Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage

We examine relations between popular variational methods in image processing and classical operator splitting methods in convex analysis. We focus on a gradient descent reprojection algorithm for image denoising and the recently proposed Split Bregman and alternating Split Bregman methods. By identifying the latter with the so-called Douglas-Rachford splitting algorithm we can guarantee its convergence. We show that for a special setting 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.

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