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Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

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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.

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  1. University of Mannheim, Mannheim, Germany

    Simon Setzer

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Setzer, S. Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing. Int J Comput Vis 92, 265–280 (2011). https://doi.org/10.1007/s11263-010-0357-3

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