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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
Today
  • Introduction to MCMC
  • Particle filters and MCMC
  • A simple example of particle filters: ellipse tracking
introduction to mcmc
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
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 chains4
Markov chains
  • Evolution of marginal distribution
  • Stationary distribution
  • Markov chain T has a stationary distribution
    • Irreducible
    • Aperiodic

Bayes’ theorem

markov chains5
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
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 hastings7
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
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
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
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
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 filter12
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
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
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
Particle filter?
  • State: Ellipse parameters
  • Measurements: Edge points
  • Particle filter
    • Non-linear dynamics
    • Non-linear measurements
      • Edge points are the measured data
motion model
Motion model
  • Simple drift with rotation

State

(x0 , y0)

θ

Could include velocity, acceleration etc.

a

b

Gaussian

likelihood

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
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
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
Initial results

Frame 1: Detect

Frame 2: Track

Frame 3: Track

Frame 4: Detect

Frame 5: Track

Frame 6: Track

future
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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