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Chapter 5AUTOMATIC IMAGE SEGMENTATION VIA BIC5.1 Pseudolikelihood for Image Models5.1.1 Potts ModelWe can model the spatial dependence in an image by using a Markov randomfield to model the true state of each pixel. We assume that each pixel has a truehidden state X i , where X i is an integer denoting one of the K states, and that thetrue state of a pixel is likely to be similar to the states of its neighbors. DefineI(X i , X j ) as an indicator function equal to 1 when X i = X j and zero otherwise.Let N(X i ) be the neighbors of X i (that is, the 8 pixels adjacent to pixel X i ),and let U(N(X i ), k) denote the number of points in N(X i ) which have state k(so U(N(X i ), X i ) is the number of neighbors of pixel i which have the same stateas pixel i). The Potts model is characterized by the joint distribution given inequation 5.1, in which the sum is over all neighbor pairs.p(X) ∝ exp(φ ∑ i jI(X i , X j )) (5.1)Equation 5.1 leads to the conditional distribution in equation 5.2.p(X i = j|N(X i ), φ) =exp(φU(N(X i), j))∑k exp(φU(N(X i ), k))(5.2)The parameter φ expresses the amount of spatial homogeneity in the model.A positive value of φ means that neighboring pixels tend to be similar, while a