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Article: Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization

TitleEfficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization
Authors
KeywordsImage restoration
Regularization
Nonsmooth and nonconvex optimization
Continuation
Interior point method
Issue Date2008
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siims.php
Citation
SIAM Journal on Imaging Sciences, 2008, v. 1 n. 1, p. 2-25 How to Cite?
AbstractWe consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies {Jε}, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each Jε is nonsmooth and is expressed as the sum of an l1 regularization term and a smooth nonconvex function. Furthermore, the local minimization of each Jε is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.
Persistent Identifierhttp://hdl.handle.net/10722/75167
ISSN
2015 Impact Factor: 2.687
2015 SCImago Journal Rankings: 2.052
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorNikolova, Men_HK
dc.contributor.authorNg, MKen_HK
dc.contributor.authorZhang, Sen_HK
dc.contributor.authorChing, WKen_HK
dc.date.accessioned2010-09-06T07:08:35Z-
dc.date.available2010-09-06T07:08:35Z-
dc.date.issued2008en_HK
dc.identifier.citationSIAM Journal on Imaging Sciences, 2008, v. 1 n. 1, p. 2-25en_HK
dc.identifier.issn1936-4954-
dc.identifier.urihttp://hdl.handle.net/10722/75167-
dc.description.abstractWe consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies {Jε}, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each Jε is nonsmooth and is expressed as the sum of an l1 regularization term and a smooth nonconvex function. Furthermore, the local minimization of each Jε is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.-
dc.languageengen_HK
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siims.php-
dc.relation.ispartofSIAM Journal on Imaging Sciencesen_HK
dc.rightsSIAM Journal on Imaging Sciences. Copyright © Society for Industrial and Applied Mathematics.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectImage restoration-
dc.subjectRegularization-
dc.subjectNonsmooth and nonconvex optimization-
dc.subjectContinuation-
dc.subjectInterior point method-
dc.titleEfficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimizationen_HK
dc.typeArticleen_HK
dc.identifier.emailNg, MK: kkpong@hku.hken_HK
dc.identifier.emailChing, WK: wching@hkucc.hku.hken_HK
dc.identifier.authorityChing, WK=rp00679en_HK
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.1137/070692285-
dc.identifier.hkuros141909en_HK
dc.identifier.volume1-
dc.identifier.issue1-
dc.identifier.spage2-
dc.identifier.epage25-
dc.identifier.isiWOS:000207567300001-
dc.publisher.placeUnited States-

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