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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

• Introduction to MCMC

• Particle filters and MCMC

• A simple example of particle filters: ellipse tracking

• 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

• 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

• Evolution of marginal distribution

• Stationary distribution

• Markov chain T has a stationary distribution

• Irreducible

• Aperiodic

Bayes’ theorem

• 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

• 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

• 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

• 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

• 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.

• 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

• 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-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

• Pupil center is a feature for eye-gaze estimation

• Track pupil boundary ellipse

Outliers

Pupil boundary edge points

Ellipse overlaid on the eye image

• 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

• State: Ellipse parameters

• Measurements: Edge points

• Particle filter

• Non-linear dynamics

• Non-linear measurements

• Edge points are the measured data

• 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

• 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.

• 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.

Frame 1: Detect

Frame 2: Track

Frame 3: Track

Frame 4: Detect

Frame 5: Track

Frame 6: Track

• Incorporate velocity, acceleration into the motion model

• Use a domain specific motion model

• Smooth pursuit

• Combination of them?

• Data association* to reduce outlier confound

* Forsyth and Ponce, “Computer Vision: A Modern Approach”, Chapter 17.