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1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision 2 Equipe Image - Laboratoire GREYC - UMR6072 GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen 3 IMB - Institut de Mathématiques de Bordeaux

Abstract : This paper develops a novel framework to compute a projected Generalized Stein Unbiased Risk Estimator GSURE for a wide class of sparsely regularized solutions of inverse problems. This class includes arbitrary convex data fidelities with both analysis and synthesis mixed L1-L2 norms. The GSURE necessitates to compute the weak derivative of a solution w.r.t.~the observations. However, as the solution is not available in analytical form but rather through iterative schemes such as proximal splitting, we propose to iteratively compute the GSURE by differentiating the sequence of iterates. This provides us with a sequence of differential mappings, which, hopefully, converge to the desired derivative and allows to compute the GSURE. We illustrate this approach on total variation regularization with Gaussian noise and to sparse regularization with poisson noise, to automatically select the regularization parameter.

keyword : Sparsity regularization inverse problems risk estimator GSURE automatic differentiation

Autor: Charles Deledalle - Samuel Vaiter - Gabriel Peyré - Jalal M. Fadili - Charles Dossal -



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