Spherical harmonic transform for minimum dimensionality regular grid sampling on the sphere

Abstract

We develop a method to compute spherical harmonic transform (SHT) of a band-limited signal on the sphere discretized over a minimum dimensionality regular sampling grid on the sphere. For the computation of SHT of a signal band-limited at L, the proposed method requires L2 number of samples on a regular grid composed of L iso-latitude rings of samples with only L samples in each ring along longitude. Since a signal band-limited at L is represented by L2 degrees of freedom in the spectral (spherical harmonic) domain, the proposed method requires the minimal number of samples for the computation of SHT. In comparison to the other schemes that require 2L - 1 samples along each iso-latitude ring, we show that the SHT can be computed, by exploiting the structure of spectral domain, from only L samples in each iso-latitude ring. We also analyse the numerical accuracy and the computational complexity of our proposed SHT for a regular grid with equiangular sampling. We demonstrate, through numerical experiments, that the proposed SHT is sufficiently accurate for band-limits of interest in diffusion magnetic resonance imaging.