Reconstruction of Undersampled MR Data using Lp (0<p<1) Spatial Constraints

Introduction: L1 norm constrained reconstruction/compressed sensing techniques [1, 2] have recently been proposed to accelerate data acquisitions in MRI by acquiring fewer data in k-space. Image artifacts due to k-space undersampling are resolved by minimizing L1 norm of the sparse image estimate while preserving fidelity to the acquired data. Using an L1 norm constraint exploits implicit sparsity in the data [3]. More recent developments to this approach include (i) using a priori information about the signal [4,5] (ii) re-weighting the signal [6] (iii) reordering the signal estimate [7,8] which enhance signal sparsity and improve the reconstructions. In a different direction, it has been suggested that an Lp (0<p<1) constraint can obtain faithful reconstructions when undersampling is more severe [9]. Theory suggests that the minimum number of measurements required to obtain good reconstructions can be reduced as p is reduced [10]. In ref. [9] results were presented using simulations in which L0.5 constraint lead to significantly improved reconstructions over the corresponding L1 norm reconstructions. Here we test the feasibility of using Lp constraint on MR Cartesian acquisitions and compare the results to those from the more standard L1 norm constraint.