dc.contributor.author Fornasier, M. dc.contributor.author Kim, Y. dc.contributor.author Langer, A. dc.contributor.author Schönlieb, C.-B. dc.date.accessioned 2016-02-28T06:44:41Z dc.date.available 2016-02-28T06:44:41Z dc.date.issued 2012-07-19 dc.identifier.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. dc.identifier.issn 1936-4954 dc.identifier.doi 10.1137/100819801 dc.identifier.uri http://hdl.handle.net/10754/600185 dc.description.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. dc.description.sponsorship 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. dc.publisher Society for Industrial & Applied Mathematics (SIAM) dc.subject Alternating minimization dc.subject Convex optimization dc.subject Image deblurring dc.subject Oblique thresholding dc.subject Total variation minimization dc.subject Wavelet decomposition method dc.title Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring dc.type Article dc.identifier.journal SIAM Journal on Imaging Sciences dc.contributor.institution Technische Universitat Munchen, Munich, Germany dc.contributor.institution UC Irvine, Irvine, United States dc.contributor.institution Karl-Franzens-Universitat Graz, Graz, Austria dc.contributor.institution University of Cambridge, Cambridge, United Kingdom kaust.grant.number KUK-I1-007-43 dc.date.published-online 2012-07-19 dc.date.published-print 2012-01
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