Representing, Sampling and Reconstructing Diffusion MRI Signal in Space and Time

Abstract

This paper presents a novel 4D-basis, sampling grid and corresponding transforms for representing, sampling and reconstructing the diffusion MRI signal that is a function of both the diffusion wave vector q and diffusion time $\tau$. The vast majority of diffusion MRI sampling schemes treat diffusion time as a constant and only focus on the wave vector q, however the incorporation of diffusion time is important for estimating axon diameter and packing density. Our proposed sampling scheme is only the second to include diffusion time, and the first to provide an accurate reconstruction transform and corresponding sampling recommendations. Unlike the first scheme proposed by Fick et al., the proposed scheme requires a number of samples which is equal to the number of degrees of freedom in the proposed basis, the smallest number required for accurate reconstruction. We demonstrate the proposed basis and sampling scheme enables machine precision reconstruction accuracy of band-limited signals and high accuracy (below the noise floor) for synthetic diffusion MRI signal measurements.