Particle Filters for Shape Correspondence Presenter: Jingting Zeng

Outline Review of Particle Filter How to Use Particle Filters in Shape Correspondence Further 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 Algorithm Initialize particles Output Output estimates 12M... Particle generation New observation Exit Normalize weights 12M... Weigth computation Resampling More observations? yes no

Demonstration lter/ lter/

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

Centroid DistanceCurvature Euclidian Distance Gaussian Distribution Normalization CenDist MatrixCurvature 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 t Particles are built by adding single correspondences at each iteration Correspondences are selected based on the updated weight matrix W t at time t

Important Steps in PF Prediction: update each particle and compute its new weight according to W t. 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. W t = 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 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 Video

Shape Correspondence Result

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 dataset 11 groups of 5 shapes each

References lter/ lter/ 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)