International Journal of Computer Vision

, Volume 92, Issue 3, pp 265–280

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Article

DOI: 10.1007/s11263-010-0357-3

Cite this article as:
Setzer, S. Int J Comput Vis (2011) 92: 265. doi:10.1007/s11263-010-0357-3

Abstract

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view—as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.

Keywords

Douglas-Rachford splittingForward-backward splittingBregman methodsAugmented Lagrangian methodAlternating split Bregman algorithmImage denoising

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of MannheimMannheimGermany