Fast iterative methods applied to tomography models with general Gibbs priors

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Abstract
Recently, we present a very fast method (BSREM) for solving regularized problems in emission tomography that is convergent to Maximum a Posteriori (MAO) solutions for convex priors. The method generalizes in a natural way the Expectation Maximization (EM) algorithm and some of its extensions to the MAP case. It consists of decomposing the likelihood function in blocks containing sets of projections plus a block that corresponds to the prior function, and iterating for each block using scaled gradient directions, resembling a smoothed iteration algorithm. In the general nonconvex case, it can be proven that the algorithm converges to a critical point, instead of the sought global maximum. In spite of this, BSREM is so fast and flexible, that it can be used to search for global optima when the model is not convex. In this article, we present an implementation of BSREM, that works as the local optimization method for each step of a Graduated Convexity approach as compared with BSREM itself without any convexifaction. We illustrate the behavior of the method with applications to emission tomography.

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