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...

Full description

Saved in:
Bibliographic Details
Published inJournal of mathematical imaging and vision Vol. 40; no. 1; pp. 120 - 145
Main Authors Chambolle, Antonin, Pock, Thomas
Format Journal Article
LanguageEnglish
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
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:0924-9907
1573-7683
DOI:10.1007/s10851-010-0251-1