Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring

Handle URI:
http://hdl.handle.net/10754/600185
Title:
Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring
Authors:
Fornasier, M.; Kim, Y.; Langer, A.; Schönlieb, C.-B.
Abstract:
In this paper, we show additional properties of the limit of a sequence produced by the subspace correction algorithm proposed by Fornasier and Schönlieb [SIAM J. Numer. Anal., 47 (2009), pp. 3397-3428 for L 2/TV-minimization problems. An important but missing property of such a limiting sequence in that paper is the convergence to a minimizer of the original minimization problem, which was obtained in [M. Fornasier, A. Langer, and C.-B. Schönlieb, Numer. Math., 116 (2010), pp. 645-685 with an additional condition of overlapping subdomains. We can now determine when the limit is indeed a minimizer of the original problem. Inspired by the work of Vonesch and Unser [IEEE Trans. Image Process., 18 (2009), pp. 509-523], we adapt and specify this algorithm to the case of an orthogonal wavelet space decomposition for deblurring problems and provide an equivalence condition to the convergence of such a limiting sequence to a minimizer. We also provide a counterexample of a limiting sequence by the algorithm that does not converge to a minimizer, which shows the necessity of our analysis of the minimizing algorithm. © 2012 Society for Industrial and Applied Mathematics.
Citation:
Fornasier M, Kim Y, Langer A, Schönlieb C-B (2012) Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring. SIAM Journal on Imaging Sciences 5: 857–885. Available: http://dx.doi.org/10.1137/100819801.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Imaging Sciences
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
17-Jul-2012
DOI:
10.1137/100819801
Type:
Article
ISSN:
1936-4954
Sponsors:
The work of the first three authors was supported by the FWF project Y 432-N15 START-Preis Sparse Approximation and Optimization in High Dimensions. The last author's work was supported by the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods, the Wissenschaftskolleg (Graduiertenkolleg, Ph.D. program) of the Faculty for Mathematics at the University of Vienna (funded by the Austrian Science Fund FWF), and the FFG project 813610 Erarbeitung neuer Algorithmen zum Image Inpainting. This publication is based on work supported by award KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). The results of this paper also contribute to the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts.
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Full metadata record

DC FieldValue Language
dc.contributor.authorFornasier, M.en
dc.contributor.authorKim, Y.en
dc.contributor.authorLanger, A.en
dc.contributor.authorSchönlieb, C.-B.en
dc.date.accessioned2016-02-28T06:44:41Zen
dc.date.available2016-02-28T06:44:41Zen
dc.date.issued2012-07-17en
dc.identifier.citationFornasier M, Kim Y, Langer A, Schönlieb C-B (2012) Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring. SIAM Journal on Imaging Sciences 5: 857–885. Available: http://dx.doi.org/10.1137/100819801.en
dc.identifier.issn1936-4954en
dc.identifier.doi10.1137/100819801en
dc.identifier.urihttp://hdl.handle.net/10754/600185en
dc.description.abstractIn this paper, we show additional properties of the limit of a sequence produced by the subspace correction algorithm proposed by Fornasier and Schönlieb [SIAM J. Numer. Anal., 47 (2009), pp. 3397-3428 for L 2/TV-minimization problems. An important but missing property of such a limiting sequence in that paper is the convergence to a minimizer of the original minimization problem, which was obtained in [M. Fornasier, A. Langer, and C.-B. Schönlieb, Numer. Math., 116 (2010), pp. 645-685 with an additional condition of overlapping subdomains. We can now determine when the limit is indeed a minimizer of the original problem. Inspired by the work of Vonesch and Unser [IEEE Trans. Image Process., 18 (2009), pp. 509-523], we adapt and specify this algorithm to the case of an orthogonal wavelet space decomposition for deblurring problems and provide an equivalence condition to the convergence of such a limiting sequence to a minimizer. We also provide a counterexample of a limiting sequence by the algorithm that does not converge to a minimizer, which shows the necessity of our analysis of the minimizing algorithm. © 2012 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThe work of the first three authors was supported by the FWF project Y 432-N15 START-Preis Sparse Approximation and Optimization in High Dimensions. The last author's work was supported by the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods, the Wissenschaftskolleg (Graduiertenkolleg, Ph.D. program) of the Faculty for Mathematics at the University of Vienna (funded by the Austrian Science Fund FWF), and the FFG project 813610 Erarbeitung neuer Algorithmen zum Image Inpainting. This publication is based on work supported by award KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). The results of this paper also contribute to the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectAlternating minimizationen
dc.subjectConvex optimizationen
dc.subjectImage deblurringen
dc.subjectOblique thresholdingen
dc.subjectTotal variation minimizationen
dc.subjectWavelet decomposition methoden
dc.titleWavelet Decomposition Method for $L_2/$/TV-Image Deblurringen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Imaging Sciencesen
dc.contributor.institutionTechnische Universitat Munchen, Munich, Germanyen
dc.contributor.institutionUC Irvine, Irvine, United Statesen
dc.contributor.institutionKarl-Franzens-Universitat Graz, Graz, Austriaen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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