On the Use of Antipodal Optimal Dimensionality Sampling Scheme on the Sphere for Recovering Intra-Voxel Fibre Structure in Diffusion MRI

In diffusion magnetic resonance imaging (dMRI), the diffusion signal can be reconstructed from measurements collected on single or multiple spheres in \(\boldsymbol{q}\)-space using a spherical harmonic expansion. The number of measurements that can be acquired is severely limited and should be as small as possible. Previous sampling schemes have focused on using antipodal symmetry to reduce the number of samples and uniform sampling to achieve rotationally invariant reconstruction accuracy, but do not allow for an accurate or computationally efficient spherical harmonic transform (SHT). The recently proposed antipodal optimal dimensionality sampling scheme on the sphere requires the minimum number of samples, equal to the number of degrees of freedom for the representation of the antipodal symmetric band-limited diffusion signal in the spherical harmonic domain. In addition, it allows for the accurate and efficient computation of the SHT. In this work, we evaluate the use of this recently proposed scheme for the reconstruction of the diffusion signal and subsequent intra-voxel fibre structure estimation in dMRI. We show, through numerical experiments, that the use of this sampling scheme allows accurate and computationally efficient reconstruction of the diffusion signal, and improved estimation of intra-voxel fibre structure, in comparison to the antipodal electrostatic repulsion and spherical code sampling schemes with the same number of samples. We also demonstrate that it achieves rotationally invariant reconstruction accuracy to the same extent as the other two sampling schemes.