Elahi, U; Khalid, Z; Kennedy, RA; McEwen, JD; (2018) An Optimal-Dimensionality Sampling for Spin-sFunctions on the Sphere. IEEE Signal Processing Letters , 25 (10) pp. 1470-1474. 10.1109/LSP.2018.2865676.

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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.

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