Introduction to Graphs and their Ability to Represent Images

Martin Cerman

•  Graph G = (V,E) •  Set of vertices V

•  Set of edges E, where E ⊂  V x V

•  Edges can be directed or undirected

Graph Definition

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V = {1, 2, 3, 4, 5} E = {(1,2), (2,3), (3,2), (4,3), (4,1), (4,5), (5,5)}

1  2  

3  

4  

5  

1

1

=  

•  Parallel / Double edges

•  Self-loops

•  Simple graph: contains NO parallel edges and NO self-loops

•  Multigraph: contains parallel edges and/or self-loops

Graph Definition

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•  Planar graphs •  Contains no crossing edges

•  Tree •  Connects all vertices without cycles

•  Forest •  All connected components are trees

Graph Definition

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•  Vertices generally stored in array or vector

•  Most common data-structures to store edges •  Adjacency list

•  Adjacency matrix

•  Incidence matrix

•  Edge data-structure defines complexity of graph operations •  Storage- and computational cost

•  Operations for example: add edge, remove edge, check adjacency,…

Graph Data-structure

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•  Collection of unordered linked lists, one for each vertex

•  Great for representation of sparse graphs

Adjacency List

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1  2  

3  

4  

5  

1 2 3 4 5

2 3 2 1 5

4 3

5

Graph  opera*on   Complexity  

Store  graph   O(|V|+|E|)  

Add  Vertex   O(1)  

Add  Edge   O(1)  

Remove  Vertex   O(|E|)  

Remove  Edge   O(|E|)  

Adjacency  check   O(|V|)  

V

•  Square matrix with side length |V|

•  Describes if two vertices are adjacent

•  Symmetric if undirected graph, binary if simple graph

Adjacency Matrix

7  

1  2  

3  

4  

5  

Graph  opera*on   Complexity  

Store  graph   O(|V|²)  

Add  Vertex   O(|V|²)  

Add  Edge   O(1)  

Remove  Vertex   O(|V|²)  

Remove  Edge   O(1)  

Adjacency  check   O(1)  

1  

1  

1   1  

1   1   1  

1  

V

V

•  Matrix of size |V| x |E|

•  Describes incidence relationship between vertices and edges

Incidence Matrix

8  

1  2  

3  

4  

5  

Graph  opera*on   Complexity  

Store  graph   O(|V|*|E|)  

Add  Vertex   O(|V|*|E|)  

Add  Edge   O(|V|*|E|)  

Remove  Vertex   O(|V|*|E|)  

Remove  Edge   O(|V|*|E|)  

Adjacency  check   O(|E|)  

1   2   3   4   5  

a   1   1  

b   1   1  

c   1   1  

d   1   1  

e   1   1  

f   1  

V

E

c

b

d

a

e f

Complexity Comparison

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Adjacency  List   Adjacency  Matrix   Incidence  Matrix  

Store  graph   O(|V|+|E|)   O(|V|²)   O(|V|*|E|)  

Add  Vertex   O(1)   O(|V|²)   O(|V|*|E|)  

Add  Edge   O(1)   O(1)   O(|V|*|E|)  

Remove  Vertex   O(|E|)   O(|V|²)   O(|V|*|E|)  

Remove  Edge   O(|E|)   O(1)   O(|V|*|E|)  

Adjacency  check   O(|V|)   O(1)   O(|E|)  

•  Choose data-structure based on algorithm needs !

•  Vertex merging (Edge contraction)

•  Edge removal

Graph Reduction Operations

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•  Built by applying reduction operations to a graph

•  Lowest level stores original graph

•  Top level stores maximally reduced graph (single vertex)

Graph Pyramid

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•  Vertices describe pixels

•  Edges describe adjacency relationships between pixels •  4-connected image graph is always planar

Graphs and Images

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•  Simplified image graph

•  Vertices describe connected regions of any shape and size

•  Edges describe adjacency relationships between regions

•  Simple graph – no multiple edges, no self-loops

Region Adjacency Graph

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•  Does not uniquely describe a “setting” of regions

•  Cannot describe inclusion relationship and multiple borders

Region Adjacency Graph

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•  Every planar graph has a dual graph

•  Primal graph •  Vertices describe regions

•  Edges describe adjacency relationships between regions

•  Dual graph •  Vertices describe faces

•  Edges describe borders of faces

Dual Graphs

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•  Primal edge contraction = dual edge removal

•  Primal edge removal = dual edge contraction

Duality of Reduction Operations

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•  Differentiates between adjacency and inclusion relationship

•  Preserves multiple borders

Dual Graphs

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1  

4  

•  Differentiates between adjacency and inclusion relationship

•  Preserves multiple borders

•  Does not uniquely describe a “setting” of regions

Dual Graphs

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1  

4  

•  Different representation of a graph

•  Combinatorial Map CM = (D, α, σ) •  Set of “darts” D

•  α involution connecting two darts

•  σ permutation connecting two darts

•  Can be stored in single array

Combinatorial Maps

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1  

2  3   4  

σ  

α  

•  Modification of the σ permutation

•  Vertex merging (edge contraction)

•  Edge removal

CM Reduction Operations

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•  Primal combinatorial map implicitly stores dual map CM’=(D, α, γ) •  Dual map can be computed anytime from the primal, and vice-versa

•  Combination of α involution and σ permutation

•  γ  = (α * σ)      or    γ  = (σ  *  α)

Dual Combinatorial Maps

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Combinatorial Maps

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•  Uniquely describe “setting” of regions •  Because of implicitly storing direction of darts around each vertex

Dual Combinatorial Maps

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•  Task: group pixels with “similar” properties •  Color, texture, higher level features,…

•  Iterative Approach: •  Represent image as graph

•  Assign weights to edges (similarity)

•  Iteratively contract edges with lowest weights

•  Other approaches: •  Minimum spanning tree based approaches

•  Graph cuts

Application: Image Segmentation

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Thank you!

Martin Cerman