1. Segmentation
Graph-Theoretic Clustering

2. Outline
Graph theory basics
Eigenvector methods for segmentation

3. Graph Theory Terminology
Graph G: Set of vertices V and edges E connecting pairs of vertices
Each edge is represented by the vertices (a, b) it joins
A weighted graph has a weight associated with each edge w(a, b)
Connectivity
Vertices are connected if there is a sequence of edges joining them
A graph is connected if all vertices are connected
Any graph can be partitioned into connected components (CC) such that each CC is a connected graph and there are no edges between vertices in different CCs

4. Graphs for Clustering Tokens are vertices
Weights on edges proportional to token similarity
Cut: “Weight” of edges joining two sets of vertices:
Segmentation: Look for minimum cut in graph
Recursively cut components until regions uniform enough A B

5. Representing Graphs As Matrices
Use N x N matrix W for N–vertex graph
Entry W(i, j) is weight on edge between vertices i and j
Undirected graphs have symmetric weight matrices
from Forsyth & Ponce 1 2 3 4 5 7 6 9 8
Example graph and its weight matrix

6. Affinity Measures
Affinity A(i, j) between tokens i and j should be proportional to similarity
Based on metric on some visual feature(s)
Position: E.g., A(i, j) = exp [-((x-y)T (x-y)/2sd2 )]
Intensity
Color
Texture
These are weights in an affinity graph A over tokens

7. Affinity by distance

8. Choice of Scale s
s=0.2
s=0.1
s=1

9. Eigenvectors and Segmentation
Given k tokens with affinities defined by A, want partition into c clusters
For a particular cluster n, denote the membership weights of the tokens with the vector wn
Require normalized weights so that
“Best” assignment of tokens to cluster n is achieved by selecting wn that maximizes objective function (highest intra-cluster affinity)
subject to weight vector normalization constraint
Using method of Lagrange multipliers, this yields system of equations
which means that wn is an eigenvector of A and a solution is obtained from the eigenvector with the largest eigenvalue

10. Eigenvectors and Segmentation
Note that an appropriate rearrangement of affinity matrix leads to block structure indicating clusters
Largest eigenvectors A of tend to correspond to eigenvectors of blocks
So interpret biggest c eigenvectors as cluster membership weight vectors
Quantize weights to 0 or 1 to make memberships definite 5 4 9 2 6 1 1 8 1 3 7
from Forsyth & Ponce

11. Example using dataset Fig 14.18

12. Next 3 Eigenvectors

13. Number of Clusters

14. Potential Problem

15. Normalized Cuts
Previous approach doesn’t work when eigenvalues of blocks are similar
Just using within-cluster similarity doesn’t account for between-cluster differences
No encouragement of larger cluster sizes
Define association between vertex subset A and full set V as
Before, we just maximized assoc(A, A); now we also want to minimize assoc(A, V). Define the normalized cut as

16. Normalized Cut Algorithm
Define diagonal degree matrix D(i, i) = Sj A(i, j)
Define integer membership vector x over all vertices such that each element is 1 if the vertex belongs to cluster A and -1 if it belongs to B (i.e., just two clusters)
Define real approximation to x as
This yields the following objective function to minimize:
which sets up the system of equations
The eigenvector with second smallest eigenvalue is the solution (smallest always 0)
Continue partitioning clusters if normcut is over some threshold

17. Example: Fig 14.23

18. Example: Fig