Transcript No Slide Title

Particle Filtering for Geometric
Active Contours
Iulian Pruteanu
3 June 2005
Paper:
Y.Rathi, A. Tannenbaum, “Particle Filtering for Geometric
Active Contours with Application to Tracking Moving and
Deforming Objects”, CVPR, 2005.
Duke University Machine Learning Discussion Group
Discussion Leader: Iulian Pruteanu
3 June 2005
Overview
• Paper presents an approach for a particle filtering algorithm
in the geometric active contour framework that can be used
for tracking moving and deforming objects.
• Framework uses both particle filters and geometric active
contours.
• Particle Filters: cannot handle changes in curve topology
• Geometric Active Contours: do not utilize the temporal
coherency of the motion or deformation.
Framework
The State Space model consists of:
the state X t  [ At , t ]
the observation-- the image at time t, Yt  Im age(t )
At  affine_ parameter_ vector
t  contour time t
relates 2 consecutive curves
Particle Filtering – Review:
General classification of filter strategies:
Gaussian models:
• Kalman filters
• Extended Kalman filters
• Linear regression filters;
Mixture of Gaussian models:
Gaussian-sum filter
Assumed density filter
Nonparametric models:
Particle filter
Histogram filter
Dynamic Systems:
• can be modeled with two equations:
1. State Transition or Evolution Equation
2. Measurement Equation
xk  f k ( xk 1 , uk 1 , vk 1 )
z k  hk ( xk , uk , nk )
f (.,.,.): evolution_ function
h(.,.,.): m easurem ent _ function
xk , xk 1   nx : current _ and _ previous_ state
vk 1   nv : state _ noise(usually_ not _ Gaussian)
uk 1   nu : known_ input
z k   nz : m easurem ent
nk   nn : m easurem ent _ noise
Assumptions:
The observations are conditionally independent given the state: p( zk / xk )
Hidden Markov Model (HMM):
p( x0 ) given and
for k>=1.
p( xk / xk 1 ) defines state transition probability
General prediction-update framework:
1. Prediction step: using Chapman-Kolmogoroff equation
prior of the state xk (at time k) without knowledge of the
measurement zk
2. Update step: compute posterior pdf from predicted prior pdf and
new measurement
likelihood  prior
posterior 
evidence
We represent the posterior probabilities by a set of randomly chosen
weighted samples.
The basic framework for most particle filter algorithms is SIS
(Sequential Importance Sampling)
{x0i :k } :set of support points (samples, particles)
i  1,..., N S
wki : associated weights, normalized to
NS
i
w
 k 1
i 1
(each particle is weighted in proportion to the likelihood of the observation at time t)
Then,
NS
p( xk / z1:k )   wki  ( x0:k  x0i :k )
i 1
(discrete true approximation to the true posterior)
i
k from
Usually, we cannot draw samples x
p(.) directly. Assume we
can sample directly from a (different) importance function q(.).
i
p
(
x
i
0:k / z1:k )
Our approximation is still correct if wk 
q( x0i :k / z1:k )
If the importance function is chosen to factorize such that
q( x0:k / z1:k )  q( xk / x0:k 1, z1:k )q( x0:k 1 / z1:k 1 )
than one can augment old particles
to get new particles
i
0:k
x
i
0:k 1 by
x
xk  q( xk / x0:k 1 , z1:k )
The weight update (after some lengthy computations):
i
i
i
p
(
z
/
x
)
p
(
x
/
x
i
i
k
k
k
k 1 )
wk  wk 1 
q( xki / x0i :k 1 , z1:k )
Furthermore, if q( xk / x0:k 1, z1:k )  q( xk / xk 1, z1:k )
NS
p( xk / z1:k )   wki  ( xk  xki )
i 1
(and we don’t need to preserve trajectories
observations z1:k 1 )
i
0:k 1
x
and history of
The prediction step for X t  [ At , t ] consists of:
1. Predicting the local deformations in the shape of the
object.
2. Predicting the affine motion of the object.
The affine motion prediction is obtained from the state dynamics for At
The prediction for local shape deformation at time t = local shape
deformation at time t-1.
~
L
Ct  Ct 1  fCE (t 1, Yt 1 )
L iterations of gradient descent
      Eimage ( , Y )
L
L
fCE (, Y )  
k
k 1
k
k 1
k=1,2,3,…,L
f AR could be any function that model the dynamics of motion of the moving
object (here, an autoregressive model is used).
At denotes the 6-dimensional affine parameter vector that relates two
consecutive curves (Ct and Ct-1). See [2].
Results:
Car sequence:
Fish sequence: interesting
Couple sequence:
Conclusions:
• the algorithm might perform poorly if the object being tracked is
completely occluded for many frames.
• for the car sequence it seems that adding the particle filtering
approach to the geometric active contours theory didn’t help too
much.
• if similar results can be obtained using level sets method and a
local initialization, why do we need some other extra steps?
• the occlusion indicated in this paper is different than the one
discussed in our meeting group (we used 2 different moving parts
in occlusion instead of a moving part and a static one).
References:
[1]. A. Blake, M. Isard, “Active Contours”, Springer, 1998.
[2]. S. Periaswamy, H. Farid, “Elastic Registration in the Presence of Intensity
Variations”, IEEE, 2003.