Outline
• Graph matching
– Definition
– Manifold application areas
• Exact matching
– Definition
– Different forms of exact matching
– Necessary concepts
• Algorithms for exact matching
– Techniques based on tree search
– Other techniques
• Conclusion
Zheng Zhang, 27.05.2010
Exact Matching
slide 1
Graph matching
• Definition:
– Graph matching is the process of finding a correspondence between the
nodes and the edges of two graphs that satisfies some (more or less
stringent) constraints ensuring that similar substructures in one graph
are mapped to similar substructures in the other.
[D. Conte, Pasquale Foggia, Carlo Sansone, and Mario Vento. Thirty
Years of Graph Matching in Pattern Recognition.]
• in comparison with matching (graph theory):
– A matching or independent edge set in a graph is a set of edges without
common vertices. It may also be an entire graph consisting of edges
without common vertices.
[Wikipedia]
Zheng Zhang, 27.05.2010
Exact Matching
slide 2
Graph matching
• Manifold application areas
– 2D and 3D image analysis
– Document processing
– Biometric identification
– Image databases
– Video analysis
– Biological and biomedical applications
• Categories of matching
– Exact matching
– Inexact matching
– Other matching problems
Zheng Zhang, 27.05.2010
Exact Matching
slide 3
Exact matching
• Definition
– Exact graph matching is characterized by the fact that the mapping
between the nodes of the two graphs must be edge-preserving in
the sense that if two nodes in the first graph are linked by an edge,
they are mapped to two nodes in the second graph that are linked
by an edge as well.
• Different forms of exact matching
– The most stringent form: graph isomorphism
– A weaker form: subgraph isomorphism
– A slightly weaker form: monomorphism
– A still weaker form: homomorphism
– Another interesting form: maximum common subgraph (MCS)
Zheng Zhang, 27.05.2010
Exact Matching
slide 4
Necessary concepts
• Morphism
– A morphism is an abstraction derived from structure-preserving
mappings between two mathematical structures.
– In comparison with homomorphism: a homomorphism is a structurepreserving map between two mathematical structures.
– If a morphism f has domain X and codomain Y, we write f :X→Y. Thus
a morphism is represented by an arrow from its domain to its
codomain. The collection of all morphisms from X to Y is denoted
Hom(X,Y) or Mor(X,Y) .
• Isomorphism
– An isomorphism is a bijective homomorphism.
– An isomorphism is a morphism f :X→Y in a category for which there
exists an "inverse" f−1 :Y→X, with the property that both f−1·f = idX
and f·f−1 =idY.
Zheng Zhang, 27.05.2010
Exact Matching
slide 5
Necessary concepts
• Epimorphism
– an surjective homomorphism
• Monomorphism
– an injective homomorphism
• Endomoprphism
– a homomorphism from an object
to itself
• Automoprphism
– an endomorphism which is also
an isomorphism
an isomorphism with itself
Zheng Zhang, 27.05.2010
Exact Matching
Homo
Mono
Iso
Epi
Auto
Endo
slide 6
Different forms of exact matching
• Graph isomorphism
– A one-to-one correspondence must be found between each node of
the first graph and each node of the second graph.
– Graphs G(VG,EG) and H(VH,EH) are isomorphic if there is an invertible
F from VG to VH such that for all nodes u and v in VG, (u,v)∈EG if and
only if (F(u),F(v)) ∈EH.
C
1
1
5
4
5
A
4
3
2
3
2
Zheng Zhang, 27.05.2010
D
Exact Matching
B
E
slide 7
Different forms of exact matching
• Subgraph isomorphism
– It requires that an isomorphism holds between one of the two
graphs and a node-induced subgraph of the other.
• Monomorphism
– It requires that each node of the first graph is mapped to a distinct
node of the second one, and each edge of the first graph has a
corresponding edge in the second one; the second graph, however,
may have both extra nodes and extra edges.
C
3
D
A
1
2
Zheng Zhang, 27.05.2010
B
Exact Matching
E
slide 8
Different forms of exact matching
• Homomorphism
– It drops the condition that nodes in the first graph are to be mapped
to distinct nodes of the other; hence, the correspondence can be
many-to-one.
– A graph homomorphism F from Graph G(VG,EG) and H(VH,EH), is a
mapping F from VG to VH such that {x,y} ∈EG implies {F(x),F(y)} ∈EH.
C
3
D
1
4
2
Zheng Zhang, 27.05.2010
A
B
Exact Matching
E
slide 9
Different forms of exact matching
• Maximum common subgraph (MCS)
– A subgraph of the first graph is mapped to an isomorphic subgraph
of the second one.
– There are two possible definitions of the problem, depending on
whether node-induced subgraphs or plain subgraphs are used.
– The problem of finding the MCS of two graphs can be reduced to the
problem of finding the maximum clique (i.e. a fully connected
subgraph) in a suitably defined association graph.
C
3
D
4
1
2
Zheng Zhang, 27.05.2010
A
B
Exact Matching
E
slide 10
Different forms of exact matching
• Properties
– The matching problems are all NP-complete except for graph
isomorphism, which has not yet been demonstrated whether in NP
or not.
– Exact graph matching has exponential time complexity in the worst
case. However, in many PR applications the actual computation time
can be still acceptable.
– Exact isomorphism is very seldom used in PR. Subgraph
isomorphism and monomorphism can be effectively used in many
contexts.
– The MCS problem is receiving much attention.
Zheng Zhang, 27.05.2010
Exact Matching
slide 11
Algorithms for exact matching
• Techniques based on tree search
– mostly based on some form of tree search with backtracking
Ullmann’s algorithm
Ghahraman’s algorithm
VF and VF2 algorithm
Bron and Kerbosh’s algorithm
Other algorithms for the MCS problem
• Other techniques
– based on A* algorithm
Demko’s algorithm
– based on group theory
Nauty algorithm
Zheng Zhang, 27.05.2010
Exact Matching
slide 12
Techniques based on tree search
• The basic idea
– A partial match (initially empty) is iteratively expanded by adding to
it new pairs of matched nodes.
– The pair is chosen using some necessary conditions, usually also
some heuristic condition to prune unfruitful search paths.
– Eventually, either the algorithm finds a complete matching, or no
further vertex pairs may be added (backtracking)
– For PR the algorithm may consider the attributes of nodes and
edges in constraining the desired matching.
Zheng Zhang, 27.05.2010
Exact Matching
slide 18
Techniques based on tree search
• The backtracking algorithm
– depth-first search(DFS):
it progresses by expanding the first child node of the search tree that
appears and thus going deeper and deeper until a goal node is found,
or until it hits a node that has no children.
– Branch and bound(B&B):
it is a BFS(breadth-first search)-like search for the optimal solution.
Branch is that a set of solution candidates is splitted into two or more
smaller sets; bound is that a procedure upper and lower bounds.
1
1
2
3
4
6
5
Zheng Zhang, 27.05.2010
7
2
8
9
12
10
11
Exact Matching
5
9
10
6
3
4
7
8
11
12
slide 19
Techniques based on tree search
• Ullmann’s algorithm
– Probably the most popular graph matching algorithm
– Application for graph isomorphism, subgraph isomorphism and
monomorphism, also for MCS problem
– A refinement procedure based on matrix of possible future matched
node pairs to prune unfruitful matches
– The simple enumeration algorithm for the isomorphisms between a
graph G and a subgraph of another graph H with the adjacency matrices
AG and AH
An M’ matrix with |VG| rows and |VH| columns can be used to permute
the rows and columns of AH to produce a further matrix P.
1) M’ specifies an isomorphism between G and
If (aG i , j  1)  ( pi , j ,then
the subgraph of H.
P  M ' ( M ' A )T
H
[J.R.Ullmann.An algorithm for subgraph isomorphism.]
Zheng Zhang, 27.05.2010
Exact Matching
slide 20
Techniques based on tree search
• Ullmann’s algorithm
– Example for permutation matrix
– The elements of M’ are 1’s and 0’s, such that each row contains 1 and
each column contains 0 or 1
P  M ' ( M ' AH )T
1
H
2
4
3
1 0 0
 0 0 1
0 1 0
1
3
4
2
1 0 0
 0 0 1
0 1 0
Zheng Zhang, 27.05.2010

0
0  1 0 0 0 
1
0   0 0 1 0.

0
0  0 1 0 0 

0
0 0 1 
0 
0 0

1 1 0 
0.
 0 0
0 0 1  
0 
 1 1
0
0
1


Exact Matching
1
0
1
1
0
1
0
0
0 

1 
0 

0 
T
1
1
0
slide 21
Techniques based on tree search
• Ullmann’s algorithm
– Construction of another matrix M(0) with the same size of M’
( 0)
i, j
m
1 if deg(VHi )  deg(VGi )

, mi , j {0,1}
0
otherweise

– Generation of all M’ by setting all but one of each row of M(0)
(aG i , j  1)  ( pi , j  1)
– A subgraph isomorphism has been found if
1
H
2
4
3
1
G
3
2
0
1
AH  
0

0
1
0
1
1
0
1
0
0
0 0 1 
AG  0 0 1
1 1 0
Zheng Zhang, 27.05.2010
Exact Matching
0
1
0

0
1 1 1 1
M 0  1 1 1 1
0 1 0 0
slide 22
Techniques based on tree search
1 1 1 1 
1 1 1 1 


0 1 0 0
• Ullmann’s algorithm
– Example
1 0 0 0
1 1 1 1


0 1 0 0
1 0 0 0
0 0 1 0 


0 1 0 0
4
1
3
1 0 0 0
0 0 0 1 


0 1 0 0
4
2
3
2
0 0 0 1 
1 1 1 1


0 1 0 0
0 0 0 1 
1 0 0 0


0 1 0 0
1
2
1
1
0 0 0 1 
0 0 1 0 


0 1 0 0
1
1
3
3
3
3
0 0 1 0 
0 0 0 1 


0 1 0 0
1
4
2
2
1
0 0 1 0 
1 1 1 1


0 1 0 0
0 0 1 0 
1 0 0 0


0 1 0 0
3
2
1
0 1 0 0 
1 1 1 1


0 1 0 0
3
3
2
P  M ' ( M ' AH )T
0 0 1 
 0 0 1
1 1 0
Zheng Zhang, 27.05.2010
0 0 1 
compared with AG  0 0 1
1 1 0
Exact Matching
1
3
2
slide 23
Techniques based on tree search
• Ghahraman’s algorithm
– Application for monomorphism
– Use of the netgraph obtained from the Cartesian product of the nodes
of two graph, monomorphisms correspond to particular subgraphs of
the netgraph
– A strong necessary condition and a weak one, then two versions of the
algorithm to detect unfruitful partial solutions
• VF and VF2 algorithm
– Application for isomorphism and subgraph isomorphism
– VF algorithm defines a heuristic based on the analysis of the sets of
nodes adjacent to the ones already considered in the partial mapping.
– VF2 algorithm reduces the memory requirement from O(N2) to O(N).
Zheng Zhang, 27.05.2010
Exact Matching
slide 24
Techniques based on tree search
• Bron and Kerbosh’s algorithm
– Application for the clique detection and the MCS problem
– Based on the use of a heuristic for pruning the search tree
– Simplicity and an acceptable performance in most cases
• Other algorithms for the MCS problem
– Balas and Yu’s alogrithm also defines a heristic, but based on graph
colouring techniques.
– McGregor’s alogrithm is not applicated for a maximum clique problem.
– Koch’s alogrithm is applicated for a slightly simplified version of the
MCS problem and suggests the use of the Bron and Kerbosh’s
algorithm.
Zheng Zhang, 27.05.2010
Exact Matching
slide 25
Other techniques
• Based on the A* algorithm
– It uses a distance-plus-cost heuristic function to determine the order
in which the search visits nodes in the tree. The heuristic is a sum of
two functions:
the path-cost function, i.e. the cost from the starting node to the
current node, and an admissible “heuristic estimate” of the distance
to the goal.
– Demko’s algorithm investigates a generalization of MCS to hypergraphs.
• Based on group theory
– McKay’s Nauty(No automorphisms yes?) algorithm deals only with the
isomorphism problem.
It constructs the automorphism group of each of the input graphs and
derives a canonical labeling. The isomorphism can be checked by
verifying the equality of the adjacency matrices.
Zheng Zhang, 27.05.2010
Exact Matching
slide 26
Conclusion
• The exact graph matching problem is of interest in a variety of
different pattern recognition contexts.
• Of the exact graph matching problems, exact isomorphism is very
seldom used in PR while subgraph isomorphism, monomorphism and
the MCS problem are popularly proposed.
• Ullmann’ algorithm, VF2 algorithm and Nauty algorithm are mostly
used algorithms, which based on the search methods, and may
outperform others. Most modified algorithms adopt some conditions
to prune the unfruitful partial matching.
Zheng Zhang, 27.05.2010
Exact Matching
slide 27
Bayes for Beginners
Reverend Thomas Bayes (1702-61)
Overview of SPM Analysis
Design matrix
fMRI time-series
Motion
Correction
Smoothing
Statistical Parametric Map
General Linear Model
Spatial
Normalisation
Anatomical Reference
Parameter Estimates
Spatial Normalisation - Overfitting
Without
Bayesian
constraints,
Template
the nonimage
linear spatial
normalisation
can introduce
unnecessary
warps.
Non-linear
registration
using
Bayes.
(2 = 302.7)
Affine
registration.
(2 = 472.1)
Non-linear
registration
without
Bayes
constraints.
(2 = 287.3)
Bayes and image segmentation
• Want to automatically classify regions into grey matter, white matter, CSF and
non-brain tissue.
• How do we use prior information? (probabilities of each voxel being of a
particular type derived from a database of 152 scans)
•Bayes!
A big advantage of a Bayesian approach
• Allows a principled approach to the
exploitation of all available data …
• with an emphasis on continually updating
one’s models as data accumulate
• as seen in the consideration of what is learned
from a positive mammogram
Bayesian Reasoning
-
Casscells, Schoenberger & Grayboys, 1978
Eddy, 1982
Gigerenzer & Hoffrage, 1995, 1999
Butterworth, 2001
Hoffrage, Lindsey, Hertwig & Gigerenzer, 2001
When PREVALENCE, SENSITIVITY, and FALSE POSITIVE
RATES are given, most physicians misinterpret the information
in a way that could be potentially disastrous for the patient.
Bayesian Reasoning
ASSUMPTIONS
1% of women aged forty who participate in a routine
screening have breast cancer
80% of women with breast cancer will get positive
tests
9.6% of women without breast cancer will also get
positive tests
EVIDENCE
A woman in this age group had a positive test in a
routine screening
PROBLEM
What’s the probability that she has breast cancer?
Bayesian Reasoning
ASSUMPTIONS
10 out of 1000 women aged forty who participate in a
routine screening have breast cancer
800 out of 1000 of women with breast cancer will get
positive tests
95 out of 1000 women without breast cancer will also
get positive tests
PROBLEM
If 1000 women in this age group undergo a routine
screening, about what fraction of women with positive
tests will actually have breast cancer?
Bayesian Reasoning
ASSUMPTIONS
100 out of 10,000 women aged forty who participate
in a routine screening have breast cancer
80 of every 100 women with breast cancer will get
positive tests
950 out of 9,900 women without breast cancer will
also get positive tests
PROBLEM
If 10,000 women in this age group undergo a routine
screening, about what fraction of women with positive
tests will actually have breast cancer?
Bayesian Reasoning
Before the screening:
100 women with breast cancer
9,900 women without breast cancer
After the screening:
A = 80 women with breast cancer and positive test
B = 20 women with breast cancer and negative test
C = 950 women without breast cancer and positive test
D = 8,950 women without breast cancer and negative test
Proportion of cancer patients with positive results, within
the group of ALL patients with positive results:
A/(A+C) = 80/(80+950) = 80/1030 = 0.078 = 7.8%
Compact Formulation
C = cancer present, T = positive test
p(A|B) = probability of A, given B, ~ = not
PRIOR PROBABILITY
p(C) = 1%
PRIORS
CONDITIONAL PROBABILITIES
p(T|C) = 80%
p(T|~C) = 9.6%
POSTERIOR PROBABILITY (or REVISED PROBABILITY)
p(C|T) = ?
Bayesian Reasoning
Before the screening:
100 women with breast cancer
9,900 women without breast cancer
After the screening:
A = 80 women with breast cancer and positive test
B = 20 women with breast cancer and negative test
C = 950 women without breast cancer and positive test
D = 8,950 women without breast cancer and negative test
Proportion of cancer patients with positive results, within
the group of ALL patients with positive results:
A/(A+C) = 80/(80+950) = 80/1030 = 0.078 = 7.8%
Bayesian Reasoning
Prior Probabilities:
100/10,000 = 1/100 = 1% = p(C)
9,900/10,000 = 99/100 = 99% = p(~C)
Conditional Probabilities:
A = 80/10,000 = (80/100)*(1/100) = p(T|C)*p(C) = 0.008
B = 20/10,000 = (20/100)*(1/100) = p(~T|C)*p(C) = 0.002
C = 950/10,000 = (9.6/100)*(99/100) = p(T|~C)*p(~C) = 0.095
D = 8,950/10,000 = (90.4/100)*(99/100) = p(~T|~C) *p(~C) = 0.895
Rate of cancer patients with positive results, within the
group of ALL patients with positive results:
A/(A+C) = 0.008/(0.008+0.095) = 0.008/0.103 = 0.078 = 7.8%
-----> Bayes’ theorem
A
p(T|C)*p(C)
p(C|T) = ______________________
P(T|C)*p(C) + p(T|~C)*p(~C)
A + C
Comments
• Common mistake: to ignore the prior probability
• The conditional probability slides the revised
probability in its direction but doesn’t replace the
prior probability
• A NATURAL FREQUENCIES presentation is one in
which the information about the prior probability is
embedded in the conditional probabilities (the
proportion of people using Bayesian reasoning rises to
around half).
• Test sensitivity issue (or: “if two conditional
probabilities are equal, the revised probability equals
the prior probability”)
• Where do the priors come from?
-----> Bayes’ theorem
p(X|A)*p(A)
p(A|X) = ______________________
P(X|A)*p(A) + p(X|~A)*p(~A)
Given some phenomenon A that we want to investigate, and an
observation X that is evidence about A, we can update the
original probability of A, given the new evidence X.
Bayes’ Theorem for a given parameter 
p (data) = p (data) p () / p (data)
1/P (data) is basically
a normalizing constant
Posterior  likelihood x prior
The prior is the probability of the parameter and represents what was
thought before seeing the data.
The likelihood is the probability of the data given the parameter and
represents the data now available.
The posterior represents what is thought given both prior information
and the data just seen.
It relates the conditional density of a parameter (posterior probability)
with its unconditional density (prior, since depends on information
present before the experiment).
Posterior Probability Distribution
precision l = 1/2
p(y|) = N(Md, ld-1)
p() = N(Mp, lp-1)
p(|y) ∝ p(y|)* p() = N(Mpost, lpost-1)
Likelihood:
Prior:
Posterior:
lpost = ld + lp
Mpost = ld Md + lp Mp
lpost
lpost-1
ld-1
lp-1
Mp
Mpost Md

Activations in fMRI….
• Classical
– ‘What is the likelihood of getting these data given
no activation occurred?’
• Bayesian option (SPM2)
– ‘What is the chance of getting these parameters, given these
data?
What use is Bayes in deciding what brain
regions are active in a particular study?
• Problems with classical frequentist approach
– All inferences relate to disproving the null hypothesis
– Never fully reject H0, only say that the effect you see is
unlikely to occur by chance
– Corrections for multiple comparisons
• significance depends on the number of contrasts you look at
– Very small effects can be declared significant with enough
data
• Bayesian Inference offers a solution through
Posterior Probability Maps (PPMs)
SPMs and PPMs
rest [2.06]
rest
contrast(s)
<
PPM 2.06
SPMresults: C:\home\spm\analysis_PET
Height threshold P = 0.95
Extent threshold k = 0 voxels
SPMmip
[0, 0, 0]
<
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
60
<
SPM{T39.0}
SPMresults: C:\home\spm\analysis_PET
Height threshold T = 5.50
Extent threshold k = 0 voxels
1 4 7 10 13 16 19 22
Design matrix
PPMs: Show activations
of a given size
3
<
4
<
SPMmip
[0, 0, 0]
<
contrast(s)
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
60
1 4 7 10 13 16 19 22
Design matrix
SPMs: show voxels with
non-zero activations
PPMs
Advantages
One can infer a cause
DID NOT elicit a response
SPMs conflate effect-size
and effect-variability
Disadvantages
Computationally
demanding (priors are
determined empirically)
Utility of Bayesian
approach is yet
to be established
Frequentist vs. Bayesian by Berry
1. Probabilities of data vs. probabilities of parameters (&
also data).
2. Evidence used:
– Frequentist measures specific to experiment.
– Posterior distribution depends on all available
information.
Makes Bayesian approach appealing, but assembling,
assessing, & quantifying information is work.
3. Depend on probabilities of results that could occur vs. did
occur:
– Frequentist measures (e.g., p values, confidence intervals)
incorporate probabilities of data that were possible but did
not occur.
– Posterior depends on data only through the likelihood,
which is calculated from observed data.
4. Flexibility:
– Frequentist measures depend on design; require that
design be followed.
– Bayesian view: update continually as data accumulate
(only requirement is honesty). Sample size: need not
choose in advance. Weigh costs/benefits; decide whether
to start experiment. After experiment starts, decide
whether to continue—stop at any time, for any reason.
5. Decision making
– Frequentist: historically avoided.
– Bayesian: tailored to decision analysis; losses
and gains considered explicitly.
References
•
Statistics: A Bayesian Perspective D. Berry, 1996, Duxbury Press.
– excellent introductory textbook, if you really want to understand what it’s all about.
•
http://ftp.isds.duke.edu/WorkingPapers/97-21.ps
– “Using a Bayesian Approach in Medical Device Development”, also by Berry
•
http://www.pnl.gov/bayesian/Berry/
– a powerpoint presentation by Berry
•
http://yudkowsky.net/bayes/bayes.html
– Extremely clear presentation of the mammography example; highly polemical and fun
too!
•
http://www.stat.ucla.edu/history/essay.pdf
– Bayes’ original essay
•
Jaynes, E. T., 1956, `Probability Theory in Science and Engineering,' (No. 4 in
`Colloquium Lectures in Pure and Applied Science,' Socony-Mobil Oil Co. USA.
http://bayes.wustl.edu/etj/articles/mobil.pdf
– A physicist’s take on Bayesian approaches. Proposes an interesting metric of probability
using decibels (yes, the unit used for sound levels!).
•
http://www.sportsci.org/resource/stats/
– a skeptical account of Bayesian approaches. The rest of the site is very informative and
sensible about basic statistical issues.
Bayes’ ending
Bunhill Fields Burial Ground
off City Road, EC1