1. From Bayesian to Particle Filter
San Francisco State University
Mathematics Department
Fang-I Chu

2. Outline Introduction Mathematical Background Literatures Review
Application – Object Tracking in Video
Conclusion

3. 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

4. Mathematical Background
Conditional Probability
Bayes’ Theorem
Prior and Posterior Probability
Markov Chain

5. 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

6. Bayes’ Theorem Law of Total Probability we obtain
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

7. 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 ,

8. 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

9. 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 before the experiment

10. Posterior Density Function
The posterior density function of , given observed value , using the definition of conditional probability, we have
In words,

11. 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

12. Literature Review Linear systems Nonlinear systems Particle Filter
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

13. Kalman Filter Linear system-Kalman Filter Recursive linear estimator
Applys only to Gaussian density
Algorithm
State-space model
Forward recursion
Smoothing
Diffuseness

14. Kalman Filter as density propagation

15. 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
Detail of state space model

16. Kalman Filter We write the state-space model as,
We derive the Best Linear Unbiased Prediction (BLUP) of based on (innovation vectors) as,
Detail of state space model

17. Kalman Filter Forward Recursions Follow 2 steps:
1. translate the response vectors to the innovation vectors
2. find the BLUP of state vector and signal vector
We get the forward recursion formula as

18. 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

19. 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

20. 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 non- Gaussian
Let the conditional density be a time-independent function

21. Non-Gaussian State-density Propagation

22. Nonlinear Systems
The rule for propagation of state density over time is
is a normalization constant that does not depend on

23. Nonlinear filtering
Apply nonlinear filter to evaluate the state density over time
Four distinct probability distributions represented in non- linear 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

24. 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

25. Extended Kalman Filter (EKF)
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

26. 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.

27. 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)

28. Unscented Transform

29. 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.

30. 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

31. 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

32. Particle Filter Monte Carlo Simulation Sequential Importance sampling
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.

33. 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.

34. 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

35. 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.

36. Observation process
The thick line is a hypothesised shaped, represented as a parametric spline curve.

37. Example: CONDENSATION
Tracking agile motion in clutter

38. 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

39. Results for Kalman Filter (data set 1)

40. Results for Kalman Filter (data set 2)

41. Results for Particle Filter (data set 1)

42. Results for Particle Filter (data set 2)

43. 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.

44. 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

45. 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.

46. 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

47. Special thanks to My thesis advisor Dr. Mohammad Kafai
My thesis committees
Dr. Yitwah Cheung
Dr. Alexandra Piryatinska
Computer Science Professor
Dr. Kaz Okada

48. Thanks for technical support from
Scott M. Shell
Network Engineer
Allen Yen
Telecommunication Engineer
Mehran Kafai
Computer Science Ph.D candidate

49. Thanks for your attending!
The End.