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

Authors  
Keywords  Image restoration Regularization Nonsmooth and nonconvex optimization Continuation Interior point method 
Issue Date  2008 
Publisher  Society 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. 225 How to Cite? 
Abstract  We 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 datafidelity 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 primaldual 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 Identifier  http://hdl.handle.net/10722/75167 
ISSN  2015 Impact Factor: 2.687 2015 SCImago Journal Rankings: 2.052 
ISI Accession Number ID 
DC Field  Value  Language 

dc.contributor.author  Nikolova, M  en_HK 
dc.contributor.author  Ng, MK  en_HK 
dc.contributor.author  Zhang, S  en_HK 
dc.contributor.author  Ching, WK  en_HK 
dc.date.accessioned  20100906T07:08:35Z   
dc.date.available  20100906T07:08:35Z   
dc.date.issued  2008  en_HK 
dc.identifier.citation  SIAM Journal on Imaging Sciences, 2008, v. 1 n. 1, p. 225  en_HK 
dc.identifier.issn  19364954   
dc.identifier.uri  http://hdl.handle.net/10722/75167   
dc.description.abstract  We 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 datafidelity 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 primaldual 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.language  eng  en_HK 
dc.publisher  Society for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siims.php   
dc.relation.ispartof  SIAM Journal on Imaging Sciences  en_HK 
dc.rights  SIAM Journal on Imaging Sciences. Copyright © Society for Industrial and Applied Mathematics.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.subject  Image restoration   
dc.subject  Regularization   
dc.subject  Nonsmooth and nonconvex optimization   
dc.subject  Continuation   
dc.subject  Interior point method   
dc.title  Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization  en_HK 
dc.type  Article  en_HK 
dc.identifier.email  Ng, MK: kkpong@hku.hk  en_HK 
dc.identifier.email  Ching, WK: wching@hkucc.hku.hk  en_HK 
dc.identifier.authority  Ching, WK=rp00679  en_HK 
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.1137/070692285   
dc.identifier.hkuros  141909  en_HK 
dc.identifier.volume  1   
dc.identifier.issue  1   
dc.identifier.spage  2   
dc.identifier.epage  25   
dc.identifier.isi  WOS:000207567300001   
dc.publisher.place  United States   