Overlapping domain decomposition methods for total variation denoising

Andreas Langer; Fernando Gaspoz

doi:10.1137/18M1173782

Overlapping domain decomposition methods for total variation denoising

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Sammanfattning

Alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented. Their derivation is based on the predual formulation of the total variation minimization problem. In particular, the predual total variation minimization problem is decomposed into overlapping domains yielding subdomain problems in the respective dual space. Subsequently these subdomain problems are again dualized, forming a splitting algorithm for the original total variation minimization problem. The convergence of the proposed domain decomposition methods to a solution of the global problem is proved. In contrast to other works, the analysis is carried out in an infinite dimensional setting. Numerical experiments are shown to support the theoretical results and to demonstrate the effectiveness of the algorithms.

Originalspråk	engelska
Sidor (från-till)	1411-1444
Antal sidor	34
Tidskrift	SIAM Journal on Numerical Analysis
Volym	57
Nummer	3
DOI	https://doi.org/10.1137/18M1173782
Status	Published - 2019
Externt publicerad	Ja

Bibliografisk information

Funding Information:
\ast Received by the editors March 5, 2018; accepted for publication (in revised form) March 21, 2019; published electronically June 25, 2019. http://www.siam.org/journals/sinum/57-3/M117378.html Funding: The work of the authors was partially supported by the Ministerium fur Wissenschaft, Forschung und Kunst Baden-Wurttemberg (Az: 7533.-30-10/56/1) through the RISC-project ``Au-tomatische Erkennung von bewegten Objekten in hochauflosenden Bildsequenzen mittels neuer Ge-bietszerlegungsverfahren."" \dagger Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany ([email protected]). \ddagger Fakult\a"t fu\"r Mathematik, Technische Universit\a"t Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany ([email protected]).

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Copyright © by SIAM.

Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

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title = "Overlapping domain decomposition methods for total variation denoising",

abstract = "Alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented. Their derivation is based on the predual formulation of the total variation minimization problem. In particular, the predual total variation minimization problem is decomposed into overlapping domains yielding subdomain problems in the respective dual space. Subsequently these subdomain problems are again dualized, forming a splitting algorithm for the original total variation minimization problem. The convergence of the proposed domain decomposition methods to a solution of the global problem is proved. In contrast to other works, the analysis is carried out in an infinite dimensional setting. Numerical experiments are shown to support the theoretical results and to demonstrate the effectiveness of the algorithms.",

keywords = "Convergence analysis, Convex optimization, Domain decomposition, Image restoration, Locally weighted total variation, Subspace correction, Total variation minimization",

author = "Andreas Langer and Fernando Gaspoz",

note = "Funding Information: \ast Received by the editors March 5, 2018; accepted for publication (in revised form) March 21, 2019; published electronically June 25, 2019. http://www.siam.org/journals/sinum/57-3/M117378.html Funding: The work of the authors was partially supported by the Ministerium fur Wissenschaft, Forschung und Kunst Baden-Wurttemberg (Az: 7533.-30-10/56/1) through the RISC-project ``Au-tomatische Erkennung von bewegten Objekten in hochauflosenden Bildsequenzen mittels neuer Ge-bietszerlegungsverfahren.{"}{"} \dagger Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany ([email protected]). \ddagger Fakult\a{"}t fu\{"}r Mathematik, Technische Universit\a{"}t Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany ([email protected]). Publisher Copyright: Copyright {\textcopyright} by SIAM. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.",

year = "2019",

doi = "10.1137/18M1173782",

language = "English",

volume = "57",

pages = "1411--1444",

journal = "SIAM Journal on Numerical Analysis",

issn = "0036-1429",

publisher = "Society for Industrial and Applied Mathematics",

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AU - Langer, Andreas

AU - Gaspoz, Fernando

N1 - Funding Information: \ast Received by the editors March 5, 2018; accepted for publication (in revised form) March 21, 2019; published electronically June 25, 2019. http://www.siam.org/journals/sinum/57-3/M117378.html Funding: The work of the authors was partially supported by the Ministerium fur Wissenschaft, Forschung und Kunst Baden-Wurttemberg (Az: 7533.-30-10/56/1) through the RISC-project ``Au-tomatische Erkennung von bewegten Objekten in hochauflosenden Bildsequenzen mittels neuer Ge-bietszerlegungsverfahren."" \dagger Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany ([email protected]). \ddagger Fakult\a"t fu\"r Mathematik, Technische Universit\a"t Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany ([email protected]). Publisher Copyright: Copyright © by SIAM. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - Alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented. Their derivation is based on the predual formulation of the total variation minimization problem. In particular, the predual total variation minimization problem is decomposed into overlapping domains yielding subdomain problems in the respective dual space. Subsequently these subdomain problems are again dualized, forming a splitting algorithm for the original total variation minimization problem. The convergence of the proposed domain decomposition methods to a solution of the global problem is proved. In contrast to other works, the analysis is carried out in an infinite dimensional setting. Numerical experiments are shown to support the theoretical results and to demonstrate the effectiveness of the algorithms.

AB - Alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented. Their derivation is based on the predual formulation of the total variation minimization problem. In particular, the predual total variation minimization problem is decomposed into overlapping domains yielding subdomain problems in the respective dual space. Subsequently these subdomain problems are again dualized, forming a splitting algorithm for the original total variation minimization problem. The convergence of the proposed domain decomposition methods to a solution of the global problem is proved. In contrast to other works, the analysis is carried out in an infinite dimensional setting. Numerical experiments are shown to support the theoretical results and to demonstrate the effectiveness of the algorithms.

KW - Convergence analysis

KW - Convex optimization

KW - Domain decomposition

KW - Image restoration

KW - Locally weighted total variation

KW - Subspace correction

KW - Total variation minimization

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JO - SIAM Journal on Numerical Analysis

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Overlapping domain decomposition methods for total variation denoising

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