IMAGE-MODELING GIBBS PRIORS

Gibbs distributions, which have been very successfully used in statist ical mechanics, have also been applied in image processing as assumed prior distributions in Bayesian (MAP) image restoration or reconstruct ion. When used in this context, the appropriateness of the Gibbs distr ibution has been judged by the success of the resulting image processi ng method; little attention has been paid to whether the Gibbs distrib ution indeed models the images that occur in the particular applicatio n area, in the sense that a randomly selected image from the distribut ion is likely to share the essential properties of those images. Indee d, many of the proposed Gibbs distributions do nothing but enforce smo othness; random samples from such distributions are likely to be unifo rmly smooth and thus probably atypical for any application area. In th is paper we investigate the possibility of finding Gibbs distributions which truly model certain properties of images and look at the potent ial usefulness of using such image-modeling distributions as priors in Bayesian image processing. Specifically, we construct a Gibbs distrib ution which models an image that consists of piecewise homogeneous reg ions. The proposed model incorporates not only the information about t he smoothness within regions in the image, but also the continuity of boundary structures which exist between regions. It is demonstrated th at by sampling the Gibbs distribution which arises from the model we o btain images with piecewise homogeneous regions resembling the global features of the image that we intend to model; hence such a Gibbs dist ribution is indeed ''image-modeling.'' Objective assessment of the mod el is accomplished by performing a goodness-of-fit test based on a chi (2) statistic computed by considering the corresponding local conditio nal distributions. Issues related to the selection of model parameters from the given data image are addressed. Importantly, the most essent ial parameter of the image model(related to the regularization paramet er associated with the penalty function in many image restoration and reconstruction methods) is estimated in the process of constructing th e image model. Comparative results are presented of the outcome of usi ng our model and an alternative model as the prior in some image resto ration problems in which noisy synthetic images were considered. (C) 1 995 Academic Press, Inc.