Handle URI:
http://hdl.handle.net/10754/597694
Title:
Bregmanized Domain Decomposition for Image Restoration
Authors:
Langer, Andreas; Osher, Stanley; Schönlieb, Carola-Bibiane
Abstract:
Computational problems of large-scale data are gaining attention recently due to better hardware and hence, higher dimensionality of images and data sets acquired in applications. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favorable due to the improved enhancement of images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N > 1 smaller problems which can be solved on parallel CPUs. The N subproblems constitute constrained minimization problems, where the constraint enforces the support of the minimizer to be the respective subdomain. In this paper we discuss a fast computational algorithm to solve domain decomposition for total variation minimization. In particular, we accelerate the computation of the subproblems by nested Bregman iterations. We propose a Bregmanized Operator Splitting-Split Bregman (BOS-SB) algorithm, which enforces the restriction onto the respective subdomain by a Bregman iteration that is subsequently solved by a Split Bregman strategy. The computational performance of this new approach is discussed for its application to image inpainting and image deblurring. It turns out that the proposed new solution technique is up to three times faster than the iterative algorithm currently used in domain decomposition methods for total variation minimization. © Springer Science+Business Media, LLC 2012.
Citation:
Langer A, Osher S, Schönlieb C-B (2012) Bregmanized Domain Decomposition for Image Restoration. Journal of Scientific Computing 54: 549–576. Available: http://dx.doi.org/10.1007/s10915-012-9603-x.
Publisher:
Springer Nature
Journal:
Journal of Scientific Computing
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
22-May-2012
DOI:
10.1007/s10915-012-9603-x
Type:
Article
ISSN:
0885-7474; 1573-7691
Sponsors:
Andreas Langer acknowledges the financial support provided by the FWF project Y 432-N15 START-Preis Sparse Approximation and Optimization in High Dimensions and the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts. Stanley Osher acknowledges the NSF grants DMS0835863 and DMS0914561, the ONR grant N000140910360, and ARO Muri subs from Rice University and the University of South Carolina. Carola-Bibiane Schonlieb acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods and the Royal Society International Exchanges Award IE110314. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
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Full metadata record

DC FieldValue Language
dc.contributor.authorLanger, Andreasen
dc.contributor.authorOsher, Stanleyen
dc.contributor.authorSchönlieb, Carola-Bibianeen
dc.date.accessioned2016-02-25T12:44:33Zen
dc.date.available2016-02-25T12:44:33Zen
dc.date.issued2012-05-22en
dc.identifier.citationLanger A, Osher S, Schönlieb C-B (2012) Bregmanized Domain Decomposition for Image Restoration. Journal of Scientific Computing 54: 549–576. Available: http://dx.doi.org/10.1007/s10915-012-9603-x.en
dc.identifier.issn0885-7474en
dc.identifier.issn1573-7691en
dc.identifier.doi10.1007/s10915-012-9603-xen
dc.identifier.urihttp://hdl.handle.net/10754/597694en
dc.description.abstractComputational problems of large-scale data are gaining attention recently due to better hardware and hence, higher dimensionality of images and data sets acquired in applications. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favorable due to the improved enhancement of images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N > 1 smaller problems which can be solved on parallel CPUs. The N subproblems constitute constrained minimization problems, where the constraint enforces the support of the minimizer to be the respective subdomain. In this paper we discuss a fast computational algorithm to solve domain decomposition for total variation minimization. In particular, we accelerate the computation of the subproblems by nested Bregman iterations. We propose a Bregmanized Operator Splitting-Split Bregman (BOS-SB) algorithm, which enforces the restriction onto the respective subdomain by a Bregman iteration that is subsequently solved by a Split Bregman strategy. The computational performance of this new approach is discussed for its application to image inpainting and image deblurring. It turns out that the proposed new solution technique is up to three times faster than the iterative algorithm currently used in domain decomposition methods for total variation minimization. © Springer Science+Business Media, LLC 2012.en
dc.description.sponsorshipAndreas Langer acknowledges the financial support provided by the FWF project Y 432-N15 START-Preis Sparse Approximation and Optimization in High Dimensions and the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts. Stanley Osher acknowledges the NSF grants DMS0835863 and DMS0914561, the ONR grant N000140910360, and ARO Muri subs from Rice University and the University of South Carolina. Carola-Bibiane Schonlieb acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods and the Royal Society International Exchanges Award IE110314. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSpringer Natureen
dc.subjectBregman distanceen
dc.subjectDomain decompositionen
dc.subjectImage restorationen
dc.subjectIterative Bregman algorithmsen
dc.subjectTotal variationen
dc.titleBregmanized Domain Decomposition for Image Restorationen
dc.typeArticleen
dc.identifier.journalJournal of Scientific Computingen
dc.contributor.institutionJohann Radon Institute for Computational and Applied Mathematics, Linz, Austriaen
dc.contributor.institutionUniversity of California, Los Angeles, Los Angeles, United Statesen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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