Stability of image restoration by minimizing regularized objective functions

Abstract

We address the general problem of the recovery of an unknown image, x/spl isin/R/sup p/, from noisy data, y/spl isin/R/sup q/, by minimizing a regularized objective function /spl epsi/(x,y). We focus on typical situations when the objective function is C/sup m/-smooth and is composed of a quadratic data-fidelity term and a general regularization term: /spl epsi/(x,y)=/spl par/Ax-y/spl par//sup 2/+/spl Phi/(x), where A is a linear operator. Many authors have shown that especially nonconvex regularizers /spl Phi/ allow the restoration of images involving both sharp edges and smoothly varying regions. The main limitation in using such regularizers is that, being highly nonconvex, the resultant objective functions are intricate to minimize. On the other hand since very few facts are known about the minimizers of such functions, the properties and in particular the stability of the resultant solutions are difficult to control. This state of the art limits the practical use of such functions. This work is devoted to the stability of the local and global minimizers x of objective functions /spl epsi/ as specified above, under the assumption that A is injective. We thus have shown that the global minimizers of /spl epsi/ are stable under small perturbations of the data.