INVERSION OF LARGE-SUPPORT ILL-POSED LINEAR-OPERATORS USING A PIECEWISE GAUSSIAN MRF

We propose a method for the reconstruction of signals and images obser ved partially through a linear operator,vith a large support (e.g., a Fourier transform on a sparse set), This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reco nstructed objects are composed of smooth regions separated by sharp tr ansitions, This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension an d the weak-membrane in two dimensions. The reconstruction is defined a s the maximum a posteriori estimate, The prerequisite for the use of s uch a prior is the success of the optimization stage, The posterior en ergy corresponding to a PG MRF is generally multimodal and its minimiz ation is particularly problematic. In this context, general forms of s imulated annealing rapidly become intractable when the observation ope rator extends over a large support. In this paper, global optimization is dealt with by extending the graduated nonconvexity (GNC) algorithm to ill-posed linear inverse problems, GNC has been Pioneered by Blake and Zisserman in the field of image segmentation. The resulting algor ithm is mathematically suboptimal but it is seen to be very efficient in practice. We show that the original GNC does not correctly apply to ill-posed problems, Our extension is based on a proper theoretical an alysis, which provides further insight into the GNC, The performance o f the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography.