An Optimal Dimensionality Sampling Scheme on the Sphere for Antipodal Signals In Diffusion Magnetic Resonance Imaging

TL;DR

This paper introduces an optimal sampling scheme on the sphere for diffusion MRI signals that leverages antipodal symmetry, reducing sample requirements and enabling highly accurate, rotationally invariant spherical harmonic transforms.

Contribution

It presents a novel sampling scheme that minimizes the number of samples needed for antipodally symmetric signals, improving efficiency over existing methods.

Findings

Reduces sample count by a factor of two or more.

Achieves near machine precision accuracy in spherical harmonic transforms.

Provides a rotationally invariant and accurate reconstruction of diffusion signals.

Abstract

We propose a sampling scheme on the sphere and develop a corresponding spherical harmonic transform (SHT) for the accurate reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI). By exploiting the antipodal symmetry, we design a sampling scheme that requires the optimal number of samples on the sphere, equal to the degrees of freedom required to represent the antipodally symmetric band-limited diffusion signal in the spectral (spherical harmonic) domain. Compared with existing sampling schemes on the sphere that allow for the accurate reconstruction of the diffusion signal, the proposed sampling scheme reduces the number of samples required by a factor of two or more. We analyse the numerical accuracy of the proposed SHT and show through experiments that the proposed sampling allows for the accurate and rotationally invariant computation of the SHT to near machine precision accuracy.