A fast compressed sensing approach to 3D MR image reconstruction.

The problem of high-resolution image volume reconstruction from reduced frequency acquisition sequences has drawn significant attention from the scientific community because of its practical importance in medical diagnosis. To address this issue, several reconstruction strategies have been recently proposed, which aim to recover the missing information either by exploiting the spatio-temporal correlations of the image series, or by imposing suitable constraints on the reconstructed image volume. The main contribution of this paper is to combine both these strategies in a compressed sensing framework by exploiting the gradient sparsity of the image volume. The resulting constrained 3D minimization problem is then solved using a penalized forward-backward splitting approach that leads to a convergent iterative two-step procedure. In the first step, the updating rule accords with the sequential nature of the data acquisitions, in the second step a truly 3D filtering strategy exploits the spatio-temporal correlations of the image sequences. The resulting NFCS-3D algorithm is very general and suitable for several kinds of medical image reconstruction problems. Moreover, it is fast, stable and yields very good reconstructions, even in the case of highly undersampled image sequences. The results of several numerical experiments highlight the optimal performance of the proposed algorithm and confirm that it is competitive with state of the art algorithms.