Smoothing of outliers in image restoration by minimizing regularized objective functions with nonsmooth data-fidelity terms

Abstract

We present a theoretical study of the recovery of images x from noisy data y by minimizing a regularized cost-function F(x, y) = /spl Psi/(x, y) + /spl alpha//spl Phi/(x), where /spl Psi/ is a data-fidelity term, /spl Phi/ is a smooth regularization term and a>0 is a parameter. Generally, /spl Psi/ is a smooth function; only a few papers make an exception. Nonsmooth data-fidelity terms are avoided in image processing. In spite of this, our ambition is to catch essential features exhibited by the local minimizers of F in relation with the smoothness of /spl Psi/. We focus on /spl Psi/(x,y)=/spl Sigma//sub i//spl psi/(a/sub i//sup T/x-y/sub i/) where a/sub i//sup T/ are linear operators and /spl psi/ is C/sup m/-smooth on R{0}. We show that if /spl psi/'(0/sup -/)</spl psi/'(0/sup +/), typical data y lead to local minimizers x/spl circ/ of F(., y) which fit exactly part of the data entries: there is a possibly large set h/spl circ/ such that a/sub i//sup T/x/spl circ/=y/sub i/ for every i/spl isin/h/spl circ/. In contrast, this effect does not occur if F is smooth. We thus have a strong mathematical property which can be used in various ways. Based on it, we construct a cost-function allowing aberrant data to be detected and selectively smoothed either from images or from noisy data, while preserving efficiently all non-aberrant pixels. This is illustrated by numerical examples. The obtained results advocate the use of non-smooth data-fidelity terms in image processing.