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# Today - PowerPoint PPT Presentation

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

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Presentation Transcript
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 ChainMonte 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
• 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
• pis 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
• Particle filter
• Non-linear dynamics
• Non-linear measurements
• Edge points are the measured data
Motion model
• Simple drift with rotation

State

(x0 , y0)

θ

Could include velocity, acceleration etc.

a

b

Gaussian

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
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

Frame 4: Detect

Frame 5: Track

Frame 6: Track

Future?
• Incorporate velocity, acceleration into the motion model
• Use a domain specific motion model
• Smooth pursuit