Markovian reconstruction using a GNC approach

This paper is concerned with the reconstruction of images (or signals) from incomplete, noisy data, obtained at the output of an observation system, T he solution is defined in maximum a posteriori (MAP) sense and it appears a s the global minimum of an energy function joining a convex data-fidelity t erm and a Markovian prior energy. The sought images are composed of nearly homogeneous zones separated by edges and the prior term accounts for this k nowledge. This term combines general nonconvex potential functions (PF's) w hich are applied to the differences between neighboring pixels. The resultant MAP energy generally exhibits numerous local minima, Calculat ing its local minimum, placed in the vicinity of the maximum likelihood est imate, is inexpensive but the resultant estimate is usually disappointing. Optimization using simulated annealing is practical only in restricted situ ations. Several deterministic suboptimal techniques approach the global min imum of special MAP energies, employed in the field of image denoising, at a reasonable numerical cost, The latter techniques are not directly applica ble to general observation systems, nor to general Markovian prior energies , This work is devoted to the generalization of one of them, the graduated no nconvexity (GNC) algorithm, in order to calculate nearly-optimal MAP soluti ons in a wide range of situations, In fact, GNC provides a solution by trac king a set of minima along a sequence of approximate energies, starting fro m a convex energy and progressing toward the original energy. In this paper , we develop a common method to derive efficient GNC-algorithms for the min imization of MAP energies which arise in the context of any observation sys tem giving rise to a convex data-fidelity term and of Markov random field ( MRF) energies involving any nonconvex and/or nonsmooth PF's. As a side-resu lt, we propose how to construct pertinent initializations which allow us to obtain meaningful solutions using local minimization of these MAP energies . Two numerical experiments-an image deblurring and an emission tomography re construction-illustrate the performance of the proposed technique.