particle filters for shape correspondence presenter: jingting zeng
TRANSCRIPT
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Particle Filters for Shape Correspondence
Presenter: Jingting Zeng
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Outline
Review of Particle FilterHow to Use Particle Filters in Shape
CorrespondenceFurther Implementation in Shape
Clustering
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Part One
Review of Particle Filter
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What Particle Filter is
Particle filter is a technique for implementing recursive Bayesian filter by Monte Carlo sampling
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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.
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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
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Demonstration
http://www.oursland.net/projects/particlefilter/
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Part Two
How to Use Particle Filters in Shape Correspondence
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Goal
The Goal of Shape Correspondence
is to find correspondences between features points in two (similar) shapes
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What is the data?
Segmentation
Boundary Tracking
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Local Feature Extraction
Centroid Distance (Relative distance to center of polygon )
Curvature (turning angle)
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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
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Centroid Distance Curvature
Euclidian Distance
Gaussian Distribution
Normalization
CenDist Matrix Curvature Matrix
joint probability
Correspondence Matrix W
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CentDist Curvature
W
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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:
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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
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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:
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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
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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:
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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.
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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:
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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.
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Demo
Video demonstration
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Shape Correspondence Result
0 100 200 300 400 500 600 700 800
0
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0 100 200 300 400 5000
50
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Shape Correspondence Result
20 40
-40
-20
0
20
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60
80
20 40
-40
-20
0
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20 40
-40
-20
0
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Part Three
Further Implementation in Shape Clustering
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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
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Step 1 & 2
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Step 3 & 4
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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.
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Experiment
55 shapes of MPEG-7 dataset11 groups of 5 shapes each
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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)