A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging

In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O (1/ N ) in finite dimensions for the complete class of problems. We further show accelerations of the proposed...

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Bibliographic Details
Published in	Journal of mathematical imaging and vision Vol. 40; no. 1; pp. 120 - 145
Main Authors	Chambolle, Antonin, Pock, Thomas
Format	Journal Article
Language	English
Published	Boston Springer US 01.05.2011
Subjects	Applications of Mathematics Computer Science Image Processing and Computer Vision Mathematical Methods in Physics Signal,Image and Speech Processing Reconstruction Total variation Dual approaches Inverse problems Image Convex optimization
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Abstract	In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O (1/ N ) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O (1/ N 2 ) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O ( ω N ) for some ω ∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.
AbstractList	In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O (1/ N ) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O (1/ N 2 ) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O ( ω N ) for some ω ∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.
Author	Pock, Thomas Chambolle, Antonin
Author_xml	– sequence: 1 givenname: Antonin surname: Chambolle fullname: Chambolle, Antonin email: antonin.chambolle@cmap.polytechnique.fr organization: CMAP, Ecole Polytechnique, CNRS – sequence: 2 givenname: Thomas surname: Pock fullname: Pock, Thomas organization: Institute for Computer Graphics and Vision, Graz University of Technology
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Cites_doi	10.1007/s10957-009-9539-y 10.1007/s10107-004-0552-5 10.1137/0314056 10.1109/WVM.1989.47097 10.1137/080716542 10.1137/0716071 10.1137/S1052623403425629 10.1007/BF01581204 10.1002/cpa.3160420503 10.1137/05064182X 10.1007/978-3-540-74936-3_22 10.1109/MCSE.2010.14 10.1016/0167-2789(92)90242-F 10.1007/11585978_10 10.1007/BF00938486 10.1109/ICIP.2009.5413571 10.1007/978-1-4613-9940-7_3 10.1007/978-1-4419-8853-9 10.1007/s10851-010-0251-1 10.1007/BF02921771 10.2140/pjm.1994.166.55
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Keywords	Reconstruction Total variation Dual approaches Inverse problems Image Convex optimization
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Snippet	In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence...
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SubjectTerms	Applications of Mathematics Computer Science Image Processing and Computer Vision Mathematical Methods in Physics Signal,Image and Speech Processing
Title	A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
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