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Proximal Splitting Methods in Signal Processing

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)

Abstract

The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.

Keywords

Alternating-direction method of multipliers Backward–backward algorithm Convex optimization Denoising Douglas–Rachford algorithm  Forward–backward algorithm Frame Landweber method Iterative thresholding  Parallel computing Peaceman–Rachford algorithm Proximal algorithm Restoration and reconstruction Sparsity Splitting 

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Notes

Acknowledgements

This work was supported by the Agence Nationale de la Recherche under grants ANR-08-BLAN-0294-02 and ANR-09-EMER-004-03.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis Lions – UMR CNRS 7598UPMC Université Paris 06ParisFrance

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