Statistical-mechanical analysis of dynamical properties of algorithms for image restoration

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In this research project, we investigated the dynamical properties of Bayesian image restoration for the blapk-while binary and gray-scaled images represented by the Ising and the Q-Ising models, respectively. We introduced the mean-field model to evaluate the accuracy or the speed of the convergence for EM algorithm and gradient descent method to obtain the maximum likelihood estimate for the hyper-parameters. We found that EM algorithm is superior to the gradient descent method in the speed of the convergence. Moreover, our analysis made it clear that the dynamical behavior of the hyper-parameters shows a kind of the "oscillating phenomena " before they converge to their true values if the time to wait until the system reaches to the equilibrium states is not enough.We also proposed a new-type of the deterministic annealing EM algorithm which was extended by means of non-extensive statistical mechanics. The method was applied to the problem of Traveling salesman and the statistical performance was investigated extensively.We also presented several applications of quantum spin glasses (random field Ising model, Shermgton-KJrkpatrick model, Ising spin glasses with p-body interaction in a transverse field) to probabilistic information processing, especially to the problems of image restoration and error-correcting codes. As a related optimization method, quantum annealing was also introduced to these research fields and its performance was investigated by using the quantum Markov chain Monte Carlo method. The Nishimori-Wong condition, on which the best possible performance of the quantum MPM estimation is achieved, was derived as a condition on the effective amplitude of the transverse field.