Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF

Abstract—We propose a method for the reconstruction of signals and images observed partially through a linear operator with 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 reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite for the use of such a prior is the success of the optimization stage. The posterior energy corresponding to a PG MRF is generally multimodal and its minimization is particularly problematic. In this context, general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support. In this paper, global optimization is dea...

• Type: Journal
• Year: 1998
• Where: TIP
• Authors: Mila Nikolova, Jérôme Idier, Ali Mohammad-Djafari
• Keywords: Ill-posed, Inverse Problems, Large Support, TIP 1998