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

TL;DR

This paper introduces an optimal sampling scheme for spin-$s$ functions on the sphere, reducing sample count to the function's degrees of freedom and improving computational accuracy and geometric properties over existing methods.

Contribution

It proposes a new sampling scheme requiring fewer samples, develops a method for spin-$s$ spherical harmonic transform, and enhances transform accuracy with a multi-pass approach.

Findings

Requires $L^2 - s^2$ samples, fewer than existing methods.

Achieves well-conditioned matrices for stable computations.

Demonstrates superior geometric properties and accuracy in numerical experiments.

Abstract

For the representation of spin-$s$ band-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 require ${\sim}2L^2$ samples for the representation of spin-$s$ functions band-limited at $L$, the proposed scheme requires $N_o=L^2-s^2$ samples for the accurate computation of the spin-$s$ spherical harmonic transform~($s$-SHT). For the proposed sampling scheme, we also develop a method to compute the $s$-SHT. We place the samples in our design scheme such that the matrices involved in the computation of $s$-SHT are well-conditioned. We also present a multi-pass $s$-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 the $s$-SHT corroborated through numerical experiments.