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

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 ~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 multipass 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.