PRIMAL AND DUAL ALTERNATING DIRECTION METHODS OF MULTIPLIERS FOR COMPRESSIVE SENSING IMAGE RECONSTRUCTION CORRUPTED BY IMPLUSIVE NOISE

摘要

It is known that, the compressive sensing theories offered the possibilities of accurately recon-structing images from highly undersampled data and simultaneously correcting the possible noise. It is also known that, if the undersampled data is corrupted by white Gaussian noise, the state-of-the-art solver RecPF can be employed successfully, but for impulsive noise, it is not quite suitable. As a remedy, in this paper, we concentrate on the impulsive noise case, and particularly focus on the cost function being the sum of a total variation regularized term, an l(1)-norm regularized term, and an l(1)-norm measured data fidelity term. However, this cost function cause a little more challenges for minimizing because of these non-differentiable terms. To tackle this difficulty, this paper presents a pair of efficient algorithms from two different aspects: (1) employing an alternating direction method of multipliers (ADMM) to solve the primal problem in a straightforward way; (2) proposing an ADMM to the dual problem whose objective function contains four blocks of variables and three blocks of non-differentiable terms. In dual cases, a symmetric Gauss-Seidel technique is employed to decompose the involved bigger subproblem into some smaller ones. It should be emphasized that the most remarkable feature of our proposed algorithms is that each subproblem is easily implementable by making full use of the favorable structures, such as the fast Fourier transforms, the proximal mapping and the Moreau decomposition of ti-norm function. We do extensive numerical sim-ulations using some magnetic resonance images which demonstrate that the algorithm based on dual model is evidently efficient.