4. Structural Patten Recognition
Instead of describing an object in terms of a feature vector, describe it by its structure
Describe complicated objects in terms of simple primitives and their relationship
Scene N L X
Object
Background Y T M Z D E M N D E L T X Y Z

8. Here is Robert Ledley’s example of the generative grammar for the two classes of chromosomes: submedian (S) and telocentric (T). A bottom-up parsing process is also illustrated by the tree.

33. 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
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 33

34. 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.
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 34

35. 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.
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 35

36. 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.
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 36

37. 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.
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 37

38. 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.
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 38

39. 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
In meinem Vortrag werde ich zunächst....
Zheng Zhang, Exact Matching 39