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