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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. 2-25 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 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 Identifier | http://hdl.handle.net/10722/75167 |
ISSN | 2023 Impact Factor: 2.1 2023 SCImago Journal Rankings: 0.960 |
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 | 2010-09-06T07:08:35Z | - |
dc.date.available | 2010-09-06T07:08:35Z | - |
dc.date.issued | 2008 | en_HK |
dc.identifier.citation | SIAM Journal on Imaging Sciences, 2008, v. 1 n. 1, p. 2-25 | en_HK |
dc.identifier.issn | 1936-4954 | - |
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 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.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 | © 2008 Society for Industrial and Applied Mathematics. First Published in SIAM Journal on Imaging Sciences in volume 1, issue 1, published by the Society for Industrial and Applied Mathematics (SIAM). | - |
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.scopus | eid_2-s2.0-84907772842 | - |
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 | - |
dc.identifier.issnl | 1936-4954 | - |