Estimation of generalized mixtures and its application in image segmentation

We introduce in this work the notion of a generalized mixture and propose some methods for estimating it, along with applications to unsupervised statistical image segmentation. A distribution mixture is said to be “generalized” when the exact nature of components is not known, but each belongs to a finite known set of families of distributions. For instance, we can consider a mixture of three distributions, each being exponential or Gaussian. The problem of estimating such a mixture contains thus a new difficulty: We have to label each of three components (there are eight possibilities). We show that the classical mixture estimation algorithms—expectationmaximization (EM), stochastic EM (SEM), and iterative conditional estimation (ICE)—can be adapted to such situations once as we dispose of a method of recognition of each component separately. That is, when we know that a sample proceeds from one family of the set considered, we have a decision rule for what family it belongs to. Considering the Pearson system, which is a set of eight families, the decision rule above is defined by the use of “skewness” and “kurtosis.” The different algorithms so obtained are then applied to the problem of unsupervised Bayesian image segmentation. We propose the adaptive versions of SEM, EM, and ICE in the case of “blind,” i.e., “pixel by pixel,” segmentation. “Global” segmentation methods require modeling by hidden random Markov fields, and we propose adaptations of two traditional parameter estimation algorithms: Gibbsian EM (GEM) and ICE allowing the estimation of generalized mixtures corresponding to Pearson’s system. The efficiency of different methods is compared via numerical studies, and the results of unsupervised segmentation of three real radar images by different methods are presented.