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- 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

- 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
- Saccades
- Combination of them?

- Data association* to reduce outlier confound

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

Thank you!