On the Generation of Sampling Schemes for Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) is probably one of the most successful application fields of compressed sensing. Despite recent advances, there is still a large discrepancy between theories and most actual implementations. Overall, many important questions related to sampling theory remain open. In this paper, we attack one of them: given a set of sampling constraints (e.g., measuring Fourier coefficients along physically plausible trajectories), how to optimally design a sampling pattern? We first outline three aspects that should be carefully designed by inspecting the literature, namely admissibility, limit of the empirical measure, and coverage speed. To address them jointly, we then propose an original approach which consists of projecting a probability distribution onto a set of admissible measures. The proposed algorithm permits handling arbitrary constraints and automatically generates efficient sampling patterns for MRI as shown on realistic simulations. We achieve a 20-fold undersampling factor at very high 2D resolution (100 $\mu$m isotropic) on physically plausible sampling trajectories with a gain in SNR of 2--3 dB on reconstructed MR images as compared to more standard sampling patterns (e.g., radial, spiral).