A novel sampling theorem on the sphere

TL;DR

This paper introduces a new sampling theorem on the sphere that reduces the number of samples needed for band-limited signals, offering faster algorithms with broad applicability and practical implementation.

Contribution

A novel sampling theorem on the sphere that requires fewer samples and provides efficient, fast algorithms applicable to scalar and spin functions without precomputation.

Findings

Requires less than half the samples of existing theorems

Algorithms are fast due to FFT-based optimizations

Applicable to high band-limits with demonstrated speed and accuracy

Abstract

We develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension. The fundamental property of any sampling theorem is the number of samples required to represent a band-limited signal. To represent exactly a signal on the sphere band-limited at L, all sampling theorems on the sphere require O(L^2) samples. However, our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical, but smaller, number of samples than the Gauss-Legendre sampling theorem. The complexity of our algorithms scale as O(L^3), however, the continual use of fast Fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably, resulting in algorithms that are fast. Furthermore, we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational complexity or computation time. We make our implementation of these algorithms available publicly and perform numerical experiments demonstrating their speed and accuracy up to very high band-limits. Finally, we highlight the advantages of our sampling theorem in the context of potential applications, notably in the field of compressive sampling.