From Bayesian to Particle Filter
San Francisco State University
Mathematics Department
Fang-I Chu
1
Outline
• Introduction
• Mathematical Background
• Literatures Review
• Application – Object Tracking in Video
• Conclusion
2
Introduction
• Why tracking moving object is important ?
• Simulate the moving path of the target
• Important applications: robot, medical (eye movement,
neurology), security systems, Wii
• What we want to do
• Simple movement: easy
• Erratic movement: need to be done
• How we approach it
• Filtering algorithms in computer science vision
3
Mathematical Background
• Conditional Probability
• Bayes’ Theorem
• Prior and Posterior Probability
• Markov Chain
4
Conditional Probability
• Conditional Probability
• Any probability that is revised to take into account the
known occurrence of other events
• Written as
, given the probability of event
happens, the probability of event
will happen
5
Bayes’ Theorem
Law of Total Probability
•
•
Let
is an event in , if
events
form a
partition of
the events
form a
partition of
and since they are disjoint, we have
substituting with the formula of conditional probability,
we obtain
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Bayes’ Thereom
• Bayes’ Theorem
• Relatively minor extension of definition of conditional
probability
• Compute
from
• The form is also known as Bayes’ Formula
• Let events
form the partition of the space,
and
, for
,
7
Prior and Posterior Probability
• Definition
• Prior Probabilities
the original probabilities of an outcome
• Posterior Probabilities
the probabilities we obtained after updating with
new information from the experiment
8
Prior Density Function
• Prior Density Function
• Let stand for the density function of , where is a random
vector with range of
• The function
is called prior density function
• represents our information about
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before the experiment
Posterior Density Function
• Posterior Density Function
• The posterior density function of , given observed
value
, using the definition of conditional
probability, we have
In words,
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Markov Chain
•
Markov Chains
• A stochastic process
• Suppose that whenever the process is in state , there is a fixed
probability
that it will next be in state , which can be
written as,
• For a Markov chain, above equation states that the conditional
distribution of any future state
given the past states
and the present state is independent
of the past states and depends only on the present state
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Literature Review
Linear systems
•
•
Kalman Filter
Nonlinear systems
•
•
•
Extended Kalman Filter
Unscented Filter
Particle Filter
•
•
•
•
•
•
Bayesian filter
Monte Carlo Simulation
Sequential Importance Sampling
Sequential Importance Re-sampling
CONDENSATION algorithm
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Kalman Filter
Linear system-Kalman Filter
•
Recursive linear estimator
Applys only to Gaussian density
Algorithm
•
•
•
•
•
•
•
State-space model
Forward recursion
Smoothing
Diffuseness
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Kalman Filter as density propagation
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Kalman Filter
• State-space model
• A special case of the signal-plus-noise model
• Under the assumption of signal-plus-noise model, as
stands for the response vector, we have
where the signal vector as
the state vectors are assumed to propagate via state
equations
15
Kalman Filter
• We write the state-space model as,
• We derive the Best Linear Unbiased Prediction (BLUP)
of
based on
(innovation vectors) as,
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Kalman Filter
• Forward Recursions
• Follow 2 steps:
1. translate the response vectors
innovation vectors
2. find the BLUP of state vector
to the
and signal vector
We get the forward recursion formula as
17
Kalman Filter
• Smoothing (Backward Recursion)
• Including the information from new response data into
predictors
• The idea of smoothing is modifying the BLUP
from being based on
to being based on
for some
• How much new information we need to incorporate
into the estimator
?
Use the most information possible
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Kalman Filter
• Diffuseness
• Assume
is a random vector with mean zero and
variance-covariance matrix
• The specific choice for
have a profound effect
on predictors
• When we produce predictions which can be computed
directly without specifying
, we need to modify
the recursions into diffuse Kalman Filter
19
Nonlinear Systems
• Nonlinear systems
• Kalman Filter is not enough because most applications
are presented in nonlinear systems
• When observation density is non-Gaussian, the
evolving state density
is also generally nonGaussian
• Let the conditional density be a time-independent
function
20
Non-Gaussian State-density Propagation
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Nonlinear Systems
• The rule for propagation of state density over time is
is a normalization constant that does not
depend on
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Nonlinear filtering
• Apply nonlinear filter to evaluate the state density over
time
• Four distinct probability distributions represented in nonlinear Bayesian Filter; three of them form part of the
problem specification and fourth constitutes the solution
• Three specified distributions are:
• The prior density
for the state
• The process density
that describes the stochastic
dynamics
• The observation density
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Nonlinear filtering
• Interested in the 2nd type, the approach to this type
of filtering is to approximate the dynamics as a
linear process and proceed as for linear Kalman filter
• The 4th type is simply to integrate the equation in
previous slide directly, using a suitable numerical
representation of the state density
• Two filtering methods of 2nd type:
• Extended Kalman Filter
• Unscented Filter
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Extended Kalman Filter (EKF)
• Extended Kalman Filter
• The most common approach
• Reliable for systems which are almost linear on the
time scale of the update interval
• Difficult to implement
• Heavy computational work required
• Algorithm scheme
• to linearize all nonlinear models, then apply the traditional
Kalman Filter
25
Example of EKF
• With linearization, the position where 1meter is, in reality it represents 96.7 cm.
• In practice, the inconsistency can be resolved by introducing additional stabilizing
noise which increase the size of the transformed covariance. The noise leads to
biased estimates.
• Why EKFs are difficult to tune ? Since sufficient noise is needed to offset the
defects of linearization.
26
Unscented Filter
• Unscented filter
• Based on the concept of the unscented transform
• The mean is calculated to a higher order of accuracy
than the EKF
• The algorithm is suitable for any process model, and
implementation is rapid (avoid the linearization
computation as in EKF)
27
Unscented Transform
28
Example of Unscented Filter
•
•
•
•
•
The true mean lies at
with a dotted covariance contour.
The unscented mean lies at
with a solid contour.
The linearized mean lies at
with a dashed contour.
The unscented mean value is the same as the true value.
The unscented transform is consistent.
29
Particle Filter
• Does not require the linearization of the relation between
the state and the measurement
• Maintains several hypotheses over time, and gives
increased robustness
• Develop our idea in following order
•
•
•
•
•
Bayesian filter
Monte Carlo Simulation
Sequential Importance Sampling
Sequential Importance Re-sampling
CONDENSATION algorithm
30
Particle Filter
• Bayesian filter
• Considering the probabilistic inference problem in
which the state variable set
is estimated from the
observed evidence ,
, the filtered pdf through
recursive estimation is
31
Particle Filter
• Monte Carlo Simulation
• The recursive estimation from Bayesian Filter needs
strong assumptions to evaluate. This problem is
resolved by using Monte Carlo methods.
• Sequential Importance sampling
• Avoid the difficulty to sampling directly from the
posterior density by sampling from an proposal
distribution.
• The posterior function can be approximated well by
drawing samples from a proposal distribution.
32
Particle Filter
• Sequential Importance Re-sampling
• A re-sampling stage is used to prune those particles
with negligible importance weights, and multiply those
with higher ones.
• A posterior density function is iteratively computed
• This pdf undergoes a diffusion-reinforcement process,
which is followed by factored sampling algorithm.
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Particle Filter
• CONDENSATION algorithm (conditional density
propagation for visual tracking)
• A particular Particle Filtering method
• Not necessary Gaussian
• The probability densities must be established for
dynamic of the object and the observation process
• Factor sampling
• generate a random variables from a distribution that approximates
the posterior
• weight coefficients are decided after a sample set is generated
from the prior density
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Steps in CONDENSATION algorithm
•
An element undergoes drift
(deterministic step)
•
Identical elements in the new
set undergo the same drift
•
Diffusion step : random and
identical elements now split
•
The sample set with new
time-step is generated without
its weight
•
The observation step from
factor sampling is applied,
generating weights from
observation density
•
The sample-set representation
of state-density is obtained.
35
Observation process
• The thick line is a hypothesised shaped, represented as a
parametric spline curve.
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Example: CONDENSATION
• Tracking agile motion in clutter
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Object tracking in video
•
Problem
• Taking a real-time video on a moving object in certain period of time, tracking
this object and marking it
•
Data
• 60 consecutive frames are selected from 800 frames with dimension 240×320
image which extracted from a 55 second real-time video (running dog image)
• 60 consecutive frames are offered by Toby Breckon from Matlab Central Forum
(dropping ball image)
•
Method
• Kalman Filter
• Particle Filter
•
Environment
• Matlab
38
Results for Kalman Filter (data set 1)
39
Results for Kalman Filter (data set 2)
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Results for Particle Filter (data set 1)
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Results for Particle Filter (data set 2)
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Discussion
• Why did data set 1 (running dog) show less precision using
Kalman Filter algorithm ?
• Possible noise factor in data set 1
• absent of initial background images
• non identical background( background changes along with the
running motion)
• multi moving object
• Why we did not see the true position (green circle) in the
images of Particle Filter for both data sets ?
• First marking the true position as green circle, and second
marking the predicted position. When the predicted position
overlapped the true position, the true position concealed.
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Comparison
Kalman Filter
Particle Filter
Prediction accuracy
Good at the starting
frames
Almost 100% close to true
position
Max lag effect
Abrupt acceleration
or bouncing
No lag effect
Noise factor
Possible reason for
increasing lag effect
Noise does not influence
the accuracy to predict true
position
Implementation
process (code)
Less iteration;
simple
More iterations;
complicated
Assumption
Used only in
Gaussian
distribution
Not necessary Gaussian
44
Conclusion
• Particle Filter is a superior algorithm than Kalman Filter in
• the requirement of assumption
• the ability to deal with noise in model
• the accuracy of its prediction
• The re-sampling stage in Particle Filter algorithm is considered
to play a vital role in increasing the accuracy of the prediction.
• The accuracy of prediction in Kalman Filter algorithm reduced
when the motions of object is abrupt acceleration or bouncing.
(lag effects)
• The trait of data could be a factor to affect the prediction result.
45
Future Research
• Design appropriate models to cope with different
types of data set
• Multi object tracking
• Moving object in non-identical background
• Tracking in clutter
•
Comparison of implementation between Extended
Kalman Filter, Unscented Filter, Kalman Filter and
Particle Filter on real-time example
46
Special thanks to
• My thesis advisor
Dr. Mohammad Kafai
• My thesis committees
Dr. Yitwah Cheung
Dr. Alexandra Piryatinska
• Computer Science Professor
Dr. Kaz Okada
47
Thanks for technical support from
• Scott M. Shell
Network Engineer
• Allen Yen
Telecommunication Engineer
• Mehran Kafai
Computer Science Ph.D candidate
48
Thanks for your attending!
The End.
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