Locally homogeneous images as minimizers of objective functions

We present a theoretical study of the reconstruction of locally homogeneous images from noisy data, by minimizing regularized objective functions. The latters are composed of a smooth data-fidelity term and a regularizer where a potential function (PF) is applied to each difference between neighbouring pixels. We confine our attention to the reconstruction of images which contain either weakly homogeneous regions (where the differences between neighbours are close to zero, but are non-zero) or strongly homogeneous regions (where these differences are zero). This distinction is suggested by the essentially different visual effect produced by the regions of these two types. We show that the recovery of regions of either type is determined by the smoothness at zero of the PF. So, a smooth at zero PF leads to minimizers-image estimates-which are only weakly homogeneous. In contrast, a nonsmooth at zero PF yields image estimates which contain large strongly homogeneous regions. In this case, the data domain is composed of volumes where the data yield minimizers are images which have the same strongly homogeneous regions. These theoretical results are eloquently illustrated by a numerical example.