Training a Boltzmann Machine for edge-preserving image restoration

The most successful methods to stabilize inverse ill-posed problems in visual reconstruction use a priori information on the local regularity of the image as well as constraints on the geometry of the discontinuities. A commonly used method to incorporate prior knowledge into the problem is to adopt a Bayesian approach in which the image is modelled by a parametric Gibbs prior and the solution is obtained by minimizing the resulting posterior energy function (MAP estimate). However, this approach presents two major difficulties: the first is related to the non-convexity of the function to be optimized; the second to the choice of the model parameters that best fit the availabie prior knowledge. Since these parameters strongly affect the quality of the reconstructions, their selection is a critical task. They are usually determined empirically by trial and error. The paper proposes a generalized Boltzmann Machine which makes it possible to learn the most appropriate parameters for a given class of images from a series of examples. The trained Boltzmann Machine is then used to implement an annealing scheme for the minimization of the non-convex posterior energy. The method is applied to the restoration of piecewise smooth images.