Segmentation

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Segmentation
Graph-Theoretic Clustering
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Outline

• Graph theory basics
• Eigenvector methods for segmentation

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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

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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

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Representing Graphs As Matrices

• Use N x N matrix W for Nvertex graph
• Entry W(i, j) is weight on edge between vertices
  i and j
• Undirected graphs have symmetric weight matrices

Example graph and its weight matrix
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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

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Affinity by distance
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Choice of Scale s
s0.2
s0.1
s1
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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

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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

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from Forsyth Ponce
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Example using dataset Fig 14.18
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Next 3 Eigenvectors
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Number of Clusters
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Potential Problem
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Normalized Cuts

• Previous approach doesnt work when eigenvalues
  of blocks are similar
• Just using within-cluster similarity doesnt
  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

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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

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Example Fig 14.23
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Example Fig. 14-24