1. Outline Graph matching Definition Manifold application areas
Exact matching
Different forms of exact matching
Necessary concepts
Algorithms for exact matching
Techniques based on tree search
Other techniques
Conclusion
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2. 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]
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3. 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
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4. 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)
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5. 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 structure- preserving 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.
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6. 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
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7. 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.
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8. 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.
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9. 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.
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10. 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.
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11. 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.
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12. 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
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18. 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.
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19. 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.
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20. 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. If , then M’ specifies an isomorphism between G and the subgraph of H.
[J.R.Ullmann.An algorithm for subgraph isomorphism.]
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21. 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
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22. Techniques based on tree search
Ullmann’s algorithm
Construction of another matrix M(0) with the same size of M’
Generation of all M’ by setting all but one of each row of M(0)
A subgraph isomorphism has been found if
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23. Techniques based on tree search
Ullmann’s algorithm
Example
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24. 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).
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25. 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.
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26. 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.
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27. 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.
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61. Statistical Parameter Estimation Reminder
Data set
Model
Parameters: Θ
The basic paradigm:
MLE / bayesian approach
Input data: series of observations X1, X2 … Xt
-We assumed observations were i.i.d (independent identical distributed)
Heads - P(H)
Tails P(H) .

62. Markov Process
• Markov Property: The state of the system at time t+1 depends only on the state of the system at time t X1 X2 X3 X4 X5
• Stationary Assumption: Transition probabilities are independent of time (t)
Bounded memory transition model

63. Markov Process Simple Example
Weather:
raining today 40% rain tomorrow
60% no rain tomorrow
not raining today 20% rain tomorrow
80% no rain tomorrow
Stochastic FSM:
rain
no rain
0.6
0.4
0.8
0.2

64. Markov Process Simple Example
Weather:
raining today 40% rain tomorrow
60% no rain tomorrow
not raining today 20% rain tomorrow
80% no rain tomorrow
The transition matrix:
Stochastic matrix:
Rows sum up to 1
Double stochastic matrix:
Rows and columns sum up to 1

65. Markov Process Gambler’s Example
– Gambler starts with $10
- At each play we have one of the following:
• Gambler wins $1 with probability p
• Gambler looses $1 with probability 1-p
– Game ends when gambler goes broke, or gains a fortune of $100
(Both 0 and 100 are absorbing states) 1 2 99
100 p
1-p
Start (10$)

66. Markov Process Markov process - described by a stochastic FSM 1 2 99
Markov chain - a random walk on this graph
(distribution over paths)
Edge-weights give us
We can ask more complex questions, like 1 2 99
100 p
1-p
Start (10$)

67. Markov Process Coke vs. Pepsi Example
Given that a person’s last cola purchase was Coke, there is a 90% chance that his next cola purchase will also be Coke.
If a person’s last cola purchase was Pepsi, there is an 80% chance that his next cola purchase will also be Pepsi.
transition matrix:
coke
pepsi
0.1
0.9
0.8
0.2

68. Markov Process Coke vs. Pepsi Example (cont)
Given that a person is currently a Pepsi purchaser, what is the probability that he will purchase Coke two purchases from now?
Pr[ Pepsi?Coke ] =
Pr[ PepsiCokeCoke ] + Pr[ Pepsi Pepsi Coke ] =
0.2 * * = 0.34
Pepsi  ?
?  Coke

69. Markov Process Coke vs. Pepsi Example (cont)
Given that a person is currently a Coke purchaser, what is the probability that he will purchase Pepsi three purchases from now?

70. Markov Process Coke vs. Pepsi Example (cont)
Assume each person makes one cola purchase per week
Suppose 60% of all people now drink Coke, and 40% drink Pepsi
What fraction of people will be drinking Coke three weeks from now?
Pr[X3=Coke] = 0.6 * * =
Qi - the distribution in week i
Q0=(0.6,0.4) - initial distribution
Q3= Q0 * P3 =(0.6438,0.3562)

71. Markov Process Coke vs. Pepsi Example (cont)
Simulation:
2/3
stationary distribution
Pr[Xi = Coke]
coke
pepsi
0.1
0.9
0.8
0.2
week - i

72. Hidden Markov Models - HMM
Hidden states H1 H2
HL-1 HL Hi X1 X2 Xi
XL-1 XL
Observed data

73. Hidden Markov Models - HMM Coin-Tossing Example
0.9
0.9
transition probabilities
0.1
fair
loaded
0.1
emission probabilities
1/2
1/2
3/4
1/4 H T H T
Fair/Loaded
Head/Tail X1 X2
XL-1 XL Xi H1 H2
HL-1 HL Hi

74. Hidden Markov Models - HMM C-G Islands Example
C-G islands: Genome regions which are very rich in C and G
q/4
q/4 P A G q
Regular
DNA P q
change P q P q
q/4 C T
q/4
p/3
p/6
(1-P)/4 A G
C-G island
(1-q)/6
(1-q)/3
p/3
P/6 C T

75. Hidden Markov Models - HMM C-G Islands ExampleT
change
C-G / Regular
{A,C,G,T} X1 X2
XL-1 XL Xi H1 H2
HL-1 HL Hi
To be continued…

90. Encoding of cone signals in the human visual system
The 3 "color-separation images are not transmitted directly to the brain
Neurons of the retina encode the color into opponent signals
Red-green signal = L-M+S
Yellow-blue = L+M-S
Luminance = L+M+S

96. -----> Bayes’ theorem
p(T|C)*p(C)
p(C|T) = ______________________
P(T|C)*p(C) + p(T|~C)*p(~C)
A C

97. 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?

98. -----> 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.

99. Posterior  likelihood x prior
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).

100. Posterior Probability Distribution
precision  = 1/2
Likelihood: p(y|) = N(Md, ld-1)
Prior: p() = N(Mp, lp-1)
Posterior: p(|y) ∝ p(y|)* p() = N(Mpost, lpost-1)
lpost = ld + lp Mpost = ld Md + lp Mp
lpost Mp
lp-1
Mpost
lpost-1
ld-1 Md

101. 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?

102. 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)

103. SPMs and PPMs PPMs: Show activations of a given size
SPMs: show voxels with
non-zero activations

104. PPMs Disadvantages Advantages Computationally
demanding (priors are determined empirically)
Utility of Bayesian
approach is yet
to be established
One can infer a cause
DID NOT elicit a response
SPMs conflate effect-size
and effect-variability

105. 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.

106. 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.

107. 5. Decision making Frequentist: historically avoided.
Bayesian: tailored to decision analysis; losses and gains considered explicitly.

108. References
Statistics: A Bayesian Perspective D. Berry, 1996, Duxbury Press.
excellent introductory textbook, if you really want to understand what it’s all about.
“Using a Bayesian Approach in Medical Device Development”, also by Berry
a powerpoint presentation by Berry
Extremely clear presentation of the mammography example; highly polemical and fun too!
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.
A physicist’s take on Bayesian approaches. Proposes an interesting metric of probability using decibels (yes, the unit used for sound levels!).
a skeptical account of Bayesian approaches. The rest of the site is very informative and sensible about basic statistical issues.

109. Bayes’ ending
Bunhill Fields Burial Ground
off City Road, EC1