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We propose a sampling scheme on the sphere and develop a correspondingspherical harmonic transform (SHT) for the accurate reconstruction of thediffusion signal in diffusion magnetic resonance imaging (dMRI). By exploitingthe antipodal symmetry, we design a sampling scheme that requires the optimalnumber of samples on the sphere, equal to the degrees of freedom required torepresent the antipodally symmetric band-limited diffusion signal in thespectral (spherical harmonic) domain. Compared with existing sampling schemeson the sphere that allow for the accurate reconstruction of the diffusionsignal, the proposed sampling scheme reduces the number of samples required bya factor of two or more. We analyse the numerical accuracy of the proposed SHTand show through experiments that the proposed sampling allows for the accurateand rotationally invariant computation of the SHT to near machine precisionaccuracy.

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