Wavelet decomposition method for L2/TV-image deblurring
Fornasier, M. ; Kim, Y. ; Langer, A. LU
and Schönlieb, C. B.
(2012)
In SIAM Journal on Imaging Sciences
5(3).
p.857-885
- 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 L2/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... (More)
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 L2/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.
(Less)
- author
- Fornasier, M.
; Kim, Y.
; Langer, A.
LU
and Schönlieb, C. B.
- publishing date
- 2012
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Alternating minimization, Convex optimization, Image deblurring, Oblique thresholding, Total variation minimization, Wavelet decomposition method
- in
- SIAM Journal on Imaging Sciences
- volume
- 5
- issue
- 3
- pages
- 29 pages
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- scopus:84867163734
- ISSN
- 1936-4954
- DOI
- 10.1137/100819801
- language
- English
- LU publication?
- no
- additional info
- Copyright: Copyright 2019 Elsevier B.V., All rights reserved.
- id
- 6f6ef39c-e4b9-4008-8d1c-c9ea41e60e73
- date added to LUP
- 2021-03-15 22:35:42
- date last changed
- 2025-10-14 09:29:33
@article{6f6ef39c-e4b9-4008-8d1c-c9ea41e60e73,
abstract = {{<p>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<sub>2</sub>/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.</p>}},
author = {{Fornasier, M. and Kim, Y. and Langer, A. and Schönlieb, C. B.}},
issn = {{1936-4954}},
keywords = {{Alternating minimization; Convex optimization; Image deblurring; Oblique thresholding; Total variation minimization; Wavelet decomposition method}},
language = {{eng}},
number = {{3}},
pages = {{857--885}},
publisher = {{Society for Industrial and Applied Mathematics}},
series = {{SIAM Journal on Imaging Sciences}},
title = {{Wavelet decomposition method for L<sub>2</sub>/TV-image deblurring}},
url = {{http://dx.doi.org/10.1137/100819801}},
doi = {{10.1137/100819801}},
volume = {{5}},
year = {{2012}},
}