Image-Modeling Gibbs Priors

Abstract

Gibbs distributions, which have been very successfully used in statistical mechanics, have also been applied in image processing as assumed prior distributions in Bayesian (MAP) image restoration or reconstruction. When used in this context, the appropriateness of the Gibbs distribution has been judged by the success of the resulting image processing method; little attention has been paid to whether the Gibbs distribution indeed models the images that occur in the particular application area, in the sense that a randomly selected image from the distribution is likely to share the essential properties of those images. Indeed, many of the proposed Gibbs distributions do nothing but enforce smoothness; random samples from such distributions are likely to be uniformly smooth and thus probably atypical for any application area. In this paper we investigate the possibility of finding Gibbs distributions which truly model certain properties of images and look at the potential usefulness of using such image-modeling distributions as priors in Bayesian image processing. Specifically, we construct a Gibbs distribution which models an image that consists of piecewise homogeneous regions. The proposed model incorporates not only the information about the smoothness within regions in the image, but also the continuity of boundary structures which exist between regions. It is demonstrated that by sampling the Gibbs distribution which arises from the model we obtain images with piecewise homogeneous regions resembling the global features of the image that we intend to model; hence such a Gibbs distribution is indeed "image-modeling." Objective assessment of the model is accomplished by performing a goodness-of-fit test based on a χ2 statistic computed by considering the corresponding local conditional distributions. Issues related to the selection of model parameters from the given data image are addressed. Importantly, the most essential parameter of the image model (related to the regularization parameter associated with the penalty function in many image restoration and reconstruction methods) is estimated in the process of constructing the image model. Comparative results are presented of the outcome of using our model and an alternative model as the prior in some image restoration problems in which noisy synthetic images were considered.