Download - Relational Dynamic Bayesian Networks to improve Multi-Target Tracking. Cristina Manfredotti and Enza Messina DISCo, University of Milano-Bicocca

Relational Dynamic Bayesian Networks to improve

Multi-Target Tracking.

Cristina Manfredotti and Enza Messina

DISCo, University of Milano-Bicocca

2C.Manfredotti, E.Messina: RDBNs to improve MTT. Mercure Chateau Chartrons, Bordeaux, France, Sept. 28 - Oct 2, 2009

Relations to improve tracking


Complex activity recognition

Y.Ke, R.Sukthankar, M.Hebert; Event Detection in Crowed Videos


Objectives

Goals: 1. To model relations and 2. To maintain beliefs over particular

relations between objects

In order to simultaneously:

• Improve tracking with informed predictions and

• Identify complex activities based on observations and prior knowledge


Relational Domain

Relational Domain: set of objects characterized by attributes1 and with relations1 between them

CarIdcolorposition(t)velocity(t)direction(t)DecreasingVelocity(t)

A

SameDirection(t)distance(t)Before(t)

Car BIdcolorposition(t)velocity(t)direction(t)DecreasingVelocity(t)SameDirection(t)distance(t)Before(t)

1Attributes and relations are predicate in FOL.


Relational State

The State of a Relational Domain is the set of the predicates that are true in the Domain.

r

a

s

ss

Relational state

State of attributes

State of relations


Relational Bayesian Networks:

Uncertainty in a Relational Domain Relational (Dynamic) Bayesian Networks

• Syntax RBN:– a set of nodes, one for each variable

– a directed, acyclic graph – a conditional distribution for each node

given its parents

This distribution must take into account the actual “complexity” of the nodes!

• Syntax RBN:– a set of nodes, one for each predicate

– a directed, graph– a conditional distribution for each node

given its parents


Dynamics

The State of a Relational Domain is the set of the predicates that are true in the Domain.

State evolves with time

We extend a RBN to a RDBN as we are used to extend a BN to a DBN.


Inference

Markov assumption andConditional independence of data on state.

bel(st) = ® p(zt|st)s p(st|st-1)bel(st-1)dst-1

Bayesian Filter

The problem of computing:

bel(st) = p(st|z1:t)


Inference

Relations in the State result in correlating the State of different objects between them

p(xt-1|z1:t-1) p(xt|z1:t-1) p(xt|z1:t)

Bel(xt-1) Bel(xt) Bel(xt)

Transition model

Sensor model

t = t+1


Sensor model (1st assumption)

part of the state relative to relations, sr, not directly observable

p(zt|st) = p(zt|sa

t)

observation zt independent by the relations between objects.

Intuitively:

Travelling Together vs Being Close


Transition model: a trick

p(st|st-1) = p(sat,sr

t|sat-1, sr

t-1)

Sat-1

Srt-1

Sat

Srt

Intuitive


* It is a technique that implements a recursive Bayesian Filter through a Monte Carlo simulation. The key idea is to represent the posterior pdf as a set of samples (particles) paired with weights and to filter the mesurament based on these weights..

Particle Filtering* (general case)


Relational Particle Filter


RPF: extraction

Xat,(m)

Xrt,(m)

Xat,(m)

~ p(xat,(m)|sa

t-1,srt-1)

Xat,(m)

~ p(xrt,(m)|sa

t = xat,(m),sr

t-1)

Xrt,(m)


RPF: weighting

The consistency of the probability function ensures the convergence of the algorithm.

Xat,(m)

Xrt,(m)

Weight ( ) ~p(zt|xat)

The weighting step is done according to the attributes part of each particle only, the relational part follows.


Experiments: FOPT


Experiments: Transition Model

• If relation true

• If relation false


Experiments: Results


Further experiments

Data: 15 simulated objects.From each cell, an object can jump to one of the n next cells where n depends by the cell.

Objects can move together. If traveling together,

two (or more) objects will always be in cells from which it is possible for one to reach the

other or vice-versa.

If traveling together, two objects will behave similarly (i.e. if one

turns left, the other will follow).


Tracking AND activity Recognition

Xat,(m)

Xrt,(m)

Xat,(m)

Xrt,(m)

Xat,(m)

Xa{t,(m)}Xo{t,(m)}

Xrt,(m)

Xat+1,(m)

1° step of sampling: prediction of the state of attributes

Xat,(m)

Xa{t,(m)}Xo{t,(m)}

Xrt,(m)

Xat+1,(m)

Xa{t,(m)}Xo{t,(m)}

Xrt+1,(m)

2° step of sampling: prediction of the state of relationsOr activity prediction


step 12step 24

True Positive Rate

False P

ositive Rate

The worst (time step 24) and the best (time step 12) ROC curve for the relation recognition task.

Further Results

0 1

1


PF: 4.6500 2.2333 3.7333 2.7667 …RPF: 4.6000 2.4667 1.3333 2.1000 …

PF : 4.7000 3.6667 5.6667 2.6000 … RPF: 4.6000 3.5333 5.2667 2.5333 …

Further Results (cont.)

Tracking error (distance) for each of the 15 objects.

Comparable behaviour of the errors BUT

for related objects RPF trackes always better than PF.

PF : 4.6667 4.6667 3.8333 1.9333 … RPF: 4.7667 2.7667 3.5333 1.5333 …

PF: 2.0667 5.9000 1.6000RPF: 2.0333 5.8333 2.2333


To conclude ...

• Modeling Relations “dynamically”:– To improve multi target tracking– To recognize complex activities

• Inference in Dynamic Relational Domain– In theory complex BUT

– Simplified by

• “smart decomposition” of the transition model

• “non-relational” sensor model

• Showed promising results

Download - Relational Dynamic Bayesian Networks to improve Multi-Target Tracking. Cristina Manfredotti and Enza Messina DISCo, University of Milano-Bicocca

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