Bregmanized Domain Decomposition for Image Restoration

Computational problems of large-scale appearing in biomedical imaging, astronomy, art restoration, and data analysis are gaining recently a lot of attention due to better hardware, higher dimensionality of images and data sets, more parameters to be measured, and an increasing number of data acquired. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favourable due to the improved enhancement of images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N > 1 smaller problems which can be solved on parallel CPUs. We discuss domain decomposition algorithms in which the N subproblems can be addressed by solving constrained minimization problems, which might be done via an iterative thresholding technique. In this paper we are interested in accelerating the computation of the solution of the subproblems by nested Bregman iterations. More precisely, we propose a Bregmanized Operator Splitting Split Bregman (BOS-SB) algorithm, which enforces the constraint by introducing a Bregman iteration that is then solved by the Split Bregman strategy. In average, this new solution technique is three times faster than the iterative oblique thresholding, which was currently used in domain decomposition methods for total variation minimization.