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In this paper we present a fast and efficient method for the reconstructionof Magnetic Resonance Images (MRI) from severely under-sampled data. From theCompressed Sensing theory we have mathematically modeled the problem as aconstrained minimization problem with a family of non-convex regularizingobjective functions depending on a parameter and a least squares data fitconstraint. We propose a fast and efficient algorithm, named Fast NonConvexReweighting (FNCR) algorithm, based on an iterative scheme where the non-convexproblem is approximated by its convex linearization and the penalizationparameter is automatically updated. The convex problem is solved by aForward-Backward procedure, where the Backward step is performed by a SplitBregman strategy. Moreover, we propose a new efficient iterative solver for thearising linear systems. We prove the convergence of the proposed FNCR method.The results on synthetic phantoms and real images show that the algorithm isvery well performing and computationally efficient, even when compared to thebest performing methods proposed in the literature.

A fast nonconvex Compressed Sensing algorithm for highly low-sampled MR images reconstruction