Optimal-Dimensionality Sampling and Robust 3D Diffusion Signal Reconstruction

This paper presents single and multi-shell sampling schemes for diffusion MRI that are optimal in terms of the number of measurements whilst enabling accurate and robust reconstruction of the diffusion signal. In diffusion MRI, it is paramount that the number of samples is as small as possible in order that scan times are practical in a clinical setting. The proposed schemes use the optimal number of measurements in that the number of samples is equal to the number of degrees of freedom in the orthonormal bases used for reconstruction. Both the single and multi-shell schemes have novel reconstruction algorithms which use smaller subsystems of linear equations compared to the standard regularized least-squares method of reconstruction. The smaller matrices used in these novel reconstruction algorithms are designed to be well-conditioned, leading to improved reconstruction accuracy. Accurate and robust reconstruction is also achieved through incorporation of regularization into the novel reconstruction algorithms and using a Rician or non-central Chi noise model. We quantitatively validate our single and multi-shell schemes against standard least- squares reconstruction methods to show that they enable more accurate reconstruction when the optimal number of samples are used. Human brain data is also used to qualitatively evaluate reconstruction.