Parameter estimation and model selection in image analysis using Gibbs-Markov random fields.

摘要

Researchers in the field of statistical image analysis are concerned with different issues, such as image restoration, boundary detection, or even object recognition, which may be used in such different contexts as images returned by satellite or medical images produced by emission tomography. There are, of course, many other issues one might address in using statistics to analyze an image. This particular research focuses on the selection of a model for a digital image.;Although model selection has been studied extensively in many areas of statistics, very little has been done within the context of image analysis. Thus this research is restricted to the most elementary images: those which are of a single texture (i.e., an image which, in its entirety, is nothing but carpet, wood grain, clouds in the sky, or some other single type of "texture"). The models under consideration are parametric Gibbs-Markov random fields.;Parameter estimation is then a critical matter. The maximum likelihood estimator (MLE) is quite intractable for such models. This research focuses on two alternatives to the MLE: a Monte Carlo maximum likelihood estimate (MCMLE), and the maximum pseudo-likelihood estimate (MPLE). Asymptotic rates for the mean square error and for a moderate deviation probability are derived for the MPLE.;The main goal of this research is the development of information criteria for choosing a model, similar to the Bayesian information criteria used in model selection for time series and for exponential families. We establish criteria based on the MLE, the MCMLE and the MPLE. We show that the criteria based on the MLE and MCMLE are both approximations to the true Bayes solution to the model selection problem; and we also show the (weak) consistency of the criterion based on the MPLE.;A simulation study of the useful parameter estimation techniques and model selection criteria is presented, using several simple models. Implementation of the model selection criteria on real textures is also discussed.