A novel sampling theorem on the sphere

TL;DR

This paper introduces a novel sampling theorem on the sphere achieved by a periodic extension to the torus, enabling exact forward and inverse spherical harmonic transforms with fewer samples than many existing equiangular schemes. The method attains about 2L^2 samples versus ~4L^2 for common equiangular or Gauss-Legendre approaches, while preserving O(L^3) asymptotic complexity and leveraging FFTs to reduce practical run-time. It handles both scalar and spin signals without precomputation, and includes a public implementation with numerical demonstrations up to L=4096, including stability advantages over competing methods. The work discusses a new quadrature rule, on-the-fly Wigner function computation, and potential applications in cosmology and compressive sensing, as well as low-bandwidth diffusion MRI.

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.

Figures (5)

• Figure 1: Exact quadrature weights corresponding to our sampling theorem. In the left column of panels the weights $v(\theta_t)$ (red squares) defined on $[0, 2\pi)$ and the quadrature weights $q(\theta_t)$ (yellow diamonds) defined on $[0, \pi]$ are plotted. These values are compared to samples of the function defined by $\sin(\theta)$ on $[0, \pi)$ and zero on $[\pi, 2\pi)$ (solid black line). In the right column of panels the difference between the quadrature weights $q(\theta_t)$ and $\sin(\theta)$ are plotted.
• Figure 2: Number of samples ${N}$ required to represent exactly a signal on the sphere of band-limit ${L}$ for the following sampling theorems: Gauss-Legendre sampling theorem (blue/dot-dashed line); Driscoll & Healy sampling theorem (green/dashed line); and the sampling theorem developed in this article (red/solid line). The inset shows very low band-limits, where the difference between Gauss-Legendre sampling and our sampling can have a large impact.
• Figure 3: Sampling schemes for the exact representation of a signal band-limited at ${L}=12$. Sample positions are shown for the following sampling theorems: Gauss-Legendre sampling theorem (blue dots); Driscoll & Healy sampling theorem (green dots); and the sampling theorem developed in this article (red dots). Notice that the Driscoll & Healy sampling theorem requires approximately twice as many samples on the sphere as the alternative samplings.
• Figure 4: Numerical accuracy of the algorithms implementing the following sampling theorems: our optimised implementation of the Gauss-Legendre sampling theorem (blue/dot-dashed line); the semi-naive algorithm in SpharmonicKit implementing the Driscoll & Healy sampling theorem (green/dashed line); and our algorithms implementing the sampling theorem developed in this article (red/solid line). $\mathcal{O}({L})$ scaling is shown by the heavy black/solid line. The algorithms implementing the Gauss-Legendre and Driscoll & Healy sampling theorems go unstable between ${L}=1024$ and ${L}=2048$, due to the enforced use of the pointwise three-term Wigner recursion. For the Gauss-Legendre and our sampling theorems, which both support spin transforms, the maximum absolute error $\epsilon$ is averaged over complex signals of spin $s\in\{0,2,10\}$ and a real spin $s=0$ signal, with one standard deviation error bars shown (in most cases differences are very small and error bars cannot be seen easily). Note that for these cases the maximum absolute error is identical (to statistical noise) for transforms of real and complex signals of different spin.
• Figure 5: Computation time of the algorithms implementing the following sampling theorems: our optimised implementation of the Gauss-Legendre sampling theorem (blue/dot-dashed line); the semi-naive algorithm in SpharmonicKit implementing the Driscoll & Healy sampling theorem (green/dashed line); and our algorithms implementing the sampling theorem developed in this article (red/solid line). $\mathcal{O}({L}^3)$ scaling is shown by the heavy black/solid line. The algorithms implementing the Gauss-Legendre and Driscoll & Healy sampling theorems go unstable between ${L}=1024$ and ${L}=2048$, due to the enforced use of the pointwise three-term Wigner recursion. For the Gauss-Legendre and our sampling theorems, which both support spin transforms, the computation time $\tau$ (seconds) is averaged over complex signals of spin $s\in\{0,2,10\}$, with one standard deviation error bars shown (in most cases differences are very small and error bars cannot be seen easily). Note that for these cases the computation time is identical (to statistical noise) for transforms of signals of different spin.