Loading in 5 sec....

Particle filters (continued…)PowerPoint Presentation

Particle filters (continued…)

- By
**zilya** - Follow User

- 111 Views
- Updated On :

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

Related searches for

Download Presentation
## PowerPoint Slideshow about '' - zilya

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Particle filters (continued…)

Recall

- Particle filters
- Track state sequence xi given the measurements (y0, y1, …., yi)
- Non-linear dynamics
- Non-linear measurements

Non-Gaussian

Non-Gaussian

Recall

- Maintain a representation of
- Two stages
- Prediction
- Correction (Bayesian)

Dynamic model (Markov)

Likelihood

Prior

Posterior

3 Useful tools

- Importance sampling
- Tool 1: Representing a distribution
- Tool 2: Marginalizing
- Tool 3: Transforming prior to posterior

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

- Marginalization
- Sampled representation
- Just retain the required components and ignore the rest!

Drop ni

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

- Prediction
- 2 steps
- Sampling from joint distribution
- Marginalization

Dynamic model (Markov)

(Notation: Chapter 2)

Drop

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*

- 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

- Targets lose identity
- Identical appearance
- Multiple peaks in the likelihood
- Best peak “hijacks” all the nearby targets

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 filters

- Sampled representation of prior
- Monte-Carlo approximation

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

- 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

- 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

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

- Sequential Importance Resampling
- Particles at time t
- Weights
- Interactions affect only the weights

Equivalent to independent particle filters

Target overlap

- Targets overlap on each other and then segregate
- Overlapped target state “hijacked”
- Probably hard to model this?

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

- 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 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

- 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

Download Presentation

Connecting to Server..