Particle Filters for Shape Correspondence

Presenter: Jingting Zeng

Outline

Review of Particle FilterHow to Use Particle Filters in Shape

CorrespondenceFurther Implementation in Shape

Clustering

Part One

Review of Particle Filter

What Particle Filter is

Particle filter is a technique for implementing recursive Bayesian filter by Monte Carlo sampling

How Particle Filters Algorithm works

1. Initialize the distribution. The initial distribution can be anything.

2. Observe the system and find a (proportional) probability for each particle that the particle is an accurate representation of the system based on that observation.

This value is refereed as a particle's importance weight.

3. Normalize the particle weights. 4. Resample the distribution to get a new distribution.

A particle is selected at a frequency proportional to its importance weight.

5. Update each particle in the filter according to the prediction of system changes.

6. Repeat from step 2.

6

Particle Filters AlgorithmInitialize particles

Output

Output estimates

1 2 M. . .

Particlegeneration

New observation

Exit

Normalize weights

1 2 M. . .

Weigthcomputation

Resampling

More observations?

yes

no

Demonstration

http://www.oursland.net/projects/particlefilter/

Part Two

How to Use Particle Filters in Shape Correspondence

Goal

The Goal of Shape Correspondence

is to find correspondences between features points in two (similar) shapes

What is the data?

Segmentation

Boundary Tracking

Local Feature Extraction

Centroid Distance (Relative distance to center of polygon )

Curvature (turning angle)

Correspondence Matrix

The correspondence matrix W measures the correspondence probability between shapes A and B

1,1 1,

,1 ,

....

..........................

....

m

n n m

w w

W

w w

Centroid Distance Curvature

Euclidian Distance

Gaussian Distribution

Normalization

CenDist Matrix Curvature Matrix

joint probability

Correspondence Matrix W

CentDist Curvature

W

Correspondence

Given two shapes S1,S2 with n1, n2 vertices, we define the set of correspondences as the set of all pairs of vertices of S1 and S2:

The matrix W defines a probability over the set of correspondences:

Grouping

A Grouping is a member of the power set of .

Each element takes the form

Further constraints on groupings (such as correspondences in order) can limit the search space to a subset

Optimal Sets of Correspondences

The weight of a grouping is defined as:

The correspondence problem is formulated as choosing the complete grouping from the set of constrained groupings with maximal weight:

About Particle Filters

A single particle contains a grouping

represents a particle at time tParticles are built by adding single

correspondences at each iterationCorrespondences are selected based on

the updated weight matrix Wt at time t

Important Steps in PF

Prediction: update each particle and compute its new weight according to Wt. The posterior distribution of at time

(iteration) t is given by eq.1:

Evaluation: Pick n updated particles according to their weights. ‘Better’ particles have a higher chance to survive.

Recede: Every m steps, n correspondences are deleted (m>n). This can be seen as an add on to the update step.

Particle Filters algorithm

1. INIT: t=1, number of particles. Wt = W. Init r for the recede-step.

2. Prepare the constraint matrices for i = 1..m and compute

3. Select a correspondence based on the distribution .

4. PREDICTION: compute posterior distribution (weight of particle) using eq.1.

5. Normalize weights:

6. EVALUATION: compute new set of particles

using residual re-sampling (RRS) preserving most probably those particles with dominant weight.

7. RECEDE: if mod(t, r) = 0 delete n < r correspondences in each particle in .

8. LOOP: if not all particles are complete: , return to step 2, else return

particle with maximum weight to represent a near optimal solution.

Demo

Video demonstration

Shape Correspondence Result

0 100 200 300 400 500 600 700 800

0

100

200

300

400

500

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

Shape Correspondence Result

20 40

-40

-20

0

20

40

60

80

20 40

-40

-20

0

20

40

60

80

20 40

-40

-20

0

20

40

60

80

Part Three

Further Implementation in Shape Clustering

A New Distance Measure

1. Computation of shape correspondence

2. Pre-Alignment using Procrustes Analysis

3. Context dependent alignment using Force Field Simulation (FFS)

4. Mapping into a feature space (Density Computation)

5. Comparison of mapped shape and cluster

Step 1 & 2

Step 3 & 4

Soft K-Means Like Clustering

(1) initialize the recursion parameter and the cluster matrix with random weights.

(2) update the weights of the matrix based on the distance of density maps.

(3) compute all new density maps

(4) decrease the recursion parameter.

(5) go back to step (2) unless convergence is achieved.

Experiment

55 shapes of MPEG-7 dataset11 groups of 5 shapes each

References

http://www.oursland.net/projects/particlefilter/

Theory and Implementation of Particle Filters.ppt by Miodrag Bolic

Finding Shape Correspondences with Particle Filters.ppt by Rolf Lakaemper

A Context Dependent Distance Measure for Shape Clustering (ISVC2008)