Segmentation by Graph by yurtgc548


Graph-Theoretic Clustering
   Graph theory basics
   Eigenvector methods for segmentation
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
     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
        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
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
            4                       9

            1                   1       8
    from Forsyth & Ponce

                               Example graph and its weight matrix
    Affinity Measures

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

     s=0.1   s=0.2   s=1
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 w n
        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 w n is an eigenvector of A and a solution is obtained
     from the eigenvector with the largest eigenvalue
Eigenvectors and Segmentation
   Note that an appropriate rearrangement of affinity matrix
    leads to block structure indicating clusters
            4                     9

            1               1           8
    1           3           7
                        from Forsyth & Ponce

   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
Example using dataset Fig 14.18
Next 3 Eigenvectors
Number of Clusters
Potential Problem
Normalized Cuts
   Previous approach doesn’t work when eigenvalues of blocks are similar
       Just using within-cluster similarity doesn’t account for between-cluster
       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
      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
Example: Fig 14.23
Example: Fig. 14-24

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