TY - JOUR

T1 - Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring

AU - Fornasier, M.

AU - Kim, Y.

AU - Langer, A.

AU - Schönlieb, C.-B.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2012/7/19

Y1 - 2012/7/19

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/600185

UR - http://epubs.siam.org/doi/10.1137/100819801

UR - http://www.scopus.com/inward/record.url?scp=84867163734&partnerID=8YFLogxK

U2 - 10.1137/100819801

DO - 10.1137/100819801

M3 - Article

VL - 5

SP - 857

EP - 885

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

SN - 1936-4954

IS - 3

ER -