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96ization to initialize (see section A.1), while my Splus implementation uses Ward’smethod.5.2.3 Marginal Segmentation via Mixture ModelsParameter Estimation by EMI use the EM algorithm to estimate the parameters of the mixture density. LetQ denote the pixel probabilities, where Q ij is the probability that pixel i is fromcomponent j. The initial segmentation is viewed as providing an initial estimateof Q; specifically, this initial estimate will consist of Q ij = 1 if pixel i is initiallyclassified in component j, and zero otherwise. After initialization, the algorithmiterates between the M-step, computing maximum likelihood estimates of the mixtureparameters θ conditional on Q, and the E-step, estimating Q conditional onθ (the name of this step comes from the fact that Q ij is the expected value ofI(Z i = j), which is an indicator function which is equal to 1 if pixel i is generatedby component j and zero otherwise).At each iteration, I compute the overall loglikelihood of the data Y given theparameters θ, using the assumption of independent pixels. In general, the EMprocess is repeated until the loglikelihood converges. My Splus implementationallows a user-specified limit on the number of iterations. The parameters θ whichmust be estimated are the density parameters from each of the K components ofthe mixture distribution and K − 1 mixture proportions (the mixture proportionssum to 1, so one of the K proportions is fixed given the other K − 1 proportions).The mixture density is shown in equation 5.70, where Y i is the observedgreyscale value of pixel i, P j is the mixture proportion of component j (also sometimescalled the prior probability of component j), Φ is the single componentdensity (e.g. Gaussian), θ j is the vector of parameters for the jth density, and K