1. Particle Filtering for Geometric Active Contours
Iulian Pruteanu
3 June 2005

2. 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

3. 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.

4. Framework The State Space model consists of: the state
the observation-- the image at time t,
relates 2 consecutive curves

5. 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

6. Dynamic Systems:
can be modeled with two equations:
1. State Transition or Evolution Equation
2. Measurement Equation

7. Assumptions:
The observations are conditionally independent given the state:
Hidden Markov Model (HMM):
given and defines state transition probability
for k>=1.

8. General prediction-update framework:
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

9. 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)
set of support points (samples, particles)
associated weights, normalized to
(each particle is weighted in proportion to the likelihood of the observation at time t)
Then,
(discrete true approximation to the true posterior)

10. Usually, we cannot draw samples from p(. ) directly
Usually, we cannot draw samples from p(.) directly. Assume we can sample directly from a (different) importance function q(.).
Our approximation is still correct if
If the importance function is chosen to factorize such that
than one can augment old particles by
to get new particles

11. The weight update (after some lengthy computations):
Furthermore, if
(and we don’t need to preserve trajectories and history of observations )

12. The prediction step for consists of:
Predicting the local deformations in the shape of the object.
Predicting the affine motion of the object.
The affine motion prediction is obtained from the state dynamics for
The prediction for local shape deformation at time t = local shape deformation at time t-1.
L iterations of gradient descent
k=1,2,3,…,L

13. could be any function that model the dynamics of motion of the moving object (here, an autoregressive model is used).
denotes the 6-dimensional affine parameter vector that relates two consecutive curves (Ct and Ct-1). See [2].

14. Results:
Car sequence:
Fish sequence: interesting
Couple sequence:

15. 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).

16. 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.