An image restoration can be often formulated as a energy minimization problem. When an energy function is expressed by using the hamiltonian of a classical spin system only with finite range interactions, the probabilistic model, which is described in the form of Gibbs distribution for the energy function, can be regarded as a Markov random field (MRF) model. In the MRF model, we have to determine not only the minimum-energy configuration but also hyper-parameters. We have a constrained optimization and a maximum likelihood (ML) estimation as mathematical frameworks to determine the hyperparameters. In this paper, some probabilistic computational methods for the search of minimum-energy configuration and the estimation of hyperparameters are proposed in the standpoint of statistical-mechanics. We summarize the mathematical framework of probabilistic computational method based on the constrained optimization and reformulate the framework of ML estimation as a hyperparameter estimation method at a finite temperature in the standpoint of the constrained optimization. The probabilistic computational algorithms for natural image restorations are con-structed from the mean-field approximation, mean-field annealing (MFA), iterative conditional modes (ICM) and cluster zero-temperature process (CZTP).
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