Unbiased Risk Estimation for Sparse Analysis Regularization

Charles Deledalle 1 Samuel Vaiter 1 Gabriel Peyré 1 Jalal M. Fadili 2 Charles Dossal 3
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 : In this paper, we propose a rigorous derivation of the expression of the projected Generalized Stein Unbiased Risk Estimator ($\GSURE$) for the estimation of the (projected) risk associated to regularized ill-posed linear inverse problems using sparsity-promoting L1 penalty. The projected GSURE is an unbiased estimator of the recovery risk on the vector projected on the orthogonal of the degradation operator kernel. Our framework can handle many well-known regularizations including sparse synthesis- (e.g. wavelet) and analysis-type priors (e.g. total variation). A distinctive novelty of this work is that, unlike previously proposed L1 risk estimators, we have a closed-form expression that can be implemented efficiently once the solution of the inverse problem is computed. To support our claims, numerical examples on ill-posed inverse problems with analysis and synthesis regularizations are reported where our GSURE estimates are used to tune the regularization parameter.
Keywords : Sparsity analysis regularization inverse problems risk estimator GSURE
Type de document :
Communication dans un congrès
Proc. ICIP'12, Sep 2012, Orlando, United States. pp.3053-3056, 2012
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Charles Deledalle, Samuel Vaiter, Gabriel Peyré, Jalal M. Fadili, Charles Dossal. Unbiased Risk Estimation for Sparse Analysis Regularization. Proc. ICIP'12, Sep 2012, Orlando, United States. pp.3053-3056, 2012. <hal-00662718>

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