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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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.

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.

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

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.

Thank you!