Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability

Abstract

In this work we wish to recover an unknown image from a blurry, or noisy-blurry version. We solve this inverse problem by energy minimiza-tion and regularization. We seek a solution of the form u + v, where u is a function of bounded variation (cartoon component), while v is an oscillatory component (texture), modeled by a Sobolev function with negative degree of differentiability. We give several results of existence and characterization of minimizers of the proposed optimization problem. Experimental results show that this cartoon + texture model better recovers textured details in natural images, by comparison with the more standard models where the unknown is restricted only to the space of functions of bounded variation. 1. Introduction. We consider in this paper one of the classical problems in image analysis: the recovery of an unknown image from its blurry version, in the presence of a known blurring operator. Suppose that we are given a blurry (and possibly noisy) gray-scale image f : Ω → R, where Ω is either R n or an open, bounded and connected subset of R n . We wish to recover a clean imag f from f . Let K be a blurring operator (a linear and continuous smoothing operator, for instance a convolution with the Gaussian kernel or with the average kernel). The standard linear degradation model that relates f tõ f is f = f + noise.