Weighted Gaussian Mixture Models

A gray-scale image is represented as a 1-D array X ={x1, x2,..., xN), where xn is an input feature for pixel n and N is the total number of pixels in the image. The input feature vector xn may be a D-dimensional vector or simply the gray-scale value of the pixel n. Let the underlying true segmentation of the image be denoted as Y ={y1, y2,..., yN). It is assumed that the number of classes is predetermined as a set of known class labels , where l e{1,...,L}, and therefore the class label of pixel n is indicated as yne{rnl}f= 1. A common assumption in modeling a density with a GMM for image segmentation is that each component m, m e{1,..., M}, will model the pdf of each class M = L. Let yn represent the estimate of the segmentation. Each component is weighted by its weight of Ymn that indicates the relationship of pixel xn to class label modeled by component m. To ensure that the parameters of each component density are learnt correctly, the weight Ymn is set to indicate the class to which data point xn belongs, thus

If Ymn = 1, then data point xn will only be considered when setting the parameters of class ml modeled by component m Using the labelled training data, a maximum likelihood (ML) estimate of all component parameters and mixing coefficients can be found.

We first describe the two modes of test image segmentation, supervised and unsupervised, in section 11.4.2. We then detail our weighted GMM/MRF models in section 11.4.3.

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