Statistical mechanics of image restoration

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Abstract
We develop the statistical mechanics formulation of the image restoration problem, pioneered by Geman and Geman (1984). Using Bayesian methods we establish the posterior probability distribution for restored images, for given data (corrupted image) and prior (assumptions about source and corruption process). In the simplest cases, studied here, the posterior is controlled by a cost function analagous to the configurational energy of an Ising model with local fields whose sense is defined by the data. Through a combination of Monte Carlo simulation and mean-field theory we address three key issues. First, we explore the sensitivity of the posterior distribution to the choice of prior parameters: we find phase transitions separating regions in which the distribution is effective (data-dominated) from regions in which it is ineffective (prior-dominated). Second, we examine the question of how best to use the posterior distribution to prescribe a single "optimal" restored image: we argue that the mean of the posterior is, in general, to be preferred over the mode, both in principle and in practice. Finally, borrowing from Monte Carlo techniques for free-energy calculations, we address the question of prior parameter estimation within the "evidence" framework of Gull (1989) and MacKay (1992): our results suggest that parameters identified by this framework provide effective priors, leading to optimal restoration, only to the extent that the forms of the priors are well matched to the processes they claim to represent.

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