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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…)

• Particle filters

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

• Non-linear dynamics

• Non-linear measurements

Non-Gaussian

Non-Gaussian

• Maintain a representation of

• Two stages

• Prediction

• Correction (Bayesian)

Dynamic model (Markov)

Likelihood

Prior

Posterior

• Importance sampling

• Tool 1: Representing a distribution

• Tool 2: Marginalizing

• Tool 3: Transforming prior to posterior

• 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!)

• Marginalization

• Sampled representation

• Just retain the required components and ignore the rest!

Drop ni

• 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

• Prediction

• 2 steps

• Sampling from joint distribution

• Marginalization

Dynamic model (Markov)

(Notation: Chapter 2)

Drop

• Correction

• Modify weights

Likelihood

Prior

Posterior

Let

Likelihood

• 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

• 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.

• Targets lose identity

• Identical appearance

• Multiple peaks in the likelihood

• Best peak “hijacks” all the nearby targets

• 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.

• Sampled representation of prior

• Monte-Carlo approximation

• 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

• 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?

• 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

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

• Sequential Importance Resampling

• Particles at time t

• Weights

• Interactions affect only the weights

Equivalent to independent particle filters

• Targets overlap on each other and then segregate

• Overlapped target state “hijacked”

• Probably hard to model this?

• 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

• 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