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Particle filters (continued…). Recall. Particle filters Track state sequence x i given the measurements ( y 0 , y 1 , …., y i ) Non-linear dynamics Non-linear measurements. Non-Gaussian. Non-Gaussian. Recall. Maintain a representation of Two stages Prediction

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Particle filters continued l.jpg

Particle filters (continued…)


Recall l.jpg
Recall

  • Particle filters

    • Track state sequence xi given the measurements (y0, y1, …., yi)

    • Non-linear dynamics

    • Non-linear measurements

Non-Gaussian

Non-Gaussian


Recall3 l.jpg
Recall

  • Maintain a representation of

  • Two stages

    • Prediction

    • Correction (Bayesian)

Dynamic model (Markov)

Likelihood

Prior

Posterior


3 useful tools l.jpg
3 Useful tools

  • Importance sampling

    • Tool 1: Representing a distribution

    • Tool 2: Marginalizing

    • Tool 3: Transforming prior to posterior


Tool 1 representing a distribution l.jpg
Tool 1: Representing a distribution

  • Have a set of samples ui with weights wi

  • (ui, wi ): Sampled representation off(u)

  • Expectation under f(u)

  • Samples used only as a means to evaluate expectations (Not true samples!)


Tool 2 marginalization l.jpg
Tool 2: Marginalization

  • Marginalization

  • Sampled representation

  • Just retain the required components and ignore the rest!

Drop ni


Tool 3 from prior to posterior l.jpg
Tool 3: From Prior to Posterior

  • Modify the weights to transform from one distribution to another

  • Similarly for going from prior to posterior

?

To

From

To

From

Scale factor is the same for all the samples


Simple particle filter l.jpg
Simple Particle filter

  • Prediction

  • 2 steps

    • Sampling from joint distribution

    • Marginalization

Dynamic model (Markov)

(Notation: Chapter 2)

Drop


Simple particle filter9 l.jpg
Simple Particle filter

  • Correction

  • Modify weights

Likelihood

Prior

Posterior

Let

Likelihood


Improved particle filter l.jpg
Improved Particle filter

  • Simple Particle filter

    • Many samples have small weights

    • Number of samples increases at every step

    • Lots of samples wasted

  • Resample (Sampling-Importance -Resampling)

    • Prior:

    • Predictions:

  • Resampling also takes care of increasing number of samples


Tracking interacting targets l.jpg
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.


Independent particle filters l.jpg
Independent Particle filters

  • Targets lose identity

  • Identical appearance

    • Multiple peaks in the likelihood

    • Best peak “hijacks” all the nearby targets


Alternate view of particle filters l.jpg
Alternate view of Particle filters

  • Notation*

Marginalization

Posterior

Likelihood

Prior

State at time t

Measurement at time t

All measurements upto time t

*Khan et al., “MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets”, PAMI, 2005.


Alternate view of particle filters14 l.jpg
Alternate view of Particle filters

  • Sampled representation of prior

  • Monte-Carlo approximation


Alternate view of particle filters15 l.jpg
Alternate view of Particle filters

  • Sequential Importance Resampling (SIR)

  • Particles at time t

  • Weights (easy to verify!)

  • Prediction and correction in one step

Particles sampled from a mixture distribution formed by previous particle set


Independent vs joint filters l.jpg
Independent vs. Joint filters

  • Multiple targets

    • Joint state space: Union of individual state spaces

  • Independent targets

    • Predictions are made independently from respective spaces

  • Interacting targets

    • Predictions are from the joint state space

    • High dimensionality: MCMC better than Importance sampling?


Interacting targets l.jpg
Interacting targets

  • Targets influence the dynamics of others

  • Particles cannot be propagated independently

  • Model interactions between targets using Markov Random Fields (MRF)

Individual dynamics

Pair wise interactions


Slide18 l.jpg
MRF

Edges are formed only when templates overlap

  • Interaction potential

  • g(Xit ,Xjt) penalizes overlap between targets

  • Takes care of “hijacking”

Overlap is penalized by the interaction potential


Joint mrf particle filter l.jpg
Joint MRF Particle filter

  • Sequential Importance Resampling

  • Particles at time t

  • Weights

  • Interactions affect only the weights

Equivalent to independent particle filters


Target overlap l.jpg
Target overlap

  • Targets overlap on each other and then segregate

  • Overlapped target state “hijacked”

  • Probably hard to model this?


Why mcmc l.jpg
Why 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 l.jpg
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 filter23 l.jpg
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


Variable number of targets l.jpg
Variable number of targets

  • Target identifiers kt is a state variable

  • Each kt determines a corresponding state space

  • State space is the union of state spaces indexed by kt

  • Particle filtering

  • RJMCMC to jump across state spaces

Prediction + Correction



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