An Optimal-Dimensionality Sampling for Spin-$s$Functions on the Sphere

摘要

For the representation of spin-n$s$nband-limited functions on the sphere, we propose a sampling scheme with optimal number of samples equal to the number of degrees of freedom of the function in harmonic space. In comparison to the existing sampling designs, which requiren${sim }2L^2$nsamples for the representation of spin-n$s$nfunctions band-limited atn$L$n, the proposed scheme requiresn$N_o=L^2-s^2$nsamples for the accurate computation of the spin-n$s$nspherical harmonic transform (n$s$n-SHT). For the proposed sampling scheme, we also develop a method to compute then$s$n-SHT. We place the samples in our design scheme such that the matrices involved in the computation ofn$s$n-SHT are well-conditioned. We also present a multipassn$s$n-SHT to improve the accuracy of the transform. We also show the proposed sampling design exhibits superior geometrical properties compared to existing equiangular and Gauss–Legendre sampling schemes, and enables accurate computation of then$s$n-SHT corroborated through numerical experiments.