A novel method and fast algorithm for MR image reconstruction with significantly under-sampled data

The aim of this work is to improve the accuracy, robustness and efficiency of the compressed sensing reconstruction technique in magnetic resonance imaging. We propose a novel variational model that enforces the sparsity of the underlying image in terms of its spatial finite differences and representation with respect to a dictionary. The dictionary is trained using prior information to improve accuracy in reconstruction. In the meantime the proposed model enforces the consistency of the underlying image with acquired data by using the maximum likelihood estimator of the reconstruction error in partial $k$-space to improve the robustness to parameter selection. Moreover, a simple and fast numerical scheme is provided to solve this model. The experimental results on both synthetic and in vivo data indicate the improvement of the proposed model in preservation of fine structures, flexibility of parameter decision, and reduction of computational cost.