OBJECT TRACKING USING PARTICLE FILTERS

1
OBJECT TRACKING USING PARTICLE FILTERS
2
Table of Contents

• Tracking
• Tracking as a probabilistic inference problem
• Applications of Tracking
• Different approaches for Object Tracking
• Particle Filter
• A Simple Particle Filter Algorithm
• Basic steps implemented in the project
• Files used in the project
• Demo

3
TRACKING

• Tracking is the problem of generating an
  inference about the motion of an object given a
  sequence of images.
• In a typical tracking problem , we have a model
  for the objects motion and some set of
  measurements from a sequence of images.

4

• The measurements could be the positions of some
  image points, the positions and moments of some
  image regions etc.
• Tracking is an inference problem. The moving
  object has some form of internal state, which is
  measured at each frame . We need to combine our
  measurements as effectively as possible to
  estimate the objects state.

5
Tracking as a probabilistic inference problem

• Prediction
• P(Xi Y0y0 , ..,Yi-1yi-1).
• Data Association
• P(Xi Y0y0 , ..,Yi-1yi-1).
• Correction
• P(Xi Y0y0 , ..., Yiyi).

6
Independence Assumptions

• Only the immediate past matters
• P(Xi X1 ,., Xi-1)P(Xi Xi-1 ).
• Measurements depend only on the current state
• P(Yi Yj ,...,YkXi )P(YiXi )P(Yj ,.,YkXi ).

7
Applications of Tracking

• Motion Capture
• Recognition from motion
• Surveillance
• Targetting

8
Different approaches for Object Tracking

• Correlation based
• Feature based
• Gradient based
• Color Histograms
• Kalman Filter
• Particle Filter

9
PARTICLE FILTER

• Particle Filters are powerful tools for bayesian
  state estimation in non-linear systems.
• The basic idea of particle filters is to
  approximate a posterior distribution over unknown
  state variables by a set of particles, drawn from
  this distribution.

10

• Particle Filters requires two types of
  information
• Data
• Controls
• Measurements
• Probabilistic model of the system
• The data is given by ztz1,z2,,zt and
• utu1,u2,.,ut .

11

• Particle Filters, like any member of the family
  of Bayes filters such as kalman filters and
  HMMs, estimate the posterior distribution of the
  state of the dynamical system conditioned on the
  data,p(xt zt ,ut ). They do so via the following
  recursive formula
• P(xtzt ,ut )ht p(ztxt) Ip(xtut ,xt-1)
  p(xt-1zt-1 ,ut-1)dxt-1

12

• To Calculate this posterior, three probability
  distributions are required ( Probabilistic model
  of the dynamical systems)
• A Measurement model,p(ztxt), which describes the
  probability of measuring zt when the system is in
  state xt .
• A Control model, p(xtut ,xt-1 ), which
  characterizes the effect of controls ut on the
  system state by specifying the probability that
  the system is in state xt after executing control
  ut in state xt-1 .
• An Intial state distribution , p(x0), which
  specifies the users knowledge about the intial
  system state.

13
Problems with probabilistic filter

• In many applications, the key concern in
  implementing this probabilistic filter is the
  continous nature of the staes x, controls u, and
  measurements z. Even in discrete versions , these
  spaces might be prohibitively large to compute
  the entire posterior.

14
Particle filter tackles the problem

• The particle filter addresses these concerns by
  approximating the posterior using the sets of
  state samples (particles)
• Xt xtii1,..,N
• The set Xt consists of N particles xt i ,for
  some large number of N. Together these particles
  approximates the posterior p(xtzt ,ut ). Xt is
  calculated recursively.

15
A Simple Particle Filter Algorithm

• Given a prior p(X1), a transition prior
  p(XtXt-1) and a likelihood
• p(YtXt ), the algorithm is as follows
• Initialization , t1
• for i1,.,N, sample (X1( i ))p(X1 ) and set
  t2.
• 2) Importance Sampling step
• For i1,.,N sample Xpt( i ) p(Xpt( i
  )Xt-1( i ))
• and set Xp1t( i ) ( Xt( i ) , X1t-1( i ) ).
• For i1,,N, Evaluate Importance weights
• wt p(YtXt( i ) )
• Normalise the importance weights.

16
Algorithm contd

• 3) Selection Step
• Resample with replacement N particles (Xit( i
  ) i1,.,N) from the set ( Xp1t( i ) i
  1,., N) according to the normalised importance
  weights.
• Set t t1 and go to step 2

17
Basic steps implemented in the project

• Intially at time t0, the particles x0i are
  generated from the initial state distribution
  p(x0 ). The t-th particle set Xt , is the
  calculated recursively from Xt-1 as follows
• Set Xt Xtaux 0
• For j1 to N do
• pick the j-th sample xt-1 j e Xt-1
• draw xt j p(xtut ,xt-1 j )
• set wt j p(ztxt j )
• add (xt j ,wt j ) to Xtaux
• End for

18
Basic Steps contd.

1. For i 1 to m do
2. draw xti from Xtaux with probability
   propotional to wti
3. add xti to Xt
4. End for

19
Files used in the project

• A set of image files, which was later converted
  to a video file.
• A matrix file consisting of data about the image.
  
• pf_proj.m , which contains the main algorithm for
  particle filters and drawing of the trajectories
  for the moving object.
• multinomialR.m , which contains the code for
  resampling.

20
Sample of the code

• for i 1N,       
• states(, t, i) A(,,i) states(,
  t-1, i) B(,,i) randn(100,1)
•    
• Evaluate importance weights.   
•     w(t, i) (exp(-0.5(states(,t,i))'
  (states (,t,i)))) 1e-99       
•        w_real(t, i) w(t, i)   
•   end     
•  w(t,) w(t,) ./ sum(w(t, ))       
  Normalise the weights.

21
DEMO

• Shown separately

22
Acknowledgement

• Thanks to Dr. Longin Jan Latecki, for providing
  me the opportunity to work on this project.