Maximum marginal likelihood estimation and constrained optimization in image restoration

In image restorations, two different mathematical frameworks for massive probabilistic models are given in the standpoint of Bayes statistics. One of them is formulated by assuming that a priori probability distribution has a form of Gibbs microcanonical distribution and can be reduced to a constrained optimization. In this framework, we have to know a quantity estimated from the original image though we do not have to know the con.guration of the original image. We give a new method to estimate the length of boundaries between areas with different grey-levels in the original image only from the degraded image in grey-level image restorations. When the length of boundaries in the original image is estimated from the given degraded image, we can construct a probabilistic model for image restorations by means of Bayes formula. This framework can be regarded as probabilistic information processing at zero temperature. The other framework is formulated by assuming the a priori probability distribution has a form of Gibbs canonical distribution. In this framework, some hyperparameters are determined so as to maximize a marginal likelihood. In this paper, it is assumed that the a priori probability distribution is a Potts model, which is one of familiar statistical-mechanical models. In this assumption, the logarithm of marginal likelihood can be expressed in terms of free energies of some probabilistic models and hence this framework can be regarded as probabilistic information processing at finite temperature. The practical algorithms in the constrained optimization and the maximum marginal likelihood estimation are given by means of cluster variation method, which is a statistical-mechanical approximation with high accuracy for massive probabilistic models. We compare them with each others in some numerical experiments for grey-level image restorations.