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Outline An optimal probabilistic graphical model for point set matching Tiberio Caetano 1,2 , Terry Caelli 1 and Dante Barone 2 1 Department of Computing Science University of Alberta, Canada 2 Instituto de Inform´ atica UFRGS, Brazil SSPR 2004 Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

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Page 1: Mlss05au Caelli Gma 01

Outline

An optimal probabilistic graphical model for pointset matching

Tiberio Caetano1,2, Terry Caelli1 and Dante Barone2

1Department of Computing ScienceUniversity of Alberta, Canada

2Instituto de InformaticaUFRGS, Brazil

SSPR 2004

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 2: Mlss05au Caelli Gma 01

Outline

Outline

1 The ProblemProblem DefinitionProperties of the FormulationComputational Complexity

2 The SolutionGraphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

3 ExperimentsExperimental SetupInexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 3: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

Outline

1 The ProblemProblem DefinitionProperties of the FormulationComputational Complexity

2 The SolutionGraphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

3 ExperimentsExperimental SetupInexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 4: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

The Problem

Scene

f: D −> C

Template

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 5: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

The Problem

Scene

f: T −> S

Template

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 6: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

The Problem

Scene

f: T −> S

Template

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 7: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

The Problem as Weighted Graph Matching

EDMs

The Euclidean Distance Matrix (EDM) of a point set uniquelydetermines the rigid conformation

As a result...

Conformations can be compared by comparing the EDMs

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 8: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

The Problem as Weighted Graph Matching

EDMs

The Euclidean Distance Matrix (EDM) of a point set uniquelydetermines the rigid conformation

As a result...

Conformations can be compared by comparing the EDMs

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 9: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

The Problem as Weighted Graph Matching

EDMs

The Euclidean Distance Matrix (EDM) of a point set uniquelydetermines the rigid conformation

As a result...

Conformations can be compared by comparing the EDMs

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 10: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

Problem Definition

Problem

Given two point sets T = {di , i = 1, . . . ,T} andS = {cj , j = 1, . . . ,S} in Rn (n ∈ N+), find the function f : T → Sthat maximizes

P(f ) =∑i ,j∈T

S(||di − dj ||, ||cf (di ) − cf (dj )||),

where S(·, ·) is a similarity function and || · || is the L2 metric

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 11: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

Properties

Properties

Handles only invariance to isometries (rotations, translations,reflexions)

f can be any function

OK NOOK

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 12: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

Computational Complexity

Complexity

There are ST possible mapping functions f

Brute force solution: test each and compute score P(f )

Exponential complexity

Question

How to solve it?

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 13: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Problem DefinitionProperties of the FormulationComputational Complexity

Computational Complexity

Complexity

There are ST possible mapping functions f

Brute force solution: test each and compute score P(f )

Exponential complexity

Question

How to solve it?

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 14: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Outline

1 The ProblemProblem DefinitionProperties of the FormulationComputational Complexity

2 The SolutionGraphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

3 ExperimentsExperimental SetupInexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 15: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Graphical Models

Graphical Models

Graphical Models are stochastic processes defined over graphs

Nodes of the graph are random variables

Edges of the graph are probabilistic dependencies betweenrandom variables

61

X2

X3

X4

X5

XX

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 16: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Types of Graphical Models

Undirected Graphical Models (= Markov Random Fields)

Symmetric probabilistic relations (undirected edges)

61

X2

X3

X4

X5

XX

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 17: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Types of Graphical Models

Directed Graphical Models (= Bayesian Networks)

Possibly Non-Symmetric probabilistic relations (directededges)

61

X2

X3

X4

X5

XX

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 18: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Features of a Graphical Model

Features

The probabilistic dependencies between neighbor nodes(“potential functions”)

The connectivity of the graph

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 19: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Undirected Graphical Models

Undirected

We consider exclusively undirected Graphical Models

61

X2

X3

X4

X5

XX

Interest

The Interest is to compute the MAP estimate for the model

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 20: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Undirected Graphical Models

Undirected

We consider exclusively undirected Graphical Models

61

X2

X3

X4

X5

XX

Interest

The Interest is to compute the MAP estimate for the model

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 21: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Hammersley-Clifford Theorem

HC Theorem

States that the joint distribution of a Graphical Model is factorizedover products of functions over maximal cliques of the graph:

p(x) =∏C∈C

ψC (xC )

Whose maximization is equivalent to minimize

U(x) = −∑C∈C

logψC (xC )

If we choose only pairwise cliques, then a suitable definition of theψC s leads to the original formulation of the problem as a weightedgraph matching one

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 22: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Our Formulation

Our Formulation

Template points are random variables

Scene points are realizations

d

Modeling domain X j= xlX i= xk

Problem domain

jd

kc

lc

i

Thus...

The best mapping becomes the MAP solution of the model.

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 23: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Our Formulation

Our Formulation

Template points are random variables

Scene points are realizations

d

Modeling domain X j= xlX i= xk

Problem domain

jd

kc

lc

i

Thus...

The best mapping becomes the MAP solution of the model.

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 24: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Constructing the model

Recall that to construct a Graphical Model one needs

A set of pairwise potential functions

A connectivity pattern

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 25: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Potential Functions

Potential Functions

We use pairwise potential functions

A potential function measures how good is a pairwise map

HIGH Potential

di

ck

dj

c l

LOW Potential

c l

di

ck

dj

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 26: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Similarity Functions

Possible Similarity Functions

A potential function ψ is build from a similarity function Slike the following ones:

l

d

GaussianHyperbolic Tanget

kld

c l

di

ck

dkdi

dj

j

ij

G(dij , dkl) = exp

(− 1

2σ2|dij − dkl |2

)H(dij , dkl) = 1−tanh

[|dij − dkl |

σ

]Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 27: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Potential Functions

Potential Functions

Each connected pair i , j can map to S2 possible pairs:

ψij(Xi ,Xj) =1

Z

S(Xi = x1,Xj = x1) . . . S(Xi = x1,Xj = xS)...

. . ....

S(Xi = xS ,Xj = x1) . . . S(Xi = xS ,Xj = xS)

i X jX

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 28: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

What is an appropriate connectivity for the graphical model?

We can’t use the fully connected graph because exactinference has exponential complexity

What is an appropriate connectivity for the graphical model?

It is possible to prove that a particular sparse graph whereexact inference is doable in polynomial time is equivalent tothe fully connected graph in the limit of exact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 29: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

What is an appropriate connectivity for the graphical model?

We can’t use the fully connected graph because exactinference has exponential complexity

What is an appropriate connectivity for the graphical model?

It is possible to prove that a particular sparse graph whereexact inference is doable in polynomial time is equivalent tothe fully connected graph in the limit of exact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 30: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

What is an appropriate connectivity for the graphical model?

We can’t use the fully connected graph because exactinference has exponential complexity

What is an appropriate connectivity for the graphical model?

It is possible to prove that a particular sparse graph whereexact inference is doable in polynomial time is equivalent tothe fully connected graph in the limit of exact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 31: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Some Geometry

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 32: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Some Geometry

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 33: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Some Geometry

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 34: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Some Geometry

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 35: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Some Geometry

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 36: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Some Geometry

Determined

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 37: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 38: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 39: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 40: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 41: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 42: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 43: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 44: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

Global Rigidity

The resulting graph is said to be globally rigid.

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 45: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Rn

There is a Lemma:

This can be generalized to any dimension, n ∈ N+

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 46: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

k-tree

The resulting graph is technically a 3-tree, in the case of R2,and a k-tree in the case of Rk−1

A k-tree has a maximal clique size fixed in k+1

4−clique3−tree,

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 47: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

There is a Theorem:

It is possible to prove that a Graphical Model whose topology isgiven by a k-tree is equivalent to the fully connected model in thelimit of exact matching

Intuition:

The potential functions depend only on the relative distances.Since a k-tree is sufficient to encode all distances, it is alsosufficient to encode all potential functions

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 48: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Connectivity

There is a Theorem:

It is possible to prove that a Graphical Model whose topology isgiven by a k-tree is equivalent to the fully connected model in thelimit of exact matching

Intuition:

The potential functions depend only on the relative distances.Since a k-tree is sufficient to encode all distances, it is alsosufficient to encode all potential functions

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 49: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

The Model

The Model

The final model looks like... (for R2)

T−1

X X X

XXX

1 2 3

4 5 TX

Optimality

It is optimal in the limit of exact matching because it isequivalent to the full model

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 50: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

The Model

The Model

The final model looks like... (for R2)

T−1

X X X

XXX

1 2 3

4 5 TX

Optimality

It is optimal in the limit of exact matching because it isequivalent to the full model

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 51: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Junction Tree Famework

Junction Tree Framework

The Junction Tree Framework provides algorithms for exact MAPcomputation in graphical models, whose complexity is exponentialonly on the size of the maximal clique

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 52: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Junction Tree Famework

Junction Tree Framework

The Junction Tree Framework provides algorithms for exact MAPcomputation in graphical models, whose complexity is exponentialonly on the size of the maximal clique

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 53: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Optimization

Optimize using JT algorithm (Hugin)

1 Construct hypergraph (a “Junction Tree”) where nodes aremaximal cliques of original graph

2 “Pass messages” in a systematic way in this hypergraph, whatmeans updating the clique potentials according to specificrules

3 After messages have been passed, the nodes contain the exactMAP estimate for the model

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 54: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Optimization

Optimize using JT algorithm (Hugin)

1 Construct hypergraph (a “Junction Tree”) where nodes aremaximal cliques of original graph

2 “Pass messages” in a systematic way in this hypergraph, whatmeans updating the clique potentials according to specificrules

3 After messages have been passed, the nodes contain the exactMAP estimate for the model

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 55: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Optimization

3X 4X 3X1 X2 X 5X 3X1 X2

X 3X1 X2

X TX3X1 X2

X3X1 X2

XXX1 X2 T−1

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 56: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Optimization

3X 4X 3X1 X2 X 5X 3X1 X2

X 3X1 X2

X TX3X1 X2

X3X1 X2

XXX1 X2 T−1

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 57: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Optimization

3X 4X 3X1 X2 X 5X 3X1 X2

X 3X1 X2

X TX3X1 X2

X3X1 X2

XXX1 X2 T−1

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 58: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Optimization

Messages

V → W:φ∗S = max

V \SψV

ψ∗W =φ∗SφSψW

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 59: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Graphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

Computational Complexity

Complexity

The overall complexity for matching tasks in Rk−1 is

O(TSk+1)

T → size of the template patternS → size of the scene pattern

k+1 → size of the maximal clique

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 60: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

Outline

1 The ProblemProblem DefinitionProperties of the FormulationComputational Complexity

2 The SolutionGraphical ModelsThe Problem as Inference in a Graphical ModelGraphical Model construction and inference

3 ExperimentsExperimental SetupInexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 61: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

Experimental Setup

Experimental Setup

We compare this approach (JT) with standard ProbabilisticRelaxation Labeling (PRL)

Matching point sets in R2

Experiments with inexact matching (for exact matching italways yields perfect results)

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 62: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

Inexact matching: varying problem size

15 20 25 30 35 40 45 500.4

0.5

0.6

0.7

0.8

0.9

1

Number of points in the codomain pattern (S)

Frac

tion

of c

orre

ct c

orre

spon

denc

eT = 10, std = 2 pixels

JTPRL

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 63: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

Inexact matching: varying noise

1 2 3 4 5 6 7 8 90.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1T = 10, S = 30

Position jitter (std, in pixels)

Frac

tion

of c

orre

ct c

orre

spon

denc

esJTPRL

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 64: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

PRL x JT

PRL

Locally optimal

Iterative

JT

Globally optimal

Non-iterative (two-pass)

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 65: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

PRL x JT

PRL

Locally optimal

Iterative

JT

Globally optimal

Non-iterative (two-pass)

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 66: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Experimental SetupInexact matching

PRL x JT

PRL

Locally optimal

Iterative

JT

Globally optimal

Non-iterative (two-pass)

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 67: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Summary

Summary

Point set matching is formulated as inference in a Graphical Model

Summary

Optimal inference is performed in polynomial time in a sparsemodel which is equivalent to the full model

Summary

The experimental results are perfect for exact matching andsignificantly better than PRL for inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 68: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Summary

Summary

Point set matching is formulated as inference in a Graphical Model

Summary

Optimal inference is performed in polynomial time in a sparsemodel which is equivalent to the full model

Summary

The experimental results are perfect for exact matching andsignificantly better than PRL for inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 69: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Summary

Summary

Point set matching is formulated as inference in a Graphical Model

Summary

Optimal inference is performed in polynomial time in a sparsemodel which is equivalent to the full model

Summary

The experimental results are perfect for exact matching andsignificantly better than PRL for inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 70: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Summary

Summary

Point set matching is formulated as inference in a Graphical Model

Summary

Optimal inference is performed in polynomial time in a sparsemodel which is equivalent to the full model

Summary

The experimental results are perfect for exact matching andsignificantly better than PRL for inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 71: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Main Lesson

Redundancies

What makes search in this problem NP-hard is redundantinformation

It can be solved optimally in polynomial time if we properlytake advantage of this redundancy

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 72: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Main Lesson

Redundancies

What makes search in this problem NP-hard is redundantinformation

It can be solved optimally in polynomial time if we properlytake advantage of this redundancy

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 73: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

To do...

Comparison

Compare with other techniques (spectral methods, continuousoptimization, etc.)

Invariance

Extend to more complex invariances

Error

Bounds for error in inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 74: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

To do...

Comparison

Compare with other techniques (spectral methods, continuousoptimization, etc.)

Invariance

Extend to more complex invariances

Error

Bounds for error in inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 75: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

To do...

Comparison

Compare with other techniques (spectral methods, continuousoptimization, etc.)

Invariance

Extend to more complex invariances

Error

Bounds for error in inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 76: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

To do...

Comparison

Compare with other techniques (spectral methods, continuousoptimization, etc.)

Invariance

Extend to more complex invariances

Error

Bounds for error in inexact matching

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 77: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Related Results

CVPR 2004

Attributed Graph Matching (n-sized cliques)

ICPR 2004

Different types of similarity functions

Tiberio’s Thesis, 2004

The theoretical development, proofs and numerous experimentalresults are available

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 78: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Related Results

CVPR 2004

Attributed Graph Matching (n-sized cliques)

ICPR 2004

Different types of similarity functions

Tiberio’s Thesis, 2004

The theoretical development, proofs and numerous experimentalresults are available

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 79: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Related Results

CVPR 2004

Attributed Graph Matching (n-sized cliques)

ICPR 2004

Different types of similarity functions

Tiberio’s Thesis, 2004

The theoretical development, proofs and numerous experimentalresults are available

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 80: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

Related Results

CVPR 2004

Attributed Graph Matching (n-sized cliques)

ICPR 2004

Different types of similarity functions

Tiberio’s Thesis, 2004

The theoretical development, proofs and numerous experimentalresults are available

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching

Page 81: Mlss05au Caelli Gma 01

The ProblemThe SolutionExperiments

Summary

More Info

More information (papers and Thesis)

www.cs.ualberta.ca/˜tcaetano

Tiberio Caetano, Terry Caelli and Dante Barone An optimal probabilistic graphical model for point set matching