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373.2 Mixture ModelsThe mixture density provides a natural way of modeling the observed “mixture”of features in an image. We can model the marginal (without spatial information)distribution of greyscale values in an image with a mixture density, shown inequation 3.15. We use the Gaussian density for Φ when the data are given inan image format, but the Poisson density is more appropriate when raw data areavailable for gamma camera images (e.g. PET). In the Poisson case, we allow thePoisson parameter λ to take on the value zero, which gives a probability mass atzero; this allows easy modeling of zero-intensity background regions, a commonand usually artifactual feature of medical images.K∑f(Y i |K, θ) = P j Φ(Y i |θ j ) (3.15)j=1Here, Y i is the greyscale value of pixel i, P j is the mixture probability of componentj, Φ is the single component density (e.g. Gaussian), θ j is the vector of parametersfor the jth density, and K is the number of components.Suppose we have a given value of K. Let Z i denote the true component whichgenerated the observation for pixel Y i , and suppose we have an initial classificationof each pixel into one of the K components (i.e. an initial estimate ˜Z i ). Considerthe Z i to be missing data; now we have cast the problem of estimating the θ jand P j parameters into a missing data problem which can be approached by theEM algorithm (Dempster, et al, 1977). Details of the estimation are presented inchapter 5.2.Once we have estimated θ j and P j , we can use equation 3.15 to compute alikelihood for each pixel. Under the assumption that all pixels are independent, thelikelihood of the whole image is the product of the pixel likelihoods, which makescomputation of the loglikelihood of the image relatively easy. The loglikelihood of