A FRIENDLY APPROACH TO PARTICLE FILTERS IN COMPUTER VISION

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A FRIENDLY APPROACH TO

PARTICLE FILTERS IN

COMPUTER VISION

Concepts, hints and examples

12/2/2011

Dr.-Ing. Marcos Nieto DoncelInvestigador/Researcher

[email protected]

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Outline

Motivation

Bayesian framework

Sampling solution

Examples

2Marcos Nieto, PhD - [email protected]/2/2011

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Motivation

For what?

Obtain estimates of a recursive/dynamic system

W

H

(x0,y0)

Let’s stay in computer vision

applications


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Motivation

Why?

Deterministicapproach vs

Probabilisticapproach


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Motivation

How?

1. Define your target

2. Define your functions

3. Select a type of filter adapted to 1) and 2)

4. Implement and run

5. Optionally: Write your paper and share : )


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Bayesian filtering

Target: xk

It evolves through time according to somedynamics, properties, interaction, etc.

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H

(x0,y0)

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H

x0y0

p(xk|xk-1)

Prior / Dynamics / Transition…


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Bayesian filtering

Observations: z1:k

Noisy, distorted, indirect

Typically, different dimensionaliy

p(zk|xk)

Likelihood / Observation model / Measurements…


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Bayesian filtering

Posterior distribution: p(xk|z1:k)

Probability density function

This is all you can expect to know

Typically we want a point-estimate of thisdistribution At each time instant: x*k

At the end


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Bayesian filtering

K-1 K K+1 TIME

MEASUREMENTS

(VISIBLE)

STATES

(HIDDEN)xk-1

zk-1

xk

zk

xk+1

zk+1

p(xk|xk-1): Dynamic model

p(zk|xk): Observation model


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Bayesian filtering

How?

PredictionUse the dynamics, guess future according to

CorrectionObtain a new observation, and apply Bayes’ rule

p(xk|xk-1)

Likelihood Prediction

Posterior


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Particle filters

Solve through sampling!

Let us approximate posterior as a set of samples

Samples / Particles / Hypotheses

WEIGHTED UNWEIGHTED


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Particle filters

Understanding particles Each particle represents a hypothesis

Remember! we will typically want just one point-estimate Best particle, mean particle, mode, median…

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H

(x0,y0)

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x0

y0


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Particle filters

How to sample?

Importance sampling

Markov Chain Monte Carlo

Gibbs sampling

Slice sampling

…

How many samples?

As much as required to track the posterior!


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Particle filters

Sequential importance sampling (SIR)


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Particle filters

Sequential importance sampling (SIR)


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SIR – example (I)

Single object tracking


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SIR – example (I)

Linear-Gaussian dynamics

Generate N samples starting fromprevious state adding estimatedvelocity

And some Gaussian noise

The noise makes thatsamples are different!

T),,,( yxyxkx

11 kkk A vxx),;()|( 111 kkkk QANp xxxx

1000

0100

010

001

k

k

A


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SIR - example (I)

Likelihood based on segmentation or color histogram

Evaluate each predicted sample according to this value

Likelihood function shouldreturn high values for

“good” hypotheses, and lowfor “bad” hypotheses


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SIR – example (II)

Eye-tracking

Linear prediction won’t work The projection of the eye movement on the screen is

difficult to predict

Define a combination of linear-Gaussian + uniform


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SIR – example (II)

• Conclusion:

Try to adapt the dynamic model you use with what youthink is the real dynamics of what you want to track

This may imply using mixture models, accelerations, etc.

Also, a good likelihood model should include somecontinuous term (like a uniform), in order to cope withocclusions, so that the track is not lost


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SIR

Problems Required number of samples increase exponentially with

problem dimension

Several objects/elements?a) Define a multimodal posterior and generate multiple point-

estimatesb) Cluster particlesc) Increase state vector dimension

Variable number of objects?a) Add external handlerb) Include the number of objects as another variable to estimate


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Particle filters

MCMC

More flexible

The problem of dimension is softened

Directly sample from the posterior

Researchers are focusing in MCMCMany excellent works that propose solutions to multiple

object, interaction, entering-exiting, number of samplesreduction, etc.


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Particle filters

MCMC

Generate a Markov chain of samples directly fromthe posterior


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MCMC

Metropolis-Hastings

Start somehow

Propose a movement

Accept with probabilityequal to the ratio betweenproposed value and previousone

Prob. = 1 if proposed is betterthan previous

Prob. = ratio if not

Metropolis-Hastings allowsobtaining samples for anarbitrary distribution bymaking a chain which acceptsor rejects movements


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MCMC

Multiple objects

State vector including all thedimensions of all objects

Metropolis-Hastings: Generate a chain of N samples

For each sample, use theinformation of all the samplesat the previous time instant

Each sample is a hypothesis of the state

of all objects

After the chain is completed, we have

the sample-basedapproximation of the

posterior


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MCMC

Marginalized proposal moves

Propose movement of a single dimension at each new sample

E.g. don’t propose a move in all dimensions for all objects

Choose a dimension randomly and update it

x W y

x WHx xW L

…

…

Burn-in period Stop when stationary function is

reached.Or when maximum number of

samples is reached.


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MCMC

Interaction between objects

Markov Random Field (MRF) factor


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MCMC

Variable number of objects

a) Add an external detector, and modify state size

b) Reversible-Jump MCMC Define an Enter move (creates an object)

Define an Exit move (removes an object)

Define an Update move (updates existing objects)


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Discussion

What should I use?

SIR

MCMC

Kalman?


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Discussion

If dynamics and observation are linear, and with Gaussian noise

Use Kalman, this is the optimum solution

If not, consider using a particle filter


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Discussion

SIR

Use it if target dimension is low (3-5)

Use it if you plan to parallelize processing Remember particles are independent one from another

Would require important design issues forManaging multiple objects

Managing variable number of objects


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Discussion

MCMCUse it if dimensions increase

It can not be parallelized Remember that particles form a chain, and each one

depends on the previous one

Adapted to multiple objectsMRF interaction is easy to insert

Metropolis-Hastings can be efficiently adapted tomultiple object


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Summary

Define your target

Determine its dynamics

Define the likelihood

Select a filter that adapts to the problem

Implement it

Run it carefully selecting the appropriateparameters of your functions, number of particles, etc.


362/2/2011 36

Dr.-Ing. Marcos Nieto DoncelInvestigador/Researcher

[email protected]

http://marcosnieto.net/

A FRIENDLY APPROACH TO PARTICLE FILTERS IN COMPUTER VISION

Technology

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