Finite Radial Reconstruction for Magnetic Resonance Imaging: A Theoretical Study

Magnetic resonance (MR) imaging is an important imaging modality for diagnostic medicine due to its ability to visualize the soft tissue of the human body in three dimensions (3D) without ionizing radiation. MR imaging however, typically relies on measurement techniques that do not exploit the geometry of discrete Fourier space, so called k-space, where the measurements are made. In this work, we firstly present a novel k-space tiling scheme that utilizes the finite geometry of discrete Fourier space via the discrete Fourier slice theorem. This produces a sampling of k-space that is pseudo-radial, to be more noise and patient-movement tolerant, and pseudo-random, further improving robustness to noise, whilst sampling with sufficient density near the central k-space region, where the majority of power of anatomical images lies. Secondly, we introduce a stable and iterative discrete reconstruction scheme for recovering images from their limited k-space measurements (in the form discrete slices) based on the maximum likelihood expectation maximization approach. We study the performance of the proposed approach using simulated MR measurements in k-space.