A Fast algorithm for Non-Convex optimization in highly under-sampled MRI.

This talk concerns a fast and efficient method for the reconstruction of Magnetic Resonance Images (MRI) from severely under-sampled data. Reduced sampled acquisitions are employed in MRI when fast imaging is essential to improve patient care and reduce the costs. The aim is to reconstruct MR images from highly under-sampled data in order to reduce the amount of acquired data without degrading the reconstructed images. Following the Compressed Sensing theory, the problem is mathematically modeled as a constrained minimization with a family of non-convex regularizing objective functions and a least squares data fit constraint. The proposed algorithm is based on an iterative scheme where the non-convex problem is substituted by a sequence of convex approximations by a reweighting scheme. Moreover the penalization parameter is automatically updated through an adaptive procedure. Each convex problem is solved by a Forward-Backward procedure, where the Backward step is solved by a Split Bregman strategy. A very efficient iterative algorithm, based on a new matrix splitting method, is introduced for the solution of the inner linear systems of the Backward steps. The results on synthetic phantoms and real images show that the algorithm is very well performing and computationally efficient when compared to the most efficient methods present in the literature.