1. Today Introduction to MCMC Particle filters and MCMC
A simple example of particle filters: ellipse tracking

2. Introduction to MCMC Sampling technique
Non-standard distributions (hard to sample)
High dimensional spaces
Origins in statistical physics in 1940s
Gained popularity in statistics around late 1980s
Markov Chain Monte Carlo

3. Markov chains* Homogeneous: T is time-invariant
Represented using a transition matrix
Series of samples
such that
* C. Andrieu et al., “An Introduction to MCMC for Machine Learning“, Mach. Learn., 2003

4. Markov chains Evolution of marginal distribution
Stationary distribution
Markov chain T has a stationary distribution
Irreducible
Aperiodic
Bayes’ theorem

5. Markov chains Detailed balance Mass transfer
Sufficient condition for stationarity of p
Mass transfer
Probability mass
Probability mass
Proportion of mass transfer
x(i)
x(i-1)
Pair-wise balance of mass transfer

6. Metropolis-Hastings Target distribution: p(x)
Set up a Markov chain with stationary p(x)
Resulting chain has the desired stationary
Detailed balance
Propose
(Easy to sample from q)
with probability
otherwise

7. Metropolis-Hastings Initial burn-in period
Drop first few samples
Successive samples are correlated
Retain 1 out of every M samples
Acceptance rate
Proposal distribution q is critical

8. Monte-Carlo simulations*
Using N MCMC samples
Target density estimation
Expectation
MAP estimation
p is a posterior
* C. Andrieu et al., “An Introduction to MCMC for Machine Learning“, Mach. Learn., 2003

9. Tracking interacting targets*
Using partilce filters to track multiple interacting targets (ants)
* Khan et al., “MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets”, PAMI, 2005.

10. Particle filter and MCMC
Joint MRF Particle filter
Importance sampling in high dimensional spaces
Weights of most particles go to zero
MCMC is used to sample particles directly from the posterior distribution

11. MCMC Joint MRF Particle filter
True samples (no weights) at each step
Stationary distribution for MCMC
Proposal density for Metropolis Hastings (MH)
Select a target randomly
Sample from the single target state proposal density

12. MCMC Joint MRF Particle filter
MCMC-MH iterations are run every time step to obtain particles
“One target at a time” proposal has advantages:
Acceptance probability is simplified
One likelihood evaluation for every MH iteration
Computationally efficient
Requires fewer samples compared to SIR

13. Particle filter for pupil (ellipse) tracking
Pupil center is a feature for eye-gaze estimation
Track pupil boundary ellipse
Outliers
Pupil boundary edge points
Ellipse overlaid on the eye image

14. Tracking Brute force: Detect ellipse every video frame
RANSAC: Computationally intensive
Better: Detect + Track
Ellipse usually does not change too much between adjacent frames
Principle
Detect ellipse in a frame
Predict ellipse in next frame
Refine prediction using data available from next frame
If track lost, re-detect and continue

15. Particle filter? State: Ellipse parameters Measurements: Edge points
Non-linear dynamics
Non-linear measurements
Edge points are the measured data

16. Motion model Simple drift with rotation State (x0 , y0 ) θ a b
Could include velocity, acceleration etc. a b
Gaussian

17. Likelihood Exponential along normal at each point
di: Approximated using focal bisector distance d1 d2 d3 d4 d5 d6 z1 z2 z3 z4 z5 z6

18. Focal bisector distance* (FBD)
Reflection property: PF’ is a reflection of PF
Favorable properties
Approximation to spatial distance to ellipse boundary along normal
No dependence on ellipse size
Foci
FBD
Focal bisector
* P. L. Rosin, “Analyzing error of fit functions for ellipses”, BMVC 1996.

19. Implementation details
Sequential importance re-sampling*
Number of particles:100
Expected state is the tracked ellipse
Possible to compute MAP estimate?
Weights:
Likelihood
Proposal distribution:
Mixture of Gaussians
* Khan et al., “MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets”, PAMI, 2005.

20. Initial results Frame 1: Detect Frame 2: Track Frame 3: Track

21. Future? Incorporate velocity, acceleration into the motion model
Use a domain specific motion model
Smooth pursuit
Saccades
Combination of them?
Data association* to reduce outlier confound
* Forsyth and Ponce, “Computer Vision: A Modern Approach”, Chapter 17.

22. Thank you!