A L1-Unified Variational Framework for Image Restoration

Abstract

Among image restoration literature, there are mainly two kinds of approach. One is based on a process over image wavelet coefficients, as wavelet shrinkage for denoising. The other one is based on a process over image gradient. In order to get an edge-preserving regularization, one usually assume that the image belongs to the space of functions of Bounded Variation (BV). An energy is minimized, composed of an observation term and the Total Variation (TV) of the image. Recent contributions try to mix both types of method. In this spirit, the goal of this paper is to define a unified-framework including together wavelet methods and energy minimization as TV. In fact, for denoising purpose, it is already shown that wavelet soft-thresholding is equivalent to choose the regularization term as the norm of the Besov space B ^11 _1. In the present work, this equivalence result is extended to the case of deconvolution problem. We propose a general functional to minimize, which includes the TV minimization, wavelet coefficients regularization, mixed (TV+wavelet) regularization or more general terms. Moreover we give a projection-based algorithm to compute the solution. The convergence of the algorithm is also stated. We show that the decomposition of an image over a dictionary of elementary shapes (atoms) is also included in the proposed framework. So we give a new algorithm to solve this difficult problem, known as Basis Pursuit. We also show numerical results of image deconvolution using TV, wavelets, or TV+wavelets regularization terms.