A convergence proof of the split Bregman method for regularized least-squares problems

The split Bregman (SB) method [14] is a fast splitting-based algorithm that solves image reconstruction problems with general l1, e.g., total-variation (TV) and compressed sensing (CS), regularizations by introducing a single variable split to decouple the data-fitting term and the regularization term, yielding simple subproblems that are separable (or partially separable) and easy to minimize. Several convergence proofs have been proposed [2, 9, 20], and these proofs either impose a “full column rank” assumption to the split or assume exact updates in all subproblems. However, these assumptions are impractical in many applications such as parallel magnetic resonance (MR) and X-ray computed tomography (CT) image reconstructions [3, 4, 14, 19], where the inner leastsquares problem usually cannot be solved efficiently due to the highly shift-variant Hessian. In this paper, we show that when the data-fitting term is quadratic, e.g., in image restoration problems with Gaussian noise, the SB method is a convergent alternating direction method of multipliers (ADMM) [1, 8, 12, 13], and a straightforward convergence proof with inexact updates is given using [8, Theorem 8]. Furthermore, since the SB method is just a special case of an ADMM algorithm, it seems likely that the ADMM algorithm will be faster than the SB method if the augmented Largangian (AL) penalty parameters are selected appropriately. To have a concrete example, we conduct a convergence rate analysis of the ADMM algorithm with two split variables (the SB method is just a special case of the two-split ADMM algorithm) for image restoration problems with quadratic data-fitting term and regularization term. According to our analysis, we can show that the two-split ADMM algorithm can be faster than the SB method if the AL penalty parameter of the SB method is suboptimal. Numerical experiments were conducted to verify our analysis.