In Magnetic Resonance Imaging (MRI) studies, for clinical applications and for research as well, reduction of scanning time is an essential issue. This time reduction could be obtained by using fast acquisition sequences, such as EPI and spiral k-space trajectories, and by acquiring less data, this being possible based on the new sampling theories that gave rise to the so called Compressed Sampling (CS for short). However the main assumption for the application of CS to Fourier data is that magnitude and phase are both sparse in some given domain. This assumption is not always true for fast acquisition sequences because of the non-homogeneities of the main magnetic field. In this article we propose a new model for MRI with different regularization penalties for magnitude and phase. Magnitude regularization exploits the sparsity assumption on the signal and the suggested penalty for phase takes into account its smoothness. We show results of numerical experiments with simulated data.
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