1 =
λ
j
j j
j j
j
˚k+1|k+1 (c,
c·D
c)dcdc.
0
The estimated clutter cardinality distribution for the jth sensor is the analog
of (18.93):
j
˜m
λ
k+1|k
j
p̂k+1 (m)
=
m!
j
˜ k+1|k )
¨ (m) (1 − λ
·G
k+1|k
(18.443)
j
j
˜ k+1
λ
ˆ k+1
λ
=
.
1
s
(18.444)
˚k+1|k+1 + ... + N
˚k+1|k+1
Nk+1|k+1 + N
18.7
THE κ-CBMEMBER FILTER
Using the techniques of this and the previous chapter, Vo, Vo, Hoseinnezhad, and
Mahler have generalized the CBMeMBer filter (Chapter 13) to situations in which
the clutter background and detection profile are not known [312], [313]. As with
background-agnostic CPHD filters, the background-agnosic CBMeMBer filters are
constructed directly from the CBMeMBer filter by:
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Advances in Statistical Multisource-Multitarget Information Fusion
• Modeling unknown probability of detection using a new state (a, x) with
conventional state x and unknown probability of detection 0 ≤ a ≤ 1.
• Modeling clutter as clutter generators of the form (c, c) with clutter probability of detection c and clutter likelihood function ˚
Lz (c).
The first approach is straightforward: in the formulas for the CBMeMBer
∫ ∫1
filter, one simply
substitutes ‘a’ wherever ‘pD (x)’ occurs, and ‘ 0 ·dadx’
∫
wherever ‘ ·dx’ occurs.
Thus in what follows, only the second approach will be described. It results
in the “κ-CBMeMBer filter.”
As with the CBMeMBer filter, it is expected that the κ-CBMeMBer filter will
be most effective when it is implemented using particle methods, for applications
involving significant motion and/or measurement nonlinearity.
As with the κ-CPHD filter, the κ-CBMeMBer filter results when one
replaces the conventional state x with a state ẍ, which can take two forms:
ẍ = x or ẍ = (c, c), with associated measurement and motion models p̈D (ẍ),
¨ z (ẍ) = fk+1 (z|ẍ), p̈S (ẍ), and M
¨ ẍ (ẍ′ ) = f¨k+1 (ẍ|ẍ′ ).
L
The section is organized as follows:
1. Section 18.7.1: Modeling assumptions for the κ-CBMeMBer filter.
2. Section 18.7.2: Time update equations for the κ-CBMeMBer filter.
3. Section 18.7.3: Measurement update equations for the κ-CBMeMBer filter.
4. Section 18.7.4: State estimation for the κ-CBMeMBer filter.
5. Section 18.7.5: Clutter estimation for the κ-CBMeMBer filter.
18.7.1
κ-CBMeMBer Filter: Modeling
The motion and measurement models for this filter are essentially the same as those
for the κ-CPHD filter (Section 18.5.1):
• Targets can transition only to targets, and clutter generators only to clutter
generators.
• Target probability of survival: pS (x) abbr.
= pS,k+1 (x).
• Target Markov density: Mx (x′ ) = fk+1|k (x|x′ ).
• Clutter-generator probability of survival: p̊S (c) abbr.
= pS,k+1 (c).
RFS Filters for Unknown Clutter
623
κ
• Clutter-generator Markov density: Mcκ (c′ ) = fk+1|k
(c|c′ ).
κ
• Markov density for clutter probability of detection: Mcκ (c′ ) = fk+1|k
(c|c′ ),
defined as in (17.12) through (17.16).
• Target probability of detection: pD (x) abbr.
= pD,k+1 (x).
• Target likelihood function: Lz (x) abbr.
= fk+1 (z|x).
κ
• Clutter-generator likelihood function: Lκz (c) abbr.
= fk+1
(z|c).
• Since clutter is modeled using clutter generators, the a priori clutter intensity
has value κk+1 (z) = 0.
Given this, the κ-CBMeMBer filter is constructed as follows. At time tk
and for i = 1, ..., ν̈k|k , the filter consists of (1) a list of joint target-clutter track
distributions s̈ik|k (ẍ) where ẍ = x or ẍ = (c, c); (2) joint target-clutter
i
probabilities of existence q̈k|k
; and (3) joint target-clutter track labels ℓ¨ik|k . Define
the density functions
sik|k (x)
=
s̈ik|k (x)
(18.445)
s̊ik|k (c, c)
=
s̈ik|k (c, c)
(18.446)
where it must be the case that
∫
i
˚i
1 = s̈ik|k (ẍ)dẍ = Nk|k
+N
k|k
(18.447)
where
i
Nk|k
=
∫
sik|k (x)dx,
˚i =
N
k|k
∫ ∫ 1
s̊ik|k (c, c)dcdc.
(18.448)
0
ν̈
k|k
Thus the joint probability distributions s̈1k|k (ẍ),...,s̈k|k
(ẍ) can be equivalently
ν̈
k|k
replaced by the target densities s1k|k (x),...,sk|k
(x)
and the clutter densities
ν̈k|k
s̊1k|k (c, c),...,s̊k|k
(c, c).
Note that (18.447) can be satisfied when sik|k (x) = 0 identically or when
(c, c) = 0 identically. Also, in general the labels ℓ¨i
have the form
s̊ik|k
k|k
ℓ¨ik|k = (ℓik|k , ˚
ℓik|k ) where ℓik|k are target track labels and ˚
ℓik|k are clutter-generator
track labels. Since it is unnecessary to propagate labels for the clutter generators,
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we can eliminate the ˚
ℓik|k and propagate only the ℓik|k . Thus a multi-Bernoulli
system will have the general form
ν̈
k|k
i
{ℓik|k , q̈k|k
, sik|k (x),s̊ik|k (c, c)}i=1
.
18.7.2
κ-CBMeMBer Filter: Time Update
We are given the prior multi-Bernoulli system
ν̈
k|k
i
T̈k|k = {ℓik|k , q̈k|k
, sik|k (x),s̊ik|k (c, c)}i=1
.
(18.449)
We are to determine the time-updated multi-Bernoulli system
ν̈
k+1|k
i
T̈k+1|k = {ℓik+1|k , q̈k+1|k
, sik+1|k (x),s̊ik+1|k (c, c)}i=1
.
(18.450)
This has the form
persist
birth
T̈k+1|k = T̈k+1|k
∪ T̈k+1|k
(18.451)
where
ν̈
persist
T̈k+1|k
=
k|k
{(ℓi , q̈i , si (x),s̊B
i (c, c))}i=1
birth
T̈k+1|k
=
bk
B B
B
{(ℓB
i , q̈i , si (x),s̊i (c, c))}i=1
(18.452)
¨
(18.453)
and where the persisting-track components are given by
ℓi
=
q̈i
=
si (x)
=
s̊B
i (c, c)
=
sik|k [pS ]
=
s̊ik|k [p̊S ]
=
ℓik|k
(18.454)
(
i
q̈k|k
· sik|k [pS ] + s̊ik|k [p̊S ]
)
(18.455)
sik|k [pS Mx ]
sik|k [pS ] + s̊ik|k [p̊S ]
(18.456)
˚(c,c) ]
s̊ik|k [p̊S M
sik|k [pS ] + s̊ik|k [p̊S ]
(18.457)
where
∫
pS (x) · sik|k (x)dx
∫ ∫ 1
p̊S (c) · s̊ik|k (c, c)dcdc.
0
(18.458)
(18.459)
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18.7.3
625
κ-CBMeMBer Filter: Measurement Update
We are given the predicted multi-Bernoulli system
ν̈
k+1|k
i
T̈k+1|k = {ℓik+1|k , q̈k+1|k
, sik+1|k (x),s̊ik+1|k (c, c)}i=1
.
(18.460)
Suppose that a new measurement set Zk+1 = {z1 , ..., zmk+1 } is collected with
|Zk+1 | = mk+1 . We are to determine the form of the measurement-updated multiBernoulli system
ν̈
k+1|k+1
i
T̈k+1|k+1 = {ℓik+1|k+1 , q̈k+1|k+1
, sik+1|k+1 (x),s̊ik+1|k+1 (c, c)}i=1
.
(18.461)
This has the form
legacy
meas
T̈k+1|k+1 = T̈k+1|k+1
∪ T̈k+1|k+1
(18.462)
where
legacy
T̈k+1|k+1
meas
T̈k+1|k+1
ν̈
=
k+1|k
L L
L
{(ℓ¨L
i , q̈i , si (x),s̊i (c, x))}i=1
(18.463)
=
mk+1
U U
U
{(ℓ¨U
j , q̈j , sj (x),s̊j (c, c))}j=1 .
(18.464)
The measurement-update equations for the legacy components are:
ℓL
i
=
ℓik+1|k
(18.465)
(
i
q̈k+1|k
· 1 − sik+1|k [pD ] − s̊ik+1|k [p̊D ]
q̈iL
=
s̈L
i (x)
=
s̈ik+1|k (x) ·
s̊L
i (c, c)
=
s̈ik+1|k (c, c) ·
)
i
i
· s̊ik+1|k [p̊D ]
1 − q̈k+1|k
· sik+1|k [sD ] − q̈k+1|k
1 − pD (x)
i
1 − sk+1|k [pD ] − s̊ik+1|k [p̊D ]
1−c
1 − sik+1|k [pD ] − s̊ik+1|k [p̊D ]
(18.466)
(18.467)
(18.468)
where
sik+1|k [pD ]
=
s̊ik+1|k [p̊D ]
=
∫
pS (x) · sik+1|k (x)dx
∫ ∫ 1
c · s̊ik+1|k (c, c)dcdc.
0
(18.469)
(18.470)
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Advances in Statistical Multisource-Multitarget Information Fusion
The measurement-update equations for the measurement-updated components are:
∗
ℓU
(18.471)
j = ℓk+1|k
and
q̈jU
i
i
˚z )
∑ν̈k+1|k q̈k+1|k
(1−q̈k+1|k
)·(sik+1|k [pD Lzj ]+s̊ik+1|k [p̊D L
j
(18.472)
i
(1−q̈k+1|k
·(sik+1|k [pD ]+s̊ik+1|k [p̊D ))2
(
)
i
˚z
· sik+1|k [pD Lzj ]+s̊ik+1|k [p̊D L
∑ν̈k+1|k q̈k+1|k
j
(
)
κk+1 (zj ) + i=1
1−q̈ i
· si
[p ]+s̊i
[p̊ ]
i=1
=
k+1|k
k+1|k
D
k+1|k
D
and
sU
j (x)
∑ν̈k+1|k
i=1
=
∑ν̈k+1|k
i=1
(18.473)
i
q̈k+1|k
· sik+1|k (x) · pD (x) · Lzj (x)
i
1−q̈k+1|k
i
q̈k+1|k
i
1−q̈k+1|k
s̊U
j (c, c)
∑ν̈k+1|k
i=1
=
∑ν̈k+1|k
i=1
(
)
· sik+1|k [pD Lzj ] + s̊ik+1|k [p̊D ˚
L zj ]
(18.474)
i
q̈k+1|k
· s̊ik+1|k (c, c) · c · ˚
Lzj (c)
i
1−q̈k+1|k
i
q̈k+1|k
i
1−q̈k+1|k
(
)
· sik+1|k [pD Lzj ] + s̊ik+1|k [p̊D ˚
L zj ]
where
sik+1|k [pD Lzj ]
=
s̊ik+1|k [p̊D ˚
L zj ]
=
∫
pD (x) · Lzj (x) · sik+1|k (x)dx
∫ ∫ 1
c·˚
Lzj (c) · s̊ik+1|k (c, c)dcdc.
(18.475)
(18.476)
0
Also, ℓ¨∗j,k+1|k is the label of the track that has the largest contribution to the
probability of existence q̈jU in (18.472).
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18.7.4
627
κ-CBMeMBer Filter: Multitarget State Estimation
Multitarget state estimation can be accomplished as follows. First, some notation
must be established. The existence probability of the ith legacy target is
q̈iL =
(
)
i
q̈k+1|k
· 1 − sik+1|k [pD ] − s̊ik+1|k [p̊D ]
i
i
1 − q̈k+1|k
· sik+1|k [sD ] − q̈k+1|k
· s̊ik+1|k [p̊D ]
(18.477)
and the existence probability of the ith measurement-updated target with measurement zj is
i
i
˚z )
∑ν̈k+1|k q̈k+1|k
(1−q̈k+1|k
)·(sik+1|k [pD Lzj ]+s̊ik+1|k [p̊D L
j
i
(1−q̈k+1|k
·(sik+1|k [pD ]+s̊ik+1|k [p̊D ))2
).
(
i
˚z
· sik+1|k [pD Lzj ]+s̊ik+1|k [p̊D L
∑ν̈k+1|k q̈k+1|k
j
(
)
κk+1 (zj ) + i=1
1−q̈ i
· si
[p ]+s̊i
[p̊ ]
i=1
q̈jU =
k+1|k
k+1|k
D
k+1|k
(18.478)
D
Given this, define the probabilities of existence of the legacy tracks to be
qiL = sik+1|k [1] · q̈iL ;
(18.479)
and the probabilities of existence of the updated tracks to be
qjU = sik+1|k [1] · q̈jU .
(18.480)
Then
ν̈k+1|k
Nk+1|k+1 =
∑
i=1
ν̈k+1|k mk+1
qiL +
∑ ∑
i=1
U
qi,j
(18.481)
j=1
is an estimate of the total expected number of targets. Second, round Nk+1|k+1
off to the nearest integer, ν. Third, find those ν target densities sk+1|k+1 (x)
i
U
with largest existence probabilities qk+1|k+1
or qi,j
. Fourth, determine the MAP
estimate of each such sk+1|k+1 (x) or, alternatively, the mean of each such
sk+1|k+1 (x)/sk+1|k+1 [1].
18.7.5
κ-CBMeMBer Filter: Clutter Estimation
Let
i
i
q̊k+1|k
= s̊k+1|k [1] · q̈k+1|k
(18.482)
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Advances in Statistical Multisource-Multitarget Information Fusion
be the probability of existence of the ith clutter component. Then the clutter
intensity function can be heuristically estimated as
ν̈k+1|k+1
κ̂k+1 (z)
=
∑
i
q̊k+1|k
∫ ∫ 1
0
i=1
ν̈k+1|k+1
=
c · Lκz (c) ·
∑
i
q̈k+1|k
∫ ∫ 1
s̊ik+1|k (c, x)
dcdc (18.483)
s̊k+1|k [1]
c · Lκz (c) · s̊ik+1|k (c, x)dcdc (18.484)
0
i=1
with associated clutter rate
ν̈k+1|k+1
ˆ k+1 =
λ
∑
i
q̈k+1|k
∫ ∫ 1
c · s̊ik+1|k (c, x)dcdc
(18.485)
0
i=1
and associated clutter cardinality distribution
pκk+1 (m) =
˜m
λ
k+1
˜ k+1 )
¨ (m) (1 − λ
·G
k+1|k
m!
(18.486)
where
ˆ k+1
λ
˜ k+1
λ
=
¨ k+1|k (z)
G
=
ˆ k+1
λ
= ∑νk+1|k k+1|k
¨k+1|k+1
N
q̈i
i=1
ν̈k+1|k (
∏
k+1|k
1 − q̈i
k+1|k
+ z · q̈i
(18.487)
)
.
(18.488)
i=1
18.8
IMPLEMENTED CLUTTER-AGNOSTIC RFS FILTERS
Four such implementations are described: two for the λ-CPHD filter (Section
18.8.1 and Section 18.8.2), one for the κ-CBMeMBer filter (Section 18.8.3), and
one for a normal-Wishart mixture (NWM) implementation of the κ-PHD filter.
18.8.1
Implemented λ-CPHD Filter
Mahler, Vo, and Vo have reported a Gaussian mixture implementation of the λCPHD filter [195], [194], as well as performance results using simulated data.
These simulations were of two types:
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629
Scenario 1: up to 12 appearing and disappearing targets, following linear
trajectories and observed by a linear-Gaussian sensor with clutter rate 50. Implementation type: beta-Gaussian mixture approach with EKF.
Scenario 2: up to 10 appearing and disappearing targets, following curvilinear
trajectories and observed by a range-bearing sensor with clutter rate 10. Implementation type: beta-Gaussian mixture approach with UKF.
The results were as follows:
Scenario 1: The λ-CPHD filter had good tracking performance, better than
that of a conventional PHD filter but not as good as a conventional CPHD filter. The
clutter rate was successfully estimated to be 50.
Scenario 2: The λ-CPHD filter had good tracking performance, though it
experienced some difficulty when targets became closely spaced. The clutter rate
was successfully estimated to be 10.
18.8.2
“Bootstrap” λ-CPHD Filter
In [18], Beard, Vo, and Vo reported certain limitations of the λ-CPHD filter, and
devised a heuristic means of correcting them. They noted that, on average, the
performance of the λ-CPHD filter is significantly worse than that of a “matched”
CPHD filter (that is, one that has been given the correct clutter rate). They attributed
this to the fact that the number of actual targets cannot be estimated from the λCPHD filter’s cardinality distribution (which, recall, is a distribution on the sum of
the number of actual and clutter targets).2
As a remedy, Beard et al. proposed a simple “bootstrap” procedure consisting
of two parallel λ-CPHD filters operating in two stages:
1. Stage 1: The first λ-CPHD filter is used to estimate the clutter rate λ.
2. Stage 2: the second λ-CPHD filter uses this estimate to detect and track the
actual targets.
This approach is somewhat questionable from a theoretical point of view,
since the algorithm double-counts the measurements. Nevertheless, Beard et al.
reported that this approach proved to be surprisingly successful: the bootstrap λCPHD filter performed nearly as well as the matched CPHD filter.
The bootstrap λ-CPHD filter employed the uniformly-distributed target-birth
model of Beard, Vo, Vo, and Arulampalam [16] (described in Section 9.5.7). Beard
2
Performance might be improved by using the target cardinality distribution to estimate target
number. The formula for this distribution was unknown at the time of writing of [18].
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Advances in Statistical Multisource-Multitarget Information Fusion
et al. applied a Gaussian mixture implementation of it in a scenario in which a
single bearing-only sensor with pD = 0.95 is carried on a platform moving
along a sinusoidal trajectory. The sensor observes five appearing, disappearing,
and slowly maneuvering targets, various amounts of uniformly distributed (in the
bearing variable) Poisson clutter.
Two simulations were considered, one with a constant clutter rate of λ = 30
and the other with a variable clutter rate, increasing from λ = 20 to λ = 40
during the middle third of the scenario. In the first scenario, the OSPA tracking
performance of the bootstrap λ-CPHD filter was as good as that of the matched
CPHD filter. It also estimated the (constant) clutter rate with good accuracy. In
the second scenario, the tracking performance of the bootstrap λ-CPHD filter was
once again as good as that of the matched CPHD filter. The bootstrap filter also
successfully estimated the variable clutter rate, but exhibited a slight (8%) upward
bias in the clutter estimate during the first third of the scenario, and a slight (5%)
downward bias during the final third.
18.8.3
Implemented λ-CBMeMBer Filter
Vo, Vo, Hoseinnazhad, and Mahler have reported simulation results for an implementation of the λ-CBMeMBer filter [312], [313]. This implementation, previously mentioned in Section 17.7, was also pD -agnostic, in that target probability of
detection a was also assumed unknown.
In this implementation, the velocities of clutter generators were ignored, so
that their states had the form (c, x, y). It was assumed that the generators followed
a random walk on the coordinates (x, y) with a linear-Gaussian Markov density
˚c (c′ ). Also, the clutter probability of detection c was assumed to be constant
M
and known.
In the simulation, up to 10 appearing and disappearing targets followed curvilinear trajectories and were observed by a range-bearing sensor. The number of
clutter measurements was binomially distributed with clutter rate of 10 returns per
scan. The clutter was spatially distributed so that it was increasingly concentrated
near the origin of the scenario, with clutter density decreasing radially from the
origin. The actual targets followed a coordinated-turn model with turn rate ω, and
thus had states of the form (x, y, vx , vy , ω) where ω is the turn rate. The Markov
transition for the target probability of detection was chosen to be a suitable beta
distribution.
The authors reported acceptable tracking performance, with accurate localization and correct track initiation and termination (with some delay).
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18.8.4
631
Implemented NWM-PHD Filter
This is the PHD filter special case of the Normal-Wishart mixture (NWM) CPHD
filter discussed in Section 18.5.8, and is due to Chen Xin, Kirubarajan, Tharmarasa,
and Pelletier in 2009 [39], [37]. These authors also proposed the integration of
a special case—a clutter estimator based on Wishart mixtures—into conventional
filters such as the MHT [38].
Chen et al. tested their NWM-κ-PHD filter in two-dimensional simulations,
one with a linear-Gaussian sensor and the other with a bearing-only sensor. In
both cases, three appearing and disappearing targets move in time-varying, spatially
nonhomogeneous clutter. The clutter process consists of several dense-clutter
subregions, of which there are two types. For the first type, the clutter spatial
distribution is uniform within an L-shaped region with clutter rates of either 10
or 18. For the second type, the spatial distribution is Gaussian with a clutter rate
9.6. Outside of these dense-clutter areas, clutter is uniform with small clutter rate.
In all cases, the probability of detection is 0.96.
For the simulation involving the linear-Gaussian sensor, the authors reported
that “the performance of the [NWM-κ-PHD filter] is comparable to the performance
obtained when the clutter’s true spatial distribution is perfectly known” ([39], p.
1227). For the bearing-only sensor, they reported that “the performance of the
[NWM-κ-PHD filter] is comparable to that obtained when the clutter’s true spatial
distribution is known...[but] brings an approximately two-scan delay for new target
initialization” ([39], p. 1227). The authors attributed this delay to the fact that
targets are more difficult to distinguish from clutter because of the low observability
of the bearing-only measurement model.
18.9
CLUTTER-AGNOSTIC PSEUDOFILTERS
Let us begin by reviewing what has been accomplished in this chapter so far. In
Section 18.2, a modeling approach for “clutter agnostic” RFS filters was described.
It was based on a Bernoulli clutter-generator model and the joint target-clutter state
space
¨ = X ⊎˚
X
C.
(18.489)
Then, in Section 18.3, the time- and measurement-update equations for a general
CPHD filter based on this general multi-Bernoulli model were described. Two
possible special cases of the joint target/clutter motion model were pointed out:
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1. A phenomenology-nonintermixing motion model, in which targets can transition only to targets and clutter generators only to clutter generators (Section
18.2.2).
2. A problematic phenomenology-intermixing motion model, in which clutter
generators can transition to targets; and/or targets can transition to clutter
generators; and in which the latter is interpreted as a model for target disappearance and the former is interpreted as a model for target appearance
(Section 18.2.3).
Finally, Sections 18.4 and 18.5 were devoted to clutter-agnostic CPHD filters
based on the nonintermixing model: a λ-CPHD filter that can estimate the clutter
cardinality distribution pκk+1 (m), and a κ-CPHD filter that can estimate both the
clutter intensity function κk+1 (z) and the clutter cardinality distribution.
But what if we had instead derived these filters assuming the phenomenologyintermixing motion model? The purpose of this section is to answer this question.
For conceptual and notational clarity, we will concentrate on the PHD filter special case of the λ-CPHD filter. As we shall see, it—and therefore the
intermixing-model analogs of the λ-CPHD and κ-CPHD filters—exhibit serious
pathologies and are therefore described as “pseudofilters.”
The section is organized as follows:
1. Section18.9.1: The λ-PHD pseudofilter (the intermixing model PHD pseudofilter).
2. Section 18.9.2: Pathological behavior of the λ-PHD pseudofilter.
18.9.1
The λ-PHD Pseudofilter
The purpose of this section is to present the time-update and measurement-update
equations for the λ-PHD pseudofilter.
18.9.1.1
λ-PHD Pseudofilter Time Update Equations
The time-update equations for the general Bernoulli clutter-generator CPHD filter
were given in (18.36) through (18.43). As with the motion models for the λ-PHD
filter (Section 18.4.6), assume that:
• Since target-to-clutter transitions model target disappearance, the target probability of survival is redundant and is therefore unity: pS (x′ ) = 1;
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633
• Since clutter-to-target transitions model target appearance, the target birth
PHD is redundant and therefore vanishes: bk+1|k (x) = 0.
• Since target-to-clutter transitions also model clutter-generator appearance,
the clutter-generator birth PHD is redundant and thus vanishes, ˚
bk+1|k (c) =
0.
• The probability that clutter generators will transition to clutter generators is
constant: p̊T (c′ ) = p̊T .
• The transition density from generators to targets does not depend on the
generator state:
⇒
fk+1|k
(x|c′ ) = sB
(18.490)
k+1|k (x),
where sB
k+1|k (x) is the spatial distribution of the appearing targets, assumed
to be known a priori.
Given this, (18.36) through (18.43) reduce to:
Dk+1|k (x)
=
˚k+1|k (c)
D
=
˚k|k · sB
(1 − p̊T ) · N
k+1|k (x)
∫
+ pT (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′
(18.491)
and
∫
⇐
(1 − pT (x′ )) · fk+1|k
(c|x′ ) · Dk|k (x′ )dx′ (18.492)
∫
˚k|k (c′ )dc′ .
+p̊T f˚k+1|k (c|c′ ) · D
Integrating both sides of (18.492) results in:
• λ-PHD pseudofilter time-update for number of clutter generators:
∫
˚k+1|k = p̊T N
˚k|k + (1 − pT (x′ )) · Dk|k (x′ )dx′ .
N
(18.493)
We also rewrite (18.491) as:
• λ-PHD pseudofilter time-update for target PHD:
∫
ˆ
Dk+1|k (x) = bk+1|k (x) + pT (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (18.494)
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where
ˆbk+1|k (x) = (1 − p̊T ) · N
˚k|k · sB
k+1|k (x)
(18.495)
is interpreted as the estimated target-birth PHD, with sB
k+1|k (x) being known
˚k|k being interpreted as the estimated target-birth
a priori and (1 − p̊T ) · N
rate.
18.9.1.2
λ-PHD Pseudofilter Measurement Update Equations
The measurement-update equations for the general Bernoulli clutter-generator
CPHD filter were given in (18.44) through (18.53). As with the measurement models for the λ-PHD filter (Section 18.4.6), assume that:
• The predicted joint target/clutter process is Poisson:
¨ k+1|k (x) = eN¨k+1|k ·(x−1) .
G
(18.496)
• The clutter-generator probability of detection is known and constant: p̊κD (c) =
p̊D .
• The clutter spatial distribution ck+1 (z) is known, and the clutter-generator
likelihood function is state-independent: ˚
Lκz (c) = ck+1 (z).
Given this, (18.53) reduces to
˚k+1|k+1 (c)
∑
D
p̊D · ck+1 (z)
= 1 − p̊D +
˚
ˆ
Dk+1|k (c)
z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)
(18.497)
and (18.52) reduces to:
• λ-PHD pseudofilter measurement-update for target PHD:
∑
Dk+1|k+1 (x)
pD (x) · Lz (x)
= 1 − pD (x) +
ˆ
Dk+1|k (x)
z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)
(18.498)
where
ˆ k+1
λ
=
τk+1 (z)
=
˚k+1|k
p̊D N
∫
pD (x) · Lz (x) · Dk+1|k (x)dx.
(18.499)
(18.500)
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635
Since the right side of (18.497) does not involve c, we can integrated this
variable out and replace it with:
• λ-PHD pseudofilter measurement-update for number of clutter generators:
˚k+1|k+1
∑
N
p̊D · ck+1 (z)
= 1 − p̊D +
.
˚
ˆ
Nk+1|k
z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)
(18.501)
Equations (18.498) and (18.501) are identical to the measurement-update
equations for the λ-PHD filter, (18.132) and (18.133).
18.9.2
Pathological Behavior of the λ-PHD Pseudofilter
The λ-PHD pseudofilter exhibits the following behaviors [153]:
1. The λ-PHD pseudofilter cannot always estimate the target-birth rate. From
˚k|k . Suppose,
(18.495), the claimed estimated birth rate is (1 − p̊T ) · N
however, that there is no clutter and therefore that there can be no clutter
˚k|k = 0 and so the estimated target-birth rate is
generators. Then N
incorrectly estimated as 0, regardless of its actual value. This behavior
should be contrasted to that of the classical PHD filter. Even when there is
no clutter, the time-update equation for the classical PHD filter is
∫
Dk+1|k (x) = bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (18.502)
and so this filter still has a target-birth model.
2. The λ-PHD pseudofilter cannot always estimate the clutter rate. This can
be demonstrated using a simple analytical counterexample. Assume (a) that
pD (x) = 1 and p̊D = 1 (that is,both targets and clutter generators are
perfectly detected); (b) pT (x) = pT is constant (that is, targets transition to
targets with constant probability); and (c) the target-to-target probability pT
and the generator-to-generator probability p̊T are “conjugate” in the sense
that, at any given time tk , they satisfy the relationship
pT + p̊T = 1.
(18.503)
Then the λ-PHD pseudofilter’s estimate of the clutter rate is always a fixed
fraction p̊T of the current number of measurements, regardless of what the
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clutter rate actually is:
ˆ k+1 = p̊T · mk+1 .
λ
(18.504)
This equation is proved as follows. From (18.498) and (18.493) it follows
˚k|k = mk+1 . But from (18.493) and the assumption p̊D = 1,
that Nk|k + N
we know that the estimated clutter rate is:
ˆ k+1
λ
=
=
˚k+1|k
p̊D · N
∫
˚k|k + (1 − pT (x′ )) · Dk|k (x′ )dx′
p̊T N
(18.505)
(18.506)
(18.507)
=
˚k|k + (1 − pT ) · Nk|k
p̊T N
˚k|k + p̊T · Nk|k
p̊T · N
=
˚k|k + Nk|k ) = p̊T · mk+1 .
p̊T · (N
(18.509)
=
(18.508)
3. The λ-PHD pseudofilter reduces to the λ-PHD filter if the intermixing
motion model is disabled. The measurement-update equations for the λPHD filter and λ-PHD pseudofilter are already identical, so we need compare
only the time-update equations. Let pT = p̊T = 1—that is, the intermixing
model has been disabled, because targets can transition only to targets and
generators only to generators. Then the pseudofilter time-update equations,
(18.498) and (18.493), reduce to:
∫
fk+1|k (x|x′ ) · Dk|k (x′ )dx′
(18.510)
Dk+1|k (x) =
˚k+1|k
N
=
˚k|k .
N
(18.511)
These are the same as the λ-PHD filter equations, (18.130) and (18.131),
when there are no target appearances or disappearances. This means that
the λ-PHD filter and λ-PHD pseudofilter will have approximately the same
performance, when the target appearance rate and the target disappearance
rate are small.
18.10
CPHD/PHD FILTERS WITH POISSON-MIXTURE CLUTTER
The Poisson-mixture model for unknown clutter was introduced in Section 16.3.3.
This section provides the filtering equations for a CPHD filter for this Poissonmixture model, and its PHD filter special case. These equations were first proposed
RFS Filters for Unknown Clutter
637
by Mahler in 2009 [155]. They are rederived in Section K.23 using Clark’s general
chain rule, (3.91).
It should also be noted that the derivation in Section K.23 can be easily
generalized to clutter generators whose p.g.fl.’s have the form
(
)
∫
κ
Gκk+1 [g|c, c] = Gk+1 1 − c + c g(z) · fk+1
(z|c)dz
(18.512)
where Gk+1 (z) is an arbitrary p.g.f. Thus the results of this section apply to clutter
models more general than Poisson-mixture models.
Since the measurement-update equations for the Poisson-mixture CPHD filter
involve combinatorial sums, they are not computationally tractable as stands. They
are presented here with the expectation that suitable approximation procedures may
eventually be discovered.
Recall that in the Poisson-mixture model, the measurement RFS has the form:
Σk+1 = Tk+1 (x1 ) ∪ ... ∪ Tk+1 (xn ) ∪ Ck+1 (c1 , c1 ), ..., Ck+1 (cν , cν ) (18.513)
where X = {x1 , ..., xn } with |X| = n is the set of target states and
˚ = {(c1 , c1 ), ..., (cν , cν )} with |C|
˚ = ν is the set of clutter generators, and
C
where the p.g.fl.’s of Tk+1 (x) and Ck+1 (c, c) are, respectively,
∫
Gk+1 [g|x] = 1 − pD (x) + pD (x) g(z) · fk+1 (z|x)dz (18.514)
( ∫
)
κ
κ
Gk+1 [g|c, c] = exp c (g(z) − 1) · fk+1 (z|c)dz
(18.515)
κ
=
ec·fk+1 [g−1|c] .
(18.516)
κ
Here κk+1 (z|c, c) = c · fk+1
(z|c) is a family of elemental clutter intensity
functions, parametrized by c and c where c > 0 is the unknown clutter rate and
κ
fk+1
(z|c) is the clutter spatial distribution. It follows that
˚ = Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (cν , cν )
Ck+1 (C)
(18.517)
is also Poisson, and its intensity function (PHD) is a mixture of the form
˚ = c1 · ck+1 (z|c1 ) + ... + cν · ck+1 (z|cν ).
κk+1 (z|C)
(18.518)
Also, the p.g.fl. of the measurement RFS, for both targets and clutter generators, is
∏
κ
˚ = (1 − pD + pD · Lg )X
Gk+1 [g|X ⊎ C]
ec·fk+1 [g−1|c] .
(18.519)
˚
(c,c)∈C
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18.10.1
Poisson-Mixture Clutter-Agnostic CPHD Filter
The Poisson clutter model was introduced in Section 16.3.3. The purpose of this
section is to describe the CPHD and PHD filters associated with this model.
The Poisson-mixture clutter CPHD filter has the form
... →
p̈k|k (n̈)
... →
Dk|k (x)
... →
˚k|k (c, c)
D
→
↑
→
↑
→
p̈k+1|k (n̈)
¨ k+1|k (x)
D
˚k+1|k (c, c)
D
→
↑↓
→
↑↓
→
p̈k+1|k+1 (n̈)
→ ...
Dk+1|k+1 (x)
→ ...
˚k+1|k+1 (c, c)
D
→ ...
where the middle filter propagates PHDs for the targets; where the bottom filter
propagates PHDs for the clutter generators; and where the top filter propagates the
probability distribution on the number n̈ = n + n̊, where n is the number of
targets and n̊ is the number of clutter generators.
The time-update equation for this filter is the same as that for the classical
CPHD filter (see Section 8.5.2). Thus we need only specify the measurementupdate equations. Let
∫
Nk+1|k =
Dk+1|k (x)dx
(18.520)
∫ ∫ ∞
˚k+1|k (c, c)dcdc
˚k+1|k =
N
D
(18.521)
0
( ∫
)
(1 − p (x)) · Dk+1|k (x)dx
∫ ∫ ∞ D−c
˚k+1|k (c, c)dcdc
+
e ·D
0
ϕ¨k =
(18.522)
˚k+1|k
Nk+1|k + N
∫
τW =
pD (x) · LW (x) · Dk+1|k (x)dx
(18.523)
∫ ∫ ∞
˚k+1||k (c, c)dcdc (18.524)
κW =
e−c · c|W | · LκW (c) · D
0
and
LW (x)
=
LκW (c)
=
{
Lz (x) if W = {z}
0
if otherwise
∏
κ
fk+1
(z|c).
z∈W
(18.525)
(18.526)
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639
Then:
• Measurement update for the joint p.g.f.:
( ∑
x
P⊟Z
∏k+1
|P|
(|P|)
¨
¨
·G
k+1|k (x · ϕk )
)
W
· W ∈P N τW +κ
˚
k+1|k +Nk+1|k
( ∑
)
¨ (|Q|) (ϕ¨k )
G
Q⊟Z
k+1|k
k+1
∏
V
· V ∈Q N τV +κ
˚
+N
¨ k+1|k+1 (x) =
G
k+1|k
(18.527)
.
k+1|k
• Measurement update for the target PHD:
=
Dk+1|k+1 (x)
(18.528)
Dk+1|k (x)


¨k )
¨ (|P|+1) (ϕ
G
1−pD (x)
k+1|k
∑
·
˚k+1|k
¨k ) 
¨ (|P|) (ϕ
+N
G
ωP  Nk+1|k
.
k+1|k
∑
pD (x)·LW (x)
+ W ∈P τW +κW
P⊟Zk+1
• Measurement update for the clutter PHD:
˚k+1|k+1 (c, c)
D
˚k+1|k (c, c)
D
=
e−c
(18.529)

∑
P⊟Zk+1

ωP 
(|P|+1)
¨k )
¨
G
(ϕ
k+1|k
1
˚k+1|k · G
¨k )
¨ (|P|) (ϕ
Nk+1|k +N
k+1|k
+
∑
c|W | ·Lκ
W (c)
W ∈P τW +κW
The summations are taken over all partitions P
Zk+1 , and

.
of the measurement set
τW +κW
¨ (|P|) (ϕ¨k ) · ∏
G
˚k+1|k
W ∈P Nk+1|k +N
ωP = ∑
∏
(|Q|)
τV +κV
¨
¨
˚
Q⊟Zk+1 Gk+1|k (ϕk ) ·
V ∈Q N
+N
k+1|k

.
(18.530)
k+1|k
• Estimate of the clutter intensity function:
κ̂k+1 (z) =
∫ ∫ ∞
0
κ
˚k+1|k (c, c)dcdc.
c · fk+1
(z|c) · D
(18.531)
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Equation (18.531) follows from an analysis similar to that presented in
Section 18.3.5 and in Section K.21. That is, from (K.436) of Section K.21, if
˚ is the p.g.fl. of the clutter RFS for a given set C
˚ of clutter generators,
Gk+1 [g|C]
then the predicted expected p.g.fl. of the clutter process is
∫
¯
˚ · f˚k+1|k (C|Z
˚ (k) )δ C.
˚
Gk+1 [g] = Gk+1 [g|C]
(18.532)
From (4.75) the predicted average intensity function is, therefore,
κ̄k+1 (z)
¯ k+1
δG
[1]
∫ δz
δGk+1 ˚
˚ (k) )δ C
˚
[1|C] · fk+1|k (C|Z
δz
∫
˚ · f˚k+1|k (C|Z
˚ (k) )δ C
˚
κk+1 (z|C)
=
=
=
(18.533)
(18.534)
(18.535)
˚ is given by (18.518). Thus
where κk+1 (z|C)


∫
∑
κ
˚ (k) )δ C
˚ (18.536)

κ̄k+1 (z) =
c · fk+1
(z|c) · f˚k+1|k (C|Z
˚
(c,c)∈C
∫ ∫ 1
=
κ
˚k+1 (c, c)dcdc
c · fk+1
(z|c) · D
(18.537)
0
where the last equation is due to Campbell’s theorem, (4.96).
18.10.2
Poisson-Mixture Clutter-Agnostic PHD Filter
The Poisson-mixture clutter (PMC) PHD filter arises when we assume that the joint
predicted target-clutter RFS is Poisson, in which case
˚k+1|k )·(ẍ−1)
¨ k+1|k (ẍ) = e(Nk+1|k +N
G
.
(18.538)
It has the form
... →
Dk|k (x)
→
¨ k+1|k (x)
D
... →
˚k|k (c, c)
D
→
˚k+1|k (c, c)
D
→
↑↓
→
Dk+1|k+1 (x)
→ ...
˚k+1|k+1 (c, c)
D
→ ...
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641
where the top filter propagates PHDs for the targets and the bottom filter propagates
PHDs for the augmented clutter generators.
The time-update equation for this filter is the same as that for the classical
PHD filter. The measurement-update equations are as follows ([155], Corollary 1):
• Measurement update for target PHD:
∑
Dk+1|k+1 (x)
= 1 − pD (x) +
Dk+1|k (x)
ωP
P⊟Zk+1
∑ pD (x) · LW (x)
. (18.539)
τW + κ W
W ∈P
• Measurement update for clutter PHD:

˚k+1|k+1 (c, c)
D
= e−c 1 +
˚k+1|k (c, c)
D
∑
P⊟Zk+1
ωP
∑ c|W | · LW (c)
τW + κ W
W ∈P

 . (18.540)
As before, the summations are taken over all partitions P of the measurement set Zk+1 . Also,
ωP = ∑
18.11
∏
W ∈P (τW + κW )
Q⊟Zk+1
∏
.
(18.541)
V ∈Q (τV + κV )
RELATED WORK
The following research efforts are related to that described in this chapter, in that
they propose a PHD filter-like structure integrated with clutter estimation. They are
as follows:
1. Section 18.11.1: The decoupled clutter-target PHD filter of Feng et al.—in
which the clutter intensity function is estimated separately and then used in
the classical PHD filter.
2. Section 18.11.2: The “dual PHD filter” of Jonsson et al.—an independently
conceived version of the λ-PHD filter of Section 18.4.6.
3. Section 18.11.3: The “iFilter”—which is identical to the λ-PHD pseudofilter
of Section 18.9.
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18.11.1
Decoupled Target-Clutter PHD Filter
This filter was proposed by Feng Lian, Chongzhao Han, and Weifeng Liu in 2010
[84]. Unlike the λ-CPHD and κ-CPHD filters, it is uncoupled: clutter estimation
and target tracking are not inherently integrated as a single recursive procedure.
Rather, the clutter intensity function is first estimated as a parametrized finitemixture model, and then employed in the classical PHD filter. As such, it can also
be applied to the CPHD filter and to classical multitarget tracking algorithms such
as MHT.
The core of the approach is a particular method for estimating the clutter
intensity function. The number of target-generated measurements is assumed to be
much smaller than the average number of clutter measurements (that is, the clutter
rate). It is also assumed that the unknown clutter intensity function does not vary
with time. Finally, it is assumed that the unknown clutter RFS is Poisson.
First the constant clutter rate is estimated as being approximately equal to the
average number of measurements:
k
1∑
λk ∼
|Zi |
=
k i=1
(18.542)
where Z1 , ..., Zk is the time sequence of measurement sets. Then the clutter spatial
distribution is modeled as a Gaussian mixture:
c(z|θ, µ) =
µ
∑
cj · NCj (z − cj ),
j=1
µ
∑
cj = 1
(18.543)
j=1
with unknown parameters
θ = (c1 , C1 , c1 , ..., cµ , Cµ , cµ )
(18.544)
with cj ∈ RN and an unknown number µ of components. Assuming conditional
independence of the measurements with respect to the parameters, the likelihood
function with respect to measurement sets Z is
∏
Lθ (Z) =
c(z|θ, µ).
(18.545)
z∈Z
The prior distribution f (θ) of θ is constructed by assuming that its parameter
variables are distributed as follows: (c1 , ..., cM ) is Dirichlet-distributed ([83], p.
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643
62); each cj is normally distributed; and each Cj is Wishart-distributed ([83],
p. 205). Given this, the posterior distribution f (θ|Z) is constructed and the MAP
estimate computed using either expectation-maximization (EM) or Markov-chain
Monte Carlo (MCMC) methods.
Feng et al. implemented both EM and MCMC versions of their algorithm, and
tested them in two-dimensional simulations. In these simulations, a linear-Gaussian
sensor with pD = 0.95 observed up to five appearing and disappearing targets
following curvilinear trajectories. The clutter process was Poisson with clutter rate
λk+1 = 50, and the clutter spatial distribution ck+1 (z) was a three-component
Gaussian mixture superimposed with a uniform spatial distribution.
The authors compared the EM and MCMC versions of their algorithm with
a conventional PHD filter—that is, one which was given the correct λk+1 and
ck+1 (z). They reported that both versions greatly outperformed the conventional
PHD filter in regard to both target-number estimation and target-localization accuracy. Their algorithms were also effective in estimating both λk+1 and ck+1 (z).
18.11.2
The “Dual PHD” Filter
In 2012 Jonsson, Degerman, Svensson, and Wintenby proposed this PHD filter as
a means of detecting and tracking multiple targets in unknown background clutter
[127].3 They also implemented it using Gaussian mixture techniques and applied
it to targets moving in Doppler clutter. The measurement-update equations (the
unnumbered equations in Section IV of [127]) for their filter correspond to the
Bernoulli clutter model of Section 18.3.5, in which p̊κD (c̊) = p̊κD (c). But since
p̊κD (c) must be assumed constant in a GM implementation, the “dual PHD filter” is
identical to the GM implementation of the λ-PHD filter of Section 18.4.6.
Jonsson et al. conducted simulations to test a Gaussian mixture (GM) implementation of their approach. The sensor is a high-flying airborne radar whose
mission is to track low-flying objects. The radar also collects Doppler clutter measurements produced by ground road traffic. Exploiting the fact that ground and
airborne objects have different dynamics, Jonsson et al. demonstrated that their PHD
filter successfully detected and tracked six appearing and disappearing targets with
traffic clutter on the order of 40 measurements per scan.
3
Though Jonsson et al. cited the paper [194] by Mahler et al., they apparently did not notice that their
filter is a special case of the λ-CPHD filter described there.
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18.11.3
The “iFilter ”
The “multitarget intensity filter” (MIF) was proposed by Streit and Stone in 2008
[281] and subsequently renamed the “iFilter.” Its measurement-update equation is
identical to the measurement-update equation for the λ-PHD filter of Section 18.4.3.
However, its time-update equations are
Dk+1|k (x)
=
˚k|k +
ψk (x|ϕ) · N
∫
ψk (x|x′ ) · Dk|k (x′ )dx′ (18.546)
˚k+1|k
N
=
˚k|k +
ψk (ϕ|ϕ) · N
∫
ψk (ϕ|x′ ) · Dk|k (x′ )dx′ (18.547)
where, in the notation of Section 18.4.1, (a) ψk (ϕ|ϕ) = p̊T ; (b) ψk (x|ϕ) =
′
′
′
′
(1 − p̊T ) · sB
k+1|k (x); (c) ψk (ϕ|x ) = 1 − pT (x ); and (d) ψk (x|x ) = pT (x ) ·
fk+1|k (x|x′ ).
The authors asserted that their claimed “Poisson point process” or “PPP”
derivation of the MIF also demonstrated that “multitarget intensity filters [that is,
the MIF and the PHD filter] can be understood in essentially elementary terms
[using]. . . PPP’s at an elementary level” ([281] Section 1, 2nd paragraph).
However, the claimed “PPP” derivations of both the MIF and the PHD filter
have serious mathematical errors and seriously restrictive hidden assumptions (see
Appendix A of [163]). A few of the mathematical errors were summarized in
footnote four in Section 8.4.6.8. As for hidden assumptions: no target spawning;
the distribution fk|k (X|Z (k) ) (and not just fk+1|k (X|Z (k) )) is Poisson; the
intensity function of the target-birth RFS is constant; and the state space is bounded.
In any case, and as can be seen from the discussion in Section 18.3, derivation
of the MIF requires only straightforward algebra. Point process theory, elementary
or otherwise, is completely unnecessary. Indeed, the MIF is identical to the λ-PHD
pseudofilter of Section 18.9. As such, it exhibits the pathological behavior identified
there (see Section 18.9.2). That is, it does not (as claimed) always estimate the
clutter rate and does not (as claimed) always estimate the target-birth rate. It does
not (as claimed) include the classical PHD filter as a special case. Finally, and
as was noted in Section 18.9.2, if the intermixing motion model is approximately
disabled (which occurs if the target appearance and target-disappearance rates are
small) then the MIF will be approximately identical to the λ-PHD filter of Section
18.4.6.
Part IV
RFS Filters for Nonstandard
Measurement Models
645
Chapter 19
RFS Filters for Superpositional Sensors
19.1
INTRODUCTION
Many if not most sensors do not conform to the assumptions of the “standard”
multitarget measurement model. Consider, for example, a mechanically rotating
radar or a phased-array radar, either of which generates a continuous real-valued
signature. The real-valued signature is the real part of a complex-valued signature,
which in turn is a sum (“superposition”) of the complex-valued signatures generated
by the individual targets and the background clutter. Those excursions of the realvalued signature that exceed a (fixed or adaptive) threshold, produce a finite set Z
(a “scan” or “frame”) of “detections.” If the targets are sufficiently distant, then
each target can be modeled as a mathematical point that generates at most a single
detection at a time (the “small target” assumption).
Detection-extraction schemes such as this tend to discard useful information,
thus tending to reduce tracking performance. For example, when the targets are
close together (relative to the angular and range resolution of the radar), amplitude
detection approaches can merge detections, thereby failing to resolve adjoining
targets.
In principle, a tracking filter that exploits the full superpositional signal model
could achieve better performance. The purpose of this chapter is to describe CPHD
filters specifically designed to operate with superpositional models. These will be
collectively described as Σ-CPHD filters.
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Examples of Superpositional Sensor Models
To emphasize the potential importance of this chapter, five different real-world
applications involving superpositional sensors will be briefly described:
1. Section 19.1.1.1: Surveillance radar.
2. Section 19.1.1.2: Time-direction-of-arrival (TDOA) for sinusoidal signals.
3. Section 19.1.1.3: Multi-user detection (MUD) in communications networks.
4. Section 19.1.1.4: Radio-frequency tomography for surveillance of interior
spaces.
5. Section 19.1.1.5: Thermopile arrays for heat-based localization.
19.1.1.1
Surveillance Radar
Suppose that targets are sufficiently distant that they can be regarded as point targets.
Also assume that the radar is narrowband, that is, its radian center frequency ωc
is much larger than its bandwidth. For a given azimuth and elevation α, θ, the
transmitted signal has the form
st = χt · eι·ωc t
(19.1)
√
where ι = −1 is the complex unit and where the complex envelope χt specifies
the shape of the radar pulse. For example, if χt = 1[0,T ] (t), then the radar transmits
a single rectangular pulse with pulsewidth T .
If the transmitted signal impinges on a point target with state x, a reflected
signal is received at the radar:
ηt (x) = At (x) · χt−τt (x) · eι·(t−τt (x))·(ωc +ωt (x)) + Vt .
(19.2)
where Vt is sensor noise, τt (x) is the time delay, ωt (x) is the Doppler shift,
and At (x) depends on factors such as radar cross section (RCS), atmospheric
absorption, and so on
In the event that there are multiple targets X = {x1 , ..., xn }, the noisy signal
received at the radar is the superposition
Zt =
∑
x∈X
ηt (x) + Vt .
(19.3)
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19.1.1.2
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Time-Direction-of-Arrival (TDOA) for Sinusoidal Signals
Balakumar, Sinha, Kirubarajan, and Reilly [14] have considered the following
TDOA problem. The sensor is a linear array of M identical receivers (for example,
antennas or microphones) with sensor positions d0 , 2d0 , ..., M d0 . Let τ0 ≜ d0 /c
where c is the signal propagation speed. Suppose that there are an unknown
number of unknown sinusoidal sources (for example, radio-frequency transmitters
or acoustic sources), which are sufficiently distant from the array that the signal
from each source impinges on it as a plane wave. Each source is characterized by
its signal amplitude α, its center frequency ω, its bandwidth β, and its angle
of arrival ϕ at the array. Assume that all sources are narrowband—that is, that
β ≪ ω—in which case β can be neglected. We wish to determine the number n
and states
xi = (αi , ωi , ϕi )
(19.4)
of the sources for i = 1, ..., n.
Towards this end, let X = {x1 , ..., xn } with |X| = n and
ηj (x)
η(x)
=
=
η(X)
=
αi · e−ι·jτ0 ωi ·sin ϕi
(η1 (x), ..., ηM (x))T
n
∑
η(xi )
(19.5)
(19.6)
(19.7)
i=1
√
where ι = −1 denotes the imaginary unit. Then the complex-valued signal from
the ith source, received at the jth receiver, is ([14], Eqs. (1,4))
Zj,i = ηj (xi ) + Wj,i
(19.8)
and Wj,i is a complex-valued zero-mean Gaussian noise vector (see Appendix H
for a brief discussion of Gaussian distributions with complex-valued arguments).
The total signal received at the jth receiver is the superposition of the signals
generated by all of the sources
Zj =
n
∑
ηj (xi ) + Wj ,
(19.9)
i=1
where Wi is also zero-mean. Thus the measurement received by the array is the
random vector
Z =η(X) + W
(19.10)
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T
where Z = (Z1 , ..., ZM ) and W = (W1 , ..., WM ) .
Equation (19.10) provides another example of a superpositional measurement
model.
To apply a PHD filter to this tracking problem, Balakumar et al. converted
this superpositional measurement model to a detection measurement model. Any
collected measurement z = (z1 , ..., zM )T is used to construct a set Z˜ =
{z̃1 , ..., z̃ñ } of pseudo-measurements, which are then employed as inputs to the
PHD filter. Let
M
1 ∑
Fz (ω) =
zj · e−ι·jωτ0
(19.11)
M j=1
be the discrete Fourier transform (DFT) of z. The DFT will have ñ peaks at
frequencies ω = ω̃1 , ..., ω̃ñ with respective amplitudes α1 , ..., αñ . Separate
the peaks and interpret each as the DFT Fz̃i (ω) of a pseudo-measurement z̃i
generated by one of the sources. An approximate likelihood Li (x) is constructed
for z̃i ([14], Eq. (11)) and used in the PHD measurement-update equation.
An obvious question presents itself: Are there PHD or CPHD filters that can
address the original superpositional data, rather than the detection data extracted
from it?
19.1.1.3
Multi-User Detection (MUD) in Communications Networks
MUD problems occur in dynamic, mobile, multiple-access wireless digital communications networks in which the goal is to detect, track, and identify system users as
they enter and exit. Current systems are typically based on the assumption that the
number of active users is constant, known, and equal to the maximum number of
registered users. In actuality, the actual number of users changes with time and is
appreciably smaller than the maximum number. Also, the set of active users is typically unknown at the receiver. System efficiency and capacity could be increased
if active users could be rapidly identified ([19], p. 54).
Besides cellphone networks, applications include ad hoc networks (in which
MUD facilitates optimal transmission strategies); and spatial multiplexing (in which
MUD facilitates the proper allocation of system power).
Biglieri, Lops, and Angelosante have written a series of publications applying
finite-set statistics techniques to MUD and related applications [5], [6], [20], [7],
[8], [21], including the monograph [19]. The purpose of this section is to sketch the
basic elements of these applications.
In MUD the time-varying state xt consists of various parameters that
potentially allow the identification of a given user. One parameter is a label ℓ
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associated with the user; and another is the message symbol dt generated by the
user (drawn from some source alphabet). Assume that there is a “reference user,”
which has known state x0 and which is always active. The signal collected at the
receiver at time t has the form
∑
Zt = ηt (x0 ) +
ηt (xt ) + Vt
(19.12)
x∈Xt
where Xt is the set of unknown users and Vt is the receiver noise. Thus the
likelihood function has the form
(
)
∑
ft (zt ) = fVt z − ηt (x0 ) −
ηt (xt ) .
(19.13)
x∈Xt
Biglieri et al. have implemented a full multitarget recursive Bayes filter that
directly incorporates this likelihood function. They also employ novel Bayesoptimal maximum a posteriori (MAP) multitarget state-estimators specifically designed for MUD. Nevertheless, it is doubtful that any approach based on the multitarget Bayes filter would be computationally tractable in a communications network
of realistic size. Consequently, CPHD filters specifically designed for superpositional models are of potential interest.
19.1.1.4
Radio-Frequency Tomography
RF tomography [236] is an approach for determining the locations and velocities
of unknown targets (such as people) in unknown, denied areas (such as closed
buildings). A number ν of transmitter-receiver units are situated around the denied
area. Each receiver measures the received signal strength (RSS) due to the signal
generated by each transmitter. At any moment, this results in the collection of
m = ν(ν − 1)/2 RSS measurements. Any object in the denied area, whether a
target of interest or otherwise, attenuates the signal along any transmitter-to-receiver
line-of-sight on which a target happens to be located.
Prior to tracking, measurements are collected with the purpose of estimating
the clutter and noise background. If ν is large enough, it is possible in principle
to detect, locate, and track all moving objects in the denied area. Algorithmic
approaches, involving techniques such as the EM algorithm and sequential Monte
Carlo (SMC) filtering, have recently been devised for this purpose [329], [326].
Since RF tomography inherently involves superpositional sensors, it would
be useful to develop PHD or CPHD filters applicable to RF tomography and related
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applications. This challenge was addressed by Thouin, Nannuru, and Coates in
2011 [290]. The model they employed is as follows.
Let j = 1, ..., m be an index that specifies each transmitter-to-receiver link.
Then the single-target measurement function for the RSS of each link at time tk+1
is ([290], Eq. (6)):
(
)
λ2j (x)
j
ηk+1 (x) = ϕ · exp −
(19.14)
2σλ2
where λj (x) is the perpendicular distance between a target located at x = (x, y)
and the jth link; and where ϕ and σλ are empirically-determined constants.
Thus the further a target at x is from the jth link, the larger the value of λj (x)
and the smaller the RSS. Since the RSS due to multiple targets X = {x1 , ..., xn }
is the superposition of the RSSs due to each target individually, the (noisy) total
attenuation on the jth link is ([290], Eq. (7))
j
ηk+1
(X) =
∑
j
ηk+1
(x) + Vk+1
(19.15)
j
NRk+1 (zj − ηk+1
(X)).
(19.16)
x∈X
with total likelihood function
L(z1 ,...,zm ) (X) =
m
∏
j=1
19.1.1.5
Thermopile Arrays
A thermopile is a low-resolution heat detector that, within its field of view, measures
the heat radiation of an object relative to ambient temperature. An array of
thermopiles can, at least in principle, be used to detect, locate, and track moving
objects (for example, humans) that are warm in comparison to their surrounding
environment (for example, rooms). Hauschildt et al. have investigated the use of
finite-set statistics techniques for this purpose [133], [104], [105].
Suppose that there are m thermopiles in the array, each regarded as a pixel in
a heat image. Let x be the state of a single target. Then the measurement function
for the jth pixel can be approximated as ([104], Eq. (57)):
j
ηk+1
(x) = aj · [arctan (bj (ϕj + ∆j ϕ + θj )) − arctan (b(ϕj − ∆j ϕ + θj ))]
(19.17)
where ϕ is the angle of arrival of the object at the pixel, ∆j ϕ is the angular size
of the target, θj is the pixel orientation, and aj , bj are calibration constants. The
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653
measurement function for the entire image is, therefore,
1
m
ηk+1 (x) = (ηk+1
(x), ..., ηk+1
(x))T .
(19.18)
Thus the noisy measurement model for multiple targets X = {x1 , ..., xn } is the
superposition
∑
Zk+1 =
ηk+1 (x) + Vk+1 .
(19.19)
x∈X
19.1.2
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• Superpositional sensors are ubiquitous, and multitarget tracking filters specifically designed for them could offer a significant improvement over conventional, detection-based tracking approaches (Section 19.1.1).
• An exact formula for a CPHD filter for general superpositional sensors can be
derived (Section 19.2). However, it is computationally intractable in general.
• An exact, closed-form Gaussian mixture solution of the exact superpositional
CPHD filter exists, but is only partially tractable (Section 19.3).
• A computationally tractable approximate superpositional CPHD filter is possible, when implemented using particle techniques (Section 19.4).
19.1.3
Organization of the Chapter
The chapter is organized as follows:
1. Section 19.2: An exact (but computationally intractable) superpositional
CPHD filter (“exact Σ-CPHD filter”).
2. Section 19.3: A semi-tractable closed-form Gaussian mixture superpositional
CPHD filter, based on an approximation due to Hauschildt, and referred to
here as the “Hauschildt Σ-CPHD filter.”
3. Section 19.4: A tractable approximate superpositional CPHD filter, based on
a Campbell’s theorem approximation originally devised by Thouin, Nannuru,
and Coates (“TNC Σ-CPHD filter”).
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19.2
EXACT SUPERPOSITIONAL CPHD FILTER
Mahler derived exact filtering equations for a general superpositional CPHD filter
in 2009 [156]. It will hereafter be referred to as the exact superpositional CPHD
filter, or exact Σ-CPHD filter for short. The purpose of this section is to summarize
the filtering equations for this filter. Since the time-update equation is the same as
that for the classical CPHD filter (Section 8.5.2), it is necessary to discuss only the
measurement-update.
Suppose that we have a single sensor with the superpositional measurement
model with likelihood function
fk+1 (z|X) = fVk+1 (z − ηk+1 (X))
where
ηk+1 (X) =
{
∑
0
x∈X ηk+1 (x)
if
if
X=∅
X ̸= ∅
(19.20)
(19.21)
where ηk+1 (x) is the (real- or complex-valued) signature-vector generated by a
single target; and where Vk+1 is a zero-mean random (real or complex) noise
vector.
Suppose that the sensor delivers a new measurement zk+1 , after having
collected a time-stream Z k : z1 , ..., zk of measurement vectors. Assume, as
with the classical CPHD filter (Chapter 8), that the predicted multitarget distribution
fk+1|k (X|Z k ) is approximately an i.i.d.c. RFS:
fk+1|k (X|Z k ) ∼
= |X|! · pk+1|k (|X||Z (k) ) ·
∏
sk+1|k (x|Z (k) )
(19.22)
x∈X
where pk+1|k (n|Z k ) is the predicted cardinality distribution with probability
generating function (p.g.f.)
Gk+1|k (x|Z (k) ) =
∑
pk+1|k (n|Z k ) · xn
(19.23)
−1
sk+1|k (x|Z (k) ) = Nk+1|k
· Dk+1|k (x|Z k )
(19.24)
n≥0
and where
is the predicted target spatial distribution, and where
∫
Nk+1|k = Dk+1|k (x|Z k )dx.
(19.25)
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655
Abbreviate
=
pk+1|k (n|Z (k) )
(19.26)
s(x)
f (z)
=
=
sk+1|k (x|Z
fVk+1 (z)
(k)
(19.27)
(19.28)
ηx
=
ηs (z)
=
ηk+1 (x)
∫
s(x) · δηx (z)dx
pn
)
(19.29)
(19.30)
where δηx (z) is the Dirac delta density function concentrated at ηk+1 (x).
Also, let
∫
(g1 ⋆ g2 )(z) =
g1 (w) · g2 (w − z)dw
(19.31)
be the convolution of the functions g1 (z) and g2 (z). Then the convolutional power
g ⋆j of g(z) is recursively defined by
g ⋆0
=
δ0
⋆1
=
=
g
g ⋆ g ⋆(j−1)
g
g ⋆j
(19.32)
(19.33)
(19.34)
(j ≥ 1) .
Given this, the measurement-update equations for the exact Σ-CPHD filter
are ([156], Theorem 1):
• Measurement update for the cardinality distribution:
pk+1|k+1 (n) = ∑
(f ⋆ ηs⋆n )(zk+1 )
⋆j
j≥0 pj · (f ⋆ ηs )(zk+1 )
· pn .
• Measurement update for the expected number of targets:
∑
⋆n
n≥1 n · pn · (f ⋆ ηs )(zk+1 )
.
Nk+1|k+1 = ∑
⋆j
j≥0 pj · (f ⋆ ηs )(zk+1 )
(19.35)
(19.36)
• Measurement update for the spatial distribution:
sk+1|k+1 (x) =
·
∑
1
Nk+1|k+1
⋆(n−1)
)(zk+1 − ηx)
n≥1 n · pn · (f ⋆ ηs
· s(x).
∑
⋆j
j≥0 pj · (f ⋆ ηs )(zk+1 )
(19.37)
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Substituting pn = e−Nk+1|k ·Nk+1|k
/n! into (19.37) yields the measurementupdate for the exact Σ-PHD filter special case of the exact Σ-CPHD filter:
Dk+1|k+1 (x) =
∑
1
⋆n
n≥0 n! · (f ⋆ ηD )(zk+1 − ηx)
· Dk+1|k (x)
∑
⋆j
1
j≥0 j! · (f ⋆ ηD )(zk+1 )
where
ηD (z) =
19.3
∫
Dk+1|k (x) · δηx (z)dx.
(19.38)
(19.39)
HAUSCHILDT’S APPROXIMATION
The measurement-update equations for the exact Σ-CPHD filter are computationally intractable in general. However, in 2011 Hauschildt proposed an approximation
that results in a closed-form and at least partially tractable solution of the exact ΣCPHD filter [104]. He called it the “superpositional sensor (SPS) CPHD filter,” but
it will be referred to here as the Hauschildt Σ-CPHD filter.1
The section is organized as follows:
1. Section 19.3.1: An overview of the Hauschildt approximation.
2. Section 19.3.2: Measurement models for the Hauschildt Σ-CPHD filter.
3. Section 19.3.3: Measurement update equations for the Hauschildt Σ-CPHD
filter.
4. Section 19.3.4: Implementations of the Hauschildt Σ-CPHD filter.
19.3.1
Hauschildt Σ-CPHD Filter: Overview
The multitarget likelihood function for the superpositional model is
fk+1 (z|X) = fVk+1 (zk+1 − ηk+1 (X)).
1
(19.40)
Note: The measurement-update equation in [104] is not quite correct, because of a minor algebra
error in its derivation. The corrected filtering equations will be presented in this section.
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The Hauschildt approximation therefore begins with the observation that the
measurement-updated cardinality distribution, PHD, and expected number of targets are given by the equations
∫
f
(z
− ηk+1 (X)) · fk+1|k (X)δX
|X|=n Vk+1 k+1
pk+1|k+1 (n) = ∫
(19.41)
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
( ∫
)
fVk+1 (zk+1 − ηk+1 (x) − ηk+1 (X))
·fk+1|k (X ∪ {x})δX
∫
Dk+1|k+1 (x) =
(19.42)
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
∫
|X| · fVk+1 (zk+1 − ηk+1 (X)) · fk+1|k (X)δX
∫
Nk+1|k+1 =
.(19.43)
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
Assume that the predicted multitarget distribution fk+1|k (X) is i.i.d.c. (as defined
in Section 4.3.2):
pk+1|k (|X|)
fk+1|k (X) = |X|! ·
|X|
X
· Dk+1|k
.
(19.44)
Nk+1|k
Let the predicted PHD be a Gaussian mixture as in Section 9.5:
νk+1|k
Dk+1|k (x) =
∑
k+1|k
k+1|k
· NP k+1|k (x − xi
wi
)
(19.45)
i
i=1
and let the single-target likelihood function fk+1 (z−ηk+1 (x)) be linear-Gaussian:
ηk+1 (x)
=
Hk+1 x
(19.46)
fVk+1 (z)
=
NRk+1 (z).
(19.47)
It follows that, for each X, the predicted multitarget distribution fk+1|k (X)
is a (very complicated) Gaussian mixture. Consequently, repeated application of
the fundamental Gaussian identity, (2.3), allows one to compute exact closedform formulas for the numerators and denominator of (19.41) through (19.43).
Furthermore, the numerator of (19.42) turns out to be a Gaussian mixture.
As with the GM-CPHD filter, it is assumed that the cardinality distribution vanishes for sufficiently large n. It then turns out that pk+1|k+1 (n) and
Dk+1|k+1 (x) can both be written in exact closed form and that, in particular,
Dk+1|k+1 (x) is a Gaussian mixture. Unscented Kalman filter (UKF) techniques
can be used to extend the approach to moderately nonlinear measurement functions
ηk+1 (x)—see [104], Section V-B.
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19.3.2
Hauschildt Σ-CPHD Filter: Models
The modeling assumptions underlying the Hauschildt Σ-CPHD filter are as follows:
• Constant probability of target survival:
pS,k+1|k (x′ ) = pS,k+1|k .
(19.48)
• Linear-Gaussian single-target motion model:
fk+1|k (x|x′ ) = NQk (x − Fk x′ ).
(19.49)
• The single-target measurement function is linear:
(19.50)
ηk+1 (x) = Hk+1 x
where Hk+1 is the measurement matrix. Define
{
0 (∑
) if
∑
Hk+1 X =
Hx
=
H
x
if
x∈X
x∈X
X=∅
.
X ̸= ∅
(19.51)
• Measurement noise is Gaussian:
(19.52)
fVk+1 (z) = NRk+1 (z).
The time-update equation for the Hauschildt Σ-CPHD filter is identical to
that for the GM-CPHD filter (Section 9.5.5.2). Thus only the measurement-update
equations need be described.
19.3.3
Hauschildt Σ-CPHD Filter: Measurement Update
Assume that the predicted PHD is a Gaussian mixture:
νk+1|k
Dk+1|k (x) =
∑
k+1|k
wi
k+1|k
· NP k+1|k (x − xi
)
(19.53)
i
i=1
with
νk+1|k
Nk+1|k =
∑
i=1
k+1|k
wi
.
(19.54)
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In what follows, let a multi-index with values in {1, ..., νk+1|k } be defined as
follows:
• For n ≥ 1, it is an n-tuple o = (o1 , ..., on ) with o1 , ..., on ∈
{1, ..., νk+1|k }.
• For n = 0, the empty multi-index is denoted as o = ().
Define
|o|
o1
=
{
=
{
0
n
if
if
o = ()
o = (o1 , ..., on )
(19.55)
()
o1
if
if
o = ()
o = (o1 , ..., on )
(19.56)
and
wok+1|k
=
{
xk+1|k
o
=
{
Pok+1|k
=
{
1
k+1|k
· · · w on
if
if
o = ()
o = (o1 , ..., on )
(19.57)
k+1|k
k+1|k
x o1
+ ... + xon
if
if
o = ()
o = (o1 , ..., on )
(19.58)
0
k+1|k
+ ... + Pon
if
if
o = ()
. (19.59)
o = (o1 , ..., on )
k+1|k
w o1
0
k+1|k
P o1
Given this, the measurement-update equations for the Hauschildt Σ-CPHD filter
are:
• Measurement updated cardinality distribution ([104], Eq. (43)):
pk+1|k+1 (n)


=
·NR



k+1|k
T
Hk+1
k+1 +Hk+1 Po
∑
·NR
(19.60)

pk+1|k (n) ∑
k+1|k
n
o:|o|=n wo
Nk+1|k
(
k+1|k
zk+1 − Hk+1 xo
pk+1|k (|o′ |)
0≤|o′ |≤n
max
|o′ |
Nk+1|k
k+1|k
T
Hk+1
k+1 +Hk+1 Po′
(
) 
k+1|k
· w o′
k+1|k
zk+1 − Hk+1 xo′

) 

where, by (19.55) through (19.59), if |o| = 0 then
T
wo · NRk+1 +Hk+1 Po Hk+1
(zk+1 − Hk+1 xo ) = NRk+1 (zk+1 ).
(19.61)
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• Measurement updated PHD ([104], Eqs. (30-42)):
∑
Dk+1|k+1 (x) =
pk+1|k+1 (|o|) · |o|
(19.62)
1≤|o|≤nmax
·NP˜ k+1|k+1 (x − x̃k+1|k+1
)
o
o
where
x̃k+1|k+1
o
=
P˜ok+1|k+1
=
Ko
=
(
)
xo1 + Ko zk+1 − Hk+1 xk+1|k
o
(19.63)
(I − Ko Hk+1 )Pok+1|k
(19.64)
1
−1
(
)
T
T
Pok+1|k
Hk+1
Hk+1 Pok+1k Hk+1
+ Rk+1
(.19.65)
1
• Measurement updated expected number of targets:
∑
Nk+1|k+1 =
pk+1|k+1 (|o|) · |o|
(19.66)
1≤|o|≤nmax
=
n∑
max
n
pk+1|k+1 (n) · n · νk+1|k
.
(19.67)
n=1
These results are established in Section K.26.
The PHD filter special case is given by the equation
∑
Dk+1|k+1 (x) =
pk+1|k+1 (|o|) · |o| · NP˜ k+1|k+1 (x − x̃o )
(19.68)
o
1≤|o|≤nmax
where
(19.69)
pk+1|k+1 (n)


=
·NR


1
n!
k+1|k
k+1 +Hk+1 Po
∑
·NR
∑

k+1|k
o:|o|=n wo
T
Hk+1
0≤|o′ |≤n
(
k+1|k
zk+1 − Hk+1 xo
k+1|k
1
|o′ |! · wo′
max (
k+1|k
T
Hk+1
k+1 +Hk+1 Po′
) 
.
k+1|k
zk+1 − Hk+1 xo′
) 
The computational complexity of these filtering equations can be reduced by
k+1|k
k+1|k
k+1|k
noticing that wo
, xo
, Po
are invariant with respect to permutations of
the components of o.
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Remark 80 (Computational complexity) Hauschildt’s Σ-CPHD filter is computationally demanding, and thus probably applicable only when the number of targets is small. However, it should be pointed out that, in many applications, only a
few targets are closely-spaced at any given time; and that superpositional models
are necessary only when targets are closely spaced. (For, otherwise, conventional
detection methods can be used to resolve them.) In such situations one could apply
the Hauschildt Σ-CPHD filter to the closely-spaced targets, and CPHD or other
nonsuperpositional filters to the remaining, less closely-spaced, targets.
19.3.4
Hauschildt Σ-CPHD Filter: Implementations
Hauschildt has applied an unscented Kalman filter (UKF) Gaussian-mixture implementation of his Σ-CPHD filter to the thermopile application discussed in Section
19.1.1.5 [104]. The thermopile was assumed to be a line array consisting of eight
pixels, observing targets that move in a one-dimensional surveillance space at constant velocity.
In the first simulation, three targets appear and disappear while moving in
the positive direction. The second target is present for most of the scenario, and is
crossed by the first target and later by the second target. The measurement noise in
each pixel is small with variance σ 2 = (0.05)2 .
The GM-Σ-CPHD filter was compared with a conventional GM-CPHD filter,
with the measurements for the latter created by applying a detection threshold to the
pixel data. While the GM-CPHD filter was unable to track the targets during the
two target crossings, the Σ-CPHD filter successfully tracked them.
In the second simulation, a fourth target was introduced at the same time as
the third, and the noise variance was increased to a much more challenging value
σ 2 = 0.25—approximately one-fourth of the maximum signal amplitude. In this
case, the GM-CPHD filter could not be used since the thresholded data contained far
too much clutter. Thus the UKF implementation was compared to an EKF implementation of the Σ-CPHD filter. It was observed that both implementations tracked
the targets largely successfully, with the UKF implementation being somewhat better.
19.4
THOUIN-NANNURU-COATES (TNC) APPROXIMATION
This very clever approximation was devised by Thouin, Nannuru, and Coates in
2011. They employed it to derive a PHD filter for superpositional sensors, which
they called the “additive likelihood moment (ALM)” filter” [290].
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The purpose of this section is to describe a generalization of the TNC
approximation, and the approximate CPHD filter for superpositional sensors that
results as a consequence. This filter, which will be called the TNC Σ-CPHD filter,
was proposed by Nannuru, Coates, and Mahler in 2013 [222].2
The section is organized as follows:
1. Section 19.4.1: An overview of the generalized TNC approximation.
2. Section 19.4.2: Modeling assumptions for the TNC Σ-CPHD filter.
3. Section 19.4.3: Measurement update equations for the TNC Σ-CPHD filter.
4. Section 19.4.4: Implementations of the TNC Σ-CPHD filter.
19.4.1
Generalized TNC Approximation: Overview
The purpose of this section is to summarize the original TNC approximation
(Section 19.4.1.1) and then its generalization (Section 19.4.1.2).
19.4.1.1
The Original TNC Approximation
As with the Hauschildt approximation, the TNC approximation begins with the
measurement-update equations
=
Dk+1|k+1 (x)
=
=
2
pk+1|k+1 (n)
(19.70)
∫
f
(z
− ηk+1 (X)) · fk+1|k (X)δX
|X|=n Vk+1 k+1
∫
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
( ∫
)
fVk+1 (zk+1 − ηk+1 (x) − ηk+1 (X))
·fk+1|k (X ∪ {x})δX
∫
(19.71)
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
Nk+1|k+1
(19.72)
∫
|X| · fVk+1 (zk+1 − ηk+1 (X)) · fk+1|k (X)δX
∫
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
The ALM filter’s measurement-update equation ([290], Eq. (12)), is valid only for discrete state
spaces. The TNC approximation itself remains valid, however. It was subsequently generalized by
Mahler and used to derive an approximate CPHD filter for superpositional sensors [188]. The PHD
filter special case of this CPHD filter is the correct form for the ALM filter for continuous spaces.
Thouin, Nannuru, and Coates report that the original and the corrected ALM filters appear to have
surprisingly similar performance, at least in an RF tomography application [223].
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where, since we are to derive a PHD filter, fk+1|k (X) is assumed to be Poisson.
The approximation is based on the recognition that the denominator of these
equations,
∫
fVk+1 (zk+1 − ηk+1 (X)) · fk+1|k (X)δX,
(19.73)
can be approximated using the following five-step process.
1. Step 1: Certain set integrals can be rewritten as ordinary integrals using the
change of variables z = η(X), using the change of variables formula for set
integrals of (3.46):
∫
∫
T (η(X)) · f (X)δX = T (z) · P (z)dz,
(19.74)
where P (z) is a conventional probability density; and where, in our
situation,
T (z)
=
η(X)
=
fVk+1 (zk+1 − z)
∑
ηk+1 (X) =
ηk+1 (x)
(19.75)
(19.76)
x∈X
f (X)
=
fk+1|k (X).
(19.77)
2. Step 2: Given that fk+1|k (X) is Poisson and that η(X) has the
superpositional form of (19.76), Campbell’s theorem (4.96) allows us to
derive explicit formulas for the expected value ok+1|k and variance Ok+1|k
of P (z):
∫
o =
ηk+1 (x) · D(x)dx
(19.78)
∫
O =
ηk+1 (x)ηk+1 (x)T · D(x)dx
(19.79)
where D(x) is the PHD of f (X) and where (19.79) is true only if f (X)
is Poisson.
3. Step 3: Approximate
P (z) ∼
= NO (z − o).
(19.80)
4. Step 4: Assume that the noise is Gaussian,
fVk+1 (z) = NRk+1 (z),
(19.81)
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in which case the right side of (19.74) can be solved, approximately, in closed
form:
∫
∫
T (z) · P (z)dz ∼
NRk+1 (zk+1 − z) · NO (z − o)dz (19.82)
=
=
NRk+1 +O (zk+1 − o)
(19.83)
and where the last equation results from the fundamental Gaussian identity,
(2.3).
5. Step 5: Employ various algebraic stratagems to bring the numerators of
(19.70) through (19.73) into a form that can be approximated using the
previous four steps.
19.4.1.2
Generalized TNC Approximation
The generalized TNC approximation exploits the fact that the quadratic form of
Campbell’s theorem, (4.102), allows us to derive an expression for O when f (X)
is arbitrary. In general, (19.79) becomes
∫
O =
ηk+1 (x)ηk+1 (x)T · D(x)dx
(19.84)
∫ ∫
[
]
+
ηk+1 (x1 )ηk+1 (x2 )T · D 2 (x1 , x2 ) − D(x1 ) · D(x2 ) dx1 dx2
where D(x) is the PHD of f (X) and where
∫
D 2 (x1 , x2 ) = f ({x1 , x2 } ∪ W )δW
(19.85)
is the second-order factorial-moment density of f (X) as defined (4.83). (Equations (19.78) and (19.84) are proved in Section K.27.
Given this, we can reformulate (19.70) and (19.71) so that they have the form
pk+1|k+1 (n)
=
Dk+1|k+1 (x)
=
∫
∫
n
fVk+1 (zk+1 − ηk+1 (X)) · f k+1|k (X)δX
(19.86)
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
( ∫
)
fVk+1 (zk+1 − ηk+1 (x) − ηk+1 (X))
x
·f k+1|k (X)δX
∫
(19.87)
fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY
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665
x
n
for suitable definitions of f k+1|k (X) and f k+1|k (X). Applying the reasoning in
Steps 1-4 and (19.78) and (19.84) to these formulas, we end up with measurementupdate equations for the TNC Σ-CPHD filter (as listed in Section 19.4.3). For the
full proof, see [188].
Example 7 (A simple illustration) A simple example illustrates the correctness of
(19.78) and (19.84). Let
Ξ = {X1 , X2 , X3 }
(19.88)
where X1 , X2 , X3 are independent random state-vectors with respective linearGaussian probability distributions
f1 (x)
f2 (x)
=
=
NP1 (x − x1 )
NP2 (x − x2 )
(19.89)
(19.90)
f3 (x)
=
NP3 (x − x3 ).
(19.91)
Then the multitarget distribution of Ξ is


f1 (x1 ) · f2 (x2 ) · f3 (x3 ) + f1 (x3 ) · f2 (x1 ) · f3 (x2 )
f (X) = δ|X|,3 ·  +f1 (x2 ) · f2 (x3 ) · f3 (x1 ) + f1 (x1 ) · f2 (x3 ) · f3 (x2 ) 
+f1 (x2 ) · f2 (x1 ) · f3 (x3 ) + f1 (x3 ) · f2 (x2 ) · f3 (x1 )
(19.92)
and its PHD and second factorial moment density are, respectively,
D(x)
D (x1 , x2 )
2
= f1 (x) + f2 (x) + f3 (x)
(19.93)
= f1 (x1 ) · f2 (x2 ) + f1 (x2 ) · f2 (x1 ) + f2 (x1 ) · f3 (x2 ) (19.94)
+f2 (x2 ) · f3 (x1 ) + f1 (x2 ) · f3 (x1 ) + f1 (x1 ) · f3 (x2 ).
Assume that ηk+1 (x) = x. Then the summed random vector is
Z = ηk+1 (Ξ) = X1 + X2 + X3 .
(19.95)
Its probability distribution is the convolution
fZ (x) = (f1 ⋆ f2 ⋆ f3 )(x) = NP1 +P2 +P3 (x − x1 − x2 − x3 ).
(19.96)
Thus the expected value and covariance of Z are o = x1 + x2 + x3 and
O = P1 + P2 + P3 , respectively. Alternatively, the expected value can be computed
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using (4.96):
o=
∫
x · D(x)dx = x1 + x2 + x3 .
(19.97)
Likewise, the covariance can be computed using (19.84). For, using (19.85),
=
D 2 (x1 , x2 ) − D(x1 ) · D(x2 )
−f1 (x1 ) · f1 (x2 ) − f2 (x1 ) · f2 (x2 ) − f3 (x1 ) · f3 (x2 ).
(19.98)
Then using (19.84) we get:
O = P1 + P2 + P3 .
19.4.2
(19.99)
TNC Σ-CPHD Filter: Models
The TNC Σ-CPHD filter requires the following measurement model:
• Gaussian superpositional likelihood function:
fk+1 (z|X) = NRk+1 (z − ηk+1 (X))
(19.100)
where ηk+1 (X) was defined in (19.21).
19.4.3
TNC Σ-CPHD Filter: Measurement Update
The predictor step is the same as that for the usual CPHD filter (see Section 8.5.2).
The corrector step is defined as follows. Suppose that we have a single sensor with
superpositional likelihood function as in (19.100). Suppose that the sensor collects
a measurement vector zk+1 , after having collected a time-stream Z k : z1 , ..., zk
of measurements. Abbreviate the predicted cardinality distribution and PHD as
pk+1|k (n) = pk+1|k (n|Z k ) and Dk+1|k (x) = Dk+1|k (x|Z k ), respectively.
Define
∫
Nk+1|k =
Dk+1|k (x|Z k )dx
(19.101)
sk+1|k (x)
=
Gk+1|k (x)
=
−1
Nk+1|k
· Dk+1|k (x|Z k )
∑
pk+1|k (n) · xn
(19.102)
(19.103)
n≥0
n
(n)
Gk+1|k (x)
=
d Gk+1|k
(x).
dxn
(19.104)
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(2)
667
(3)
2
Let σk+1|k
, Gk+1|k (1), and Gk+1|k (1) be, respectively, the variance, second
factorial moment, and third factorial moment of pk+1|k (n). Assume that:
• The likelihood function has the form
fk+1 (z|X) = NRk+1 (z − ηk+1 (X)) = NRk+1
(
z−
∑
)
ηk+1 (x) .
x∈X
(19.105)
• There exists an n0 ≥ 0 such that pk+1|k (n) < 1/n for all n > n0 (which
will be true, for example, if pk+1|k (n) = 0 for all n > n0 ).
Then the corrector equations for the TNC Σ-CPHD filter are ([188], Theorem
1):
N
n
Rk+1 +O k
pk+1|k+1 (n)
∝
(zk+1 − nôk )
NRk+1 +Ok (zk+1 − Nk+1|k ôk )
· pk+1|k (n) (19.106)
NRk+1 +O
˚k (zk+1 − ηk+1 (x) −o̊k )
Dk+1|k+1 (x)
=
(19.107)
NRk+1 +Ok (zk+1 − Nk+1|k ôk )
·Dk+1|k (x)
where
=
(
)
ˆ k − ôk ôTk
n· O
(19.108)
=
T
ˆ k + (σ 2
Nk+1|k · O
k+1|k − Nk+1|k ) · ôk ôk
(19.109)
n
Ok
Ok
(2)
Gk+1|k (1)
o̊k
=
Nk+1|k
· ôk
(19.110)
and where
˚k
O
=
ôk
=
ˆk
O
=


(3)
(2)
Gk+1|k (1) Gk+1|k (1)2
G(2) (1) ˆ
 · ôk ôTk (19.111)
· Ok + 
−
2
Nk+1|k
Nk+1|k
Nk+1|k
∫
ηk+1 (x) · sk+1|k (x)dx
(19.112)
∫
ηk+1 (x)ηk+1 (x)T · sk+1|k (x)dx.
(19.113)
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The measurement-update for the corresponding PHD filter—that is, the corrected ALM filter—is
Dk+1|k+1 (x)
(19.114)
NRk+1 +Nk+1|k Oˆ k (zk+1 − ηk+1 (x) − Nk+1|k ôk )
=
NRk+1 +Nk+1|k Oˆ k (zk+1 − Nk+1|k ôk )
·Dk+1|k+1 (x).
(Note: This equation does not require the assumption that pk+1|k (n) < 1/n for
all n > n0 .) Somewhat surprisingly, the performance of the corrected and original
ALM filters do not appear to be significantly different [223].
19.4.4
TNC Σ-CPHD Filter: Implementations
Because of the nonlinear, non-Gaussian nature of the measurement update formulas,
both the TNC Σ-CPHD filter and the TNC Σ-PHD filter must be implemented
using sequential Monte Carlo (SMC, also known as particle) techniques. In [222],
Nannuru, Coates, and Mahler devised auxiliary particle filter implementations of
both filters.
The SMC implementation of the TNC Σ-CPHD filter was shown to be
significantly faster than that of the TNC Σ-PHD filter: O(νM 2 +nmax M 3 +nmax ν)
for the former versus O(νM 2 + M 3 + n2max ν 2 ) for the latter, where ν is the
number of particles, nmax is the maximum possible number of targets, and M
is the dimension of the measurement space. (For the passive-acoustic application,
M is the number of passive-acoustic sensors. For the RF tomography application,
M = 12 ν(ν − 1) where ν is the number of transmitter-receiver nodes.)
As a baseline for comparison, a conventional Markov Chain Monte Carlo
(MCMC) algorithm was implemented, based on an approach due to Septier, Pang,
Carmi, and Godsill [268]. This algorithm was much more computationally intensive than the TNC Σ-CPHD filter.
The authors applied three filters—MCMC, TNC Σ-CPHD, and TNC ΣPHD—to two applications: multitarget tracking using passive-acoustic sensors, and
multitarget detection and tracking using RF tomography sensor arrays. The results
for each are described in turn.
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19.4.4.1
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TNC Σ-CPHD Filter: Application to Passive Acoustics
In these simulations, targets are observed by a field of 25 omnidirectional passive
acoustic-amplitude sensors, which are located at the nodes of a rectangular grid.
Four acoustically active targets follow trajectories that are somewhat separated and
slightly curvilinear. The performance of the three filters was compared, for 100
Monte Carlo trials, using the OSPA metric (Section 6.2.2).
The authors reported that the TNC Σ-CPHD filter outperformed both the
MCMC algorithm (which had roughly two times more OSPA error) and the TNC
Σ-PHD filter (roughly three times more OSPA error). The TNC Σ-CPHD filter was
about 87 times faster than the MCMC filter and about 27 times faster than the TNC
Σ-PHD filter.
19.4.4.2
TNC Σ-CPHD Filter: Application to Radio-Frequency Tomography
RF tomography was described in Section 19.1.1.4. In these simulations, 24 RF
receiver/transmitter sensors are located on the perimeter of the monitoring region,
resulting in a total of 276 unique bidirectional sensor-to-sensor links. The tomography array observes four targets moving along slightly curvilinear trajectories, with
the first pair approaching and diverging, and the second pair approaching and diverging.
As in the passive-acoustic application, the TNC Σ-CPHD filter outperformed
both the MCMC algorithm (which had roughly two times more OSPA error) and
the TNC Σ-PHD filter (roughly three times more OSPA error). However, in all
cases the OSPA error was roughly half that of the passive-acoustic application.
This is because the measurement dimension and the signal-to-noise ratio were much
higher.
The TNC Σ-CPHD filter was about 30 times faster than the MCMC filter and
about 14 times faster than the TNC Σ-PHD filter.
Chapter 20
RFS Filters for Pixelized Images
20.1
INTRODUCTION
This chapter addresses the problem of detecting and tracking multiple targets that
move within a time-series of pixelized images. Examples of sensors that produce
such data include electro-optical (EO) cameras, infrared cameras, synthetic aperture
radars (SARs), and inverse SARs (ISARs).
Conventional algorithms for processing such data are typically based on a
detection paradigm. For example, preprocessing algorithms extract “blobs” from
the images using techniques such as thresholding (which identifies higher-intensity
pixels) or edge detection (which identifies circumscribed regions in an image).
Features, such as the centroids of the blobs, are then used as the input measurements
to some kind of multitarget tracking algorithm.
However, this approach can waste potentially valuable information. When
signal-to-noise ratio (SNR) is small, feature-detection algorithms will often eliminate target pixels along with clutter pixels; and clutter pixels can be spuriously
declared to be target pixels. When SNR is very low, targets will not even be identifiable in any given image, thus making feature extraction ineffective.
Consequently, performance could be improved if target detection, and target
tracking were performed jointly in a single, unified nonlinear algorithm, using entire
images rather than the features extracted from them.
The purpose of this chapter is to describe innovative RFS filters, proposed by
B.-N. Vo and B.-T. Vo, that accomplish this task. The most notable of these filters is
the image-observation multitarget multi-Bernoulli (IO-MeMBer) filter, which will
be described in Section 20.5. It has been shown to outperform the previously best
track-before-detect filters in this application; and has been successfully applied to
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real grayscale and color video recordings of complex, rapidly evolving multitarget
scenarios such as hockey games (see Section 20.6).
The discussion will be at a relatively high level. For greater detail, see the
original papers cited in the chapter.
20.1.1
Summary of Major Lessons Learned
The following are the major concepts, results, and formulas that the reader will learn
in this chapter:
• All of the RFS filters in this chapter are based on B.-N. Vo’s imageobservation (IO) measurement model (Section 20.2). It models targetilluminated regions of an image as adaptively parametrized, area-filling templates.
• These filters can be applied to either grayscale or color images.
• These RFS filters are based on the assumption that targets have physical
extent in two dimensions. In particular, it is assumed that targets cannot
overlap or pass over or through each other.
• These RFS filters are exact closed-form solutions of the multitarget Bayes
filter. They require no approximations other than (a) those in the modeling
assumptions; and (b) the simplifying assumption that the predicted-target
RFS has a specific form (multi-Bernoulli, i.i.d.c., or Poisson).
• Because the IO measurement model is usually highly nonlinear, these RFS
filters must typically be implemented using sequential Monte Carlo (SMC,
also known as particle) techniques.
• The IO-MeMBer filter has been shown to outperform the best previously
known track-before-detect filter, the histogram-PMHT (Section 20.6.1).
• Because of the difficulties associated with state estimation for multitarget
SMC filters, the IO-PHD filter and IO-CPHD filter are computationally more
demanding than the IO multi-Bernoulli filter. As a consequence, they are
probably of lesser practical significance.
20.1.2
Organization of the Chapter
The chapter is organized as follows:
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1. Section 20.2: B.-N. Vo’s image-observation (IO) multitarget measurement
model.
2. Section 20.3: An approximate multitarget motion model for the IO model.
3. Section 20.4: The IO-CPHD filter.
4. Section 20.5: The IO-MeMBer filter.
5. Section 20.6: Implementations of the IO-MeMBer filter.
20.2
THE IO MULTITARGET MEASUREMENT MODEL
The measurement model described in this section is due to B.-N. Vo [315], [316].
Suppose that an imaging sensor presents its measurement as an m1 × m2 array
of pixels. The pixel measurement can be a real number representing a gray-scale
intensity; or a three-dimensional vector (R, G, B)T , representing the intensities of
the red, green, and blue color channels. Whatever their form, the pixel measurements are packaged into an image-vector
z = (z1,1 , ..., zm1 ,m2 )T = (z 1 , ..., z M )T
(20.1)
of dimension M = m1 m2 .
Assume:
• Targets have a physical extent. The state-vector x of a target can include,
besides position and velocity variables, variables that specify target shape,
size, orientation, and identity-class.
• Some pixels are “illuminated” by the target, whereas all other pixels are
“background.” Let Jk+1 (x) be the set of indices of all pixels that are
illuminated by a target with state x.
• Targets move on a surface and thus, because they have physical extent, cannot
overlap or pass over each other (see Figure 20.1). Thus if x1 and x2 are
the states of two distinct targets then
Jk+1 (x1 ) ∩ Jk+1 (x2 ) = ∅.
(20.2)
j
• fk+1
(z|x) is the probability density of z in the jth pixel, given that it is
target-illuminated; and it assumed to be known a priori.
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Figure 20.1 A schematic diagram of the image-observation (IO) measurement
model. Shown are “blobs” consisting of pixels illuminated by targets or landmarks.
It is assumed that targets—and therefore blobs—cannot physically overlap each
other.
j
• fk+1
(z) is the probability density of z in the jth pixel, given that it is not
target-illuminated; and it is assumed to be known a priori.
• Pixels are conditionally independent with respect to target state.
Given this, the sensor likelihood function of a target-containing image is

LZ (x) = fk+1 (z|x) = 
∏
j ∈J
/ k+1 (x)

j
fk+1
(z j ) 
∏
j∈Jk+1 (x)

j
fk+1
(z j |x) (20.3)
whereas the likelihood of an image with no targets is
ℓz =
m
∏
j=1
j
fk+1
(z j ).
(20.4)
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Assume that a multitarget state X = {x1 , ..., xn } with |X| = n is
“physically realizable” in the sense that, for all i, l = 1, ..., n with i ̸= l,
(20.5)
Jk+1 (xi ) ∩ Jk+1 (xl ) = ∅.
Then the multitarget likelihood function is ([316], Eq. (3)):
fk+1 (z|X) = ℓz · χX
z
(20.6)
where the power-functional notation χX was defined in (3.5) and where
∏
χz (x) =
j∈Jk+1 (x)
j
fk+1
(z j |x)
j
fk+1
(z j )
(20.7)
.
If X = ∅ then fk+1 (z|X) = ℓz . Otherwise, let
(20.8)
Jk+1 (X) = Jk+1 (x1 ) ⊎ ... ⊎ Jk+1 (xn )
be the set of indices of all target-illuminated pixels. Then because of (20.5),



∏
∏
j
j
j
fk+1
(z|X) = 
fk+1
(z j ) 
fk+1
(z j |x1 ) (20.9)
j ∈J
/ k+1 (X)

···

=
=


∏
j∈Jk+1 (xn )
M
∏
j=1
ℓz ·
j∈Jk+1 (x1 )
j
fk+1
(z j |xn )

j
fk+1
(z j ) 
n
∏
n
∏
∏
i=1 j∈Jk+1 (xi )
χz (xi ) = ℓz · χX
z .
j
fk+1
(z j |xi )
j
fk+1
(z j )

 (20.10)
(20.11)
i=1
Remark 81 This measurement model is in some respects similar to that employed
for SAR images in the DARPA MSTAR program of the 1990s [118]. The algorithms
developed under MSTAR addressed a different problem than that considered here:
automatic target recognition (ATR) of single, motionless ground targets. However,
the MSTAR algorithms were also based on the presumptions that (1) the probability
distribution of a target in any pixel is known (because of radar physics using CAD
modeling of the targets), and (2) pixels are statistically independent.
676
Advances in Statistical Multisource-Multitarget Information Fusion
20.3
IO MOTION MODEL
Because targets cannot overlap, their motion is not statistically independent and
thus the standard multitarget motion model of Section 7.4 is not strictly applicable.
Assume, however, that targets are sufficiently small that they do not occupy large
regions of any given image. Then the motion model for the conventional multiBernoulli filter (see Section 13.4.2) can be adopted as an approximation.
Consequently, the time-update for the IO-CPHD filter is the same as that
for the classical CPHD filter (Section 8.5.2). Thus we need only specify the
measurement-update equations.
20.4
IO-CPHD FILTER
We are given the predicted PHD Dk+1|k (x) and the predicted cardinality
distri∫
bution pk+1|k (n) or predicted p.g.f. Gk+1|k (x). Let Nk+1|k = Dk+1|k (x)dx.
Then if the new measurement at time tk+1 is zk+1 , the exact closed-form
measurement-updated versions of these are ([316], Eqs. (10,11)):
• Measurement updates for the p.g.f. and cardinality distribution:
Gk+1|k+1 (x)
=
Gk+1|k (x · ϕk )
Gk+1|k (ϕk )
(20.12)
pk+1|k+1 (n)
=
ϕn · pk+1|k (n)
∑ k l
l≥0 ϕk · pk+1|k (l)
(20.13)
χzk+1 (x) · Dk+1|k (x)dx.
(20.14)
where
∫
1
ϕk =
Nk+1|k
• Measurement update for the expected number of targets:
(1)
Gk+1|k (ϕk )
Nk+1|k+1 =
Gk+1|k (ϕk )
· ϕk .
(20.15)
• Measurement update for the PHD:
(1)
Gk+1|k (ϕk )
1
Dk+1|k+1 (x) =
Nk+1|k
·
Gk+1|k (ϕk )
· χzk+1 (x) · Dk+1|k (x)
(20.16)
RFS Filters for Pixelized Images
677
The PHD filter special case is ([316], Corollary 1):
Dk+1|k+1 (x) = χzk+1 (x) · Dk+1|k (x).
(20.17)
These equations are proved in Section K.28. Because the pseudolikelihood
χzk+1 (x) is highly nonlinear in general, they must be implemented using sequential
Monte Carlo (SMC) techniques.
20.5
IO-MEM