Target Tracking with Binary Proximity
Sensors: Fundamental Limits, Minimal
Descriptions, and Algorithms
N. Shrivastava, R. Mudumbai, U. Madhow, and S. Suri
Univ. of California Santa Barbara
SENSYS 2006
Presented by Jeffrey Hsiao
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Introduction
• Binary Proximity Sensors
– outputs a 1 when the target of interest is
within its sensing range R
– 0 otherwise
0
R
1
Fundamental Limit Of
Spatial Resolution
• Spatial Resolution
– worst-case deviation between the estimated
and the actual paths
• Ideal achievable resolution  is of the
order of 1/R
– R is the sensing range of individual sensors
–  is the sensor density per unit area
Minimal Representations For The
Target’s Trajectory
• Analogy between binary sensing and the
sampling theory and quantization
– “high-frequency” variations in the target’s
trajectory are invisible to the sensor field
– estimate the shape or velocity for a “low-pass”
version of the trajectory.
– consider piecewise linear approximations to
the trajectory that can be described
economically
Occam’s Razor Approach
• Explanation of any phenomenon should
make as few assumptions as possible
• Entities should not be multiplied beyond
necessity
• All things being equal, the simplest
solution tends to be the best one
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Geometry of Binary Sensing
Geometry of Binary Sensing
• Signature
Geometry of Binary Sensing
• Localization
Patch
Geometry of Binary Sensing
• Localization
Arc
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Fundamental Limits
• An Upper Bound on Spatial Resolution
• Achievability of Spatial Resolution
Bound
• Remarks on Spatial Resolution
Theorems
• Sampling and Velocity Estimation
An Upper Bound on Spatial
Resolution
• THEOREM 1
• If a network of binary proximity sensors
has average sensor density  and each
sensor has sensing radius R
• Then the worst-case error in localizing the
target is at least W(1/  R)
An Upper Bound on Spatial
Resolution
Achievability of Spatial
Resolution Bound
• THEOREM 2
– Consider a network of binary proximity
sensors, distributed according to the Poisson
distribution of density , where each sensor
has sensing radius R.
– Then the localization error at any point in the
plane is of order 1/R .
Remarks on Spatial Resolution
Theorems
• The more sensors we have, the better the
spatial accuracy one should be able to
achieve
• Having a large sensing radius may seem
like a disadvantage
• A quadratic increase in the number of
patches into which the sensor field is
partitioned by the sensing disks
Sampling and Velocity
Estimation
Low-pass Trajectories
Velocity Estimation Error
• THEOREM 3
– Suppose a portion of the trajectory is approximated
by a straight line segment of length L to within spatial
resolution 
– Then, the maximum variation in the velocity estimate
due to the choice of different candidate straight line
approximations is at most
– Furthermore, this also bounds the relative velocity
error if the true trajectory is well approximated as a
straight line over the segment under consideration
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Tracking Algorithms
• I
– be the subset of sensors whose binary output
is 1 during the relevant interval
• Z
– be the remaining sensors whose binary output
is 0 during this interval
Spatial Band B
Analysis of OCCAMTRACK
• THEOREM 4
– The algorithm OCCAMTRACK computes a
piecewise linear path that visits the
localization arcs in order and uses at most
twice the optimal number of segments in the
worst-case
– If there are m arcs in the sequence, then the
worst-case time complexity of OCCAMTRACK
is O(m3)
Robust Tracking With Non-ideal
Sensors
Particle Filtering Algorithm
• At any time n, we have K particles (or candidate
trajectories)
– with the current location for the kth particle denoted
by xk[n]
• At the next time instant n+1, suppose that the
localization patch is F
– Choose m candidates for xk [n+1] uniformly at random
from F
• We now have mK candidate trajectories
• Pick the K particles with the best cost functions
Cost Function
• Chose an additive cost function that
penalizes changes in the vector velocity
Geometric Postprocessing
• Particle filtering algorithm described above
gives a robust estimate of the trajectory
consistent with the sensor observations
• But it provides no guarantees of a “clean”
or minimal description
Geometric Postprocessing
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Simulation Results
• The code for OCCAMTRACK was Written
in C and C++
• The code for PARTICLE-FILTER was
written in Matlab
• The experiments were performed on an
AMD Athlon 1.8 Ghz PC with 350 MB
RAM
OCCAMTRACK with ideal sensing
• A 1000×1000 unit field
• Containing 900 sensors in a regular 30×30 grid
• Sensing range for each sensor was set to 100
units
• Geometric random walks to generate a variety of
trajectories
– Each walk consists of 10 to 50 steps, where each
step chooses a random direction and walks in that
direction for some length, before making the next turn
• Each trajectory has the same total length
• Generated 50 such trajectories randomly
Quality Of Trajectory Approximation
Velocity Estimation Performance
Spatial Resolution As A Function Of
Density
Spatial Resolution As A Function Of
Sensing Range
Tracking with Non-Ideal Sensing
Tracking with Non-Ideal Sensing
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Mote Experiments
• 16 MICA2 motes arranged in a 4×4 grid
with 30 centimeter separation
Probability Of Target Detection With
Distance
Mote Experiments
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Closing Remarks
• Have identified the fundamental limits of
tracking performance possible with binary
proximity sensors
• Have provided algorithms that approach
these limits
Closing Remarks
• Results show that the binary proximity
model, despite its minimalism, does
indeed provide enough information to
achieve respectable tracking accuracy
– assuming that the product of the sensing
radius and sensor density is large enough
Future Works
• An in-depth understanding, and
accompanying algorithms, for multiple
targets is therefore an important topic for
future investigation
• To incorporate additional information (e.g.,
velocity, distance) if available
• To embed Occam’s razor criteria in the
particle filtering algorithm
Outline
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Introduction
Geometry of Binary Sensing
Fundamental Limits
Tracking Algorithms
Simulation Results
Mote Experiments
Closing Remarks
Comments
Comments
• Strength
– A simple yet robust algorithm proposed for target
tracking in sensor networks
– Complete work
• Analysis
• Simulation
• Experiments
• Weakness
– For non-ideal case, performance for particle filtering
and geometric postprocessing is not mentioned
• Time complexity could be high
Thank You Very Much For
Your Attention!
Any More Questions?