Reconstruction of locally homogeneous images

The reconstruction of images involving large homogeneous zones from noisy data, given at the output of an observation system, is a common problem arising in various applications. A popular approach for its resolution is regularized estimation: the sought image is defined as the minimizer of an energy function combining a data-fidelity term and a regularization prior term. The latter term results from applying a set of potential functions (PFs) to the differences between neighbouring pixels and it can be seen as a Markovian energy. We formalize and perform a mathematical study of the possibility to obtain images comprising either strongly homogeneous regions or weakly homogeneous zones, using regularized estimation. Our results reveal that the recovery of zones of either type in an estimated image depends uniquely on the smoothness at zero of the PFs involved in the prior term. These theoretical results are illustrated on the deblurring of an image.