Motivating Markov Chain Monte Carlo for Multiple Target Tracking

Krishna

Overview

• Single Target Tracking : Bayes filter.

• Multiple Target Tracking : Extending Bayes filter to Joint Probabilistic Data Association Filter (JPDAF).

• JPDAF is NP Hard. Extend JPDAF to MCMC.

Prior

Posterior

Basic Concepts

Observation

Law of Total Probability

Markov Process

Bayes Rule

Locating an Object

Single -Target Tracking : Problem Definition

k -1 k k + 1 k + 2k -2

Consider tracking 1 Object.

is the sequence of all measurements upto time k

state of a single object at time k

Noisy observation- time k

How to estimate the state for observations ?

Bayes Filters

Motion Model

Observation Model

Predict :

Update : P(Current State | previous observations)P(Current State | Previous State)

Motion Model !

P(Previous State | Previous Observations)

P(Current State | Current & previous observations)P(Current Observation | Current State)

Observation Model !

P(Current State | previous observations)

Predicted State Observation

Kalman Filter : Specialization of Baye’s Filter

Assumptions of Kalman Filter:

1 , where (0, )

, where (0, )t t t t t t

t t t t t t

x A x w w N Q

z C x v v N R

Multi-Target Tracking : Problem Definition

k -1 k k + 1 k + 2k -2

State of these objects at time k :

Consider tracking T Objects.

is the state space of a single object.

is observation at time k is one such observation.is the sequence of all observations upto time k

How to assign the observed observations to individual objects ?Simultaneously Assign and Track

Predict :

Update :

?

JPDAF Framework

Predict :

Update :

1

2

3

Observation Model

Thank You

Markov Process

Recall

Approximation by the belief about predicted state of objects

Chicken egg problem : State of objects θ

State of objectsθ

Likelihood of assignments given current states are constant for all Objects