# A l1-unified variational framework for image restoration

1 ARIANA - Inverse problems in earth monitoring
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - SIS - Signal, Images et Systèmes
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.
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https://hal-supelec.archives-ouvertes.fr/hal-00217251
Contributor : Julien Bect Connect in order to contact the contributor
Submitted on : Friday, January 25, 2008 - 10:20:07 AM
Last modification on : Friday, January 21, 2022 - 3:09:42 AM

### Citation

Julien Bect, Laure Blanc-Féraud, Gilles Aubert, Antonin Chambolle. A l1-unified variational framework for image restoration. 8th European Conference on Computer Vision (ECCV 2004), May 2004, Prague, Czech Republic. pp.1-13, ⟨10.1007/b97873⟩. ⟨hal-00217251⟩

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