Provable Deterministic Sampling Strategies for Fourier Encoding in Magnetic Resonance Imaging

Abstract

There is a constant demand for acceleration of magnetic resonance (MR) imaging to alleviate motion artifacts, and more generally, due to the time sensitive nature of certain imaging applications. One way to speed up MR imaging is to reduce the image acquisition time by subsampling the data domain (k-space). There are several methods available to reconstruct the MR image from undersampled k-space, e.g., those based on the theory of Compressive Sensing. Standard methods employ random undersampling of k-space; however, these methods provide only probabilistic guarantees on the quality of reconstruction. We present a method to reconstruct MR images from deterministically undersampled k-space, and provide analytical guarantees on the quality of MR image reconstruction. Our approach uses sampling constructions formed by deterministic selection of rows of Fourier matrices; coupled with sparsity assumptions on the finite differences of MR images, we formulate the reconstruction problem as a Total Variation (TV) minimization problem. We demonstrate the utility of our TV minimization based approach for MR image reconstruction by reconstructing MR brain scan data, and compare our reconstructions with those obtained via random sampling. Our results suggest that accurate MR reconstructions are possible by deterministic undersampling the k-space, and the quality of deterministic reconstructions are on par with those of reconstructions from randomly acquired data.