Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation
| hal.structure.identifier | CEntre de REcherches en MAthématiques de la DEcision [CEREMADE] | |
| dc.contributor.author | VAITER, Samuel | |
| hal.structure.identifier | CEntre de REcherches en MAthématiques de la DEcision [CEREMADE] | |
| dc.contributor.author | DELEDALLE, Charles | |
| hal.structure.identifier | CEntre de REcherches en MAthématiques de la DEcision [CEREMADE] | |
| dc.contributor.author | PEYRÉ, Gabriel | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | DOSSAL, Charles | |
| hal.structure.identifier | Equipe Image - Laboratoire GREYC - UMR6072 | |
| dc.contributor.author | FADILI, Jalal M. | |
| dc.date.accessioned | 2024-04-04T02:24:24Z | |
| dc.date.available | 2024-04-04T02:24:24Z | |
| dc.date.created | 2012-04 | |
| dc.date.issued | 2013 | |
| dc.identifier.issn | 1063-5203 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/189813 | |
| dc.description.abstractEn | In this paper, we aim at recovering an unknown signal x0 from noisy L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we impose an analysis sparsity prior. More precisely, the recovery is cast as a convex optimization program where the objective is the sum of a quadratic data fidelity term and a regularization term formed of the L1-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The L1-sparsity analysis prior is weighted by a regularization parameter lambda>0. In this paper, we prove that any minimizers of this problem is a piecewise-affine function of the observations y and the regularization parameter lambda. As a byproduct, we exploit these properties to get an objectively guided choice of lambda. In particular, we develop an extension of the Generalized Stein Unbiased Risk Estimator (GSURE) and show that it is an unbiased and reliable estimator of an appropriately defined risk. The latter encompasses special cases such as the prediction risk, the projection risk and the estimation risk. We apply these risk estimators to the special case of L1-sparsity analysis regularization. We also discuss implementation issues and propose fast algorithms to solve the L1 analysis minimization problem and to compute the associated GSURE. We finally illustrate the applicability of our framework to parameter(s) selection on several imaging problems. | |
| dc.description.sponsorship | Adaptivité pour la représentation des images naturelles et des textures - ANR-08-EMER-0009 | |
| dc.language.iso | en | |
| dc.publisher | Elsevier | |
| dc.subject.en | sparsity | |
| dc.subject.en | analysis regularization | |
| dc.subject.en | inverse problems | |
| dc.subject.en | L1 minimization | |
| dc.subject.en | local variation | |
| dc.subject.en | degrees of freedom | |
| dc.subject.en | SURE | |
| dc.subject.en | GSURE | |
| dc.subject.en | unbiased risk estimation | |
| dc.title.en | Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1016/j.acha.2012.11.006 | |
| dc.subject.hal | Informatique [cs]/Traitement du signal et de l'image | |
| dc.subject.hal | Sciences de l'ingénieur [physics]/Traitement du signal et de l'image | |
| dc.subject.hal | Mathématiques [math]/Statistiques [math.ST] | |
| dc.subject.hal | Statistiques [stat]/Théorie [stat.TH] | |
| dc.subject.hal | Mathématiques [math]/Théorie de l'information et codage [math.IT] | |
| dc.subject.hal | Informatique [cs]/Théorie de l'information [cs.IT] | |
| dc.identifier.arxiv | 1204.3212 | |
| dc.description.sponsorshipEurope | Sparsity, Image and Geometry to Model Adaptively Visual Processings | |
| bordeaux.journal | Applied and Computational Harmonic Analysis | |
| bordeaux.page | 433-451 | |
| bordeaux.volume | 35 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 3 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00687751 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00687751v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Applied%20and%20Computational%20Harmonic%20Analysis&rft.date=2013&rft.volume=35&rft.issue=3&rft.spage=433-451&rft.epage=433-451&rft.eissn=1063-5203&rft.issn=1063-5203&rft.au=VAITER,%20Samuel&DELEDALLE,%20Charles&PEYR%C3%89,%20Gabriel&DOSSAL,%20Charles&FADILI,%20Jalal%20M.&rft.genre=article |
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