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The framework is presented of Bayesian image restoration for multi-valued images by means of the multi-state classical spin systems. Hyperparameters in the probabilistic models are determined so as to maximize the marginal likelihood. A practical algorithm is described for multi-valued image restoration based on loopy belief propagations in probabilistic(More)
By RCA 0 , we denote the system of second order arithmetic based on recursive comprehension axioms and Σ 0 1 induction. WKL 0 is defined to be RCA 0 plus weak König's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. In this paper, we first show that for any countable model M of RCA 0 , there exists a count-able model M of WKL 0(More)
Boltzmann machines can be regarded as Markov random fields. For binary cases, they are equivalent to the Ising spin model in statistical mechanics. Learning systems in Boltzmann machines are one of the NP-hard problems. Thus, in general we have to use approximate methods to construct practical learning algorithms in this context. In this letter, we propose(More)