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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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Today l.jpg

Today

  • Introduction to MCMC

  • Particle filters and MCMC

  • A simple example of particle filters: ellipse tracking


Introduction to mcmc l.jpg

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


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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

    • pis 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)

θ

Could include velocity, acceleration etc.

a

b

Gaussian


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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


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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 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.


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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!


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