Today

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 ChainMonte 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 • 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 • pis 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 • Particle filter • Non-linear dynamics • Non-linear measurements • Edge points are the measured data

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

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

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 Frame 4: Detect Frame 5: Track Frame 6: 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!