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Inverse imaging covers a wide range of imaging applications, including super-resolution, deblurring, and compressive sensing. We propose a novel scheme to solve such problems by combining duality and the alternating direction method of multipliers (ADMM). In addition to a conventional ADMM process, we introduce a second one that solves the dual problem to find the estimated nontrivial lower bound of the objective function, and the related iteration results are used in turn to guide the primal iterations. We call this D-ADMM, and show that it converges to the global minimum when the regularization function is convex and the optimization problem has at least one optimizer. Furthermore, we show how the scheme can give rise to two specific algorithms, called D-ADMM-L2 and D-ADMM-TV, by having different regularization functions. We compare D-ADMM-TV with other methods on image super-resolution and demonstrate comparable or occasionally slightly better quality results. This paves the way of incorporating advanced operators and strategies designed for basic ADMM into the D-ADMM method as well to further improve the performances of those methods.

Dual Alternating Direction Method of Multipliers for Inverse Imaging