Today

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Today

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

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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

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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
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Markov chains

• Evolution of marginal distribution
• Stationary distribution
• Markov chain T has a stationary distribution
• Irreducible
• Aperiodic

Bayes theorem
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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
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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
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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

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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
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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.
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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

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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

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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

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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
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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

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Particle filter?

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

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Motion model

• Simple drift with rotation

State
(x0 , y0 )
?
Could include velocity, acceleration etc.
a
b
Gaussian
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Likelihood

• Exponential along normal at each point
• di Approximated using focal bisector distance

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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.
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Implementation details

• Sequential importance re-sampling
• Number of particles100
• Expected state is the tracked ellipse
• Possible to compute MAP estimate?

Proposal distribution Mixture of Gaussians
Weights Likelihood
Khan et al., MCMC-Based Particle Filtering for
Tracking a Variable Number of Interacting
Targets, PAMI, 2005.
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Initial results
Frame 1 Detect
Frame 2 Track
Frame 3 Track
Frame 4 Detect
Frame 5 Track
Frame 6 Track
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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.
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Thank you!