A Novel Sampling Theorem on the Sphere

Abstract

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 bandlimits. Finally, we highlight the advantages of our sampling theorem in the context of potential applications, notably in the field of compressive sampling.

S2LET: A code to perform fast wavelet analysis on the sphere

“…(4) and (6), for the input parameters (L, λ, J 0 ). To reconstruct signals on the sphere, by default S2LET uses the exact spherical harmonic transform of the MW sampling theorem (McEwen & Wiaux 2011) implemented in the SSHT 9 code. In this case all transforms are theoretically exact and one can analyse and synthesise real and complex signals at floating-point precision.…”

“…These examples cover multiple combinations of parameters and types of signals. S2LET requires SSHT, which implements fast and exact algorithms to perform the forward and inverse spherical harmonic transforms corresponding to the MW sampling theorem (McEwen & Wiaux 2011). SSHT in turn requires the FFTW 11 package for the computation of fast Fourier transforms.…”

“…Order of magnitude of the accuracy of the HEALPix spherical harmonic transform, averaged over the parameter N side . have been widely studied in the past (see, e.g., Reinecke 2011;Doroshkevich et al 2011;Reinecke & Seljebotn 2013), and that of the MW sampling were presented in McEwen & Wiaux (2011). We do not compile the entirety of these results here, but we have reproduced the essential results on our machine; Table 1 summarises the orders of accuracy of the HEALPix iterative spherical harmonic transform.…”

“…S2LET aims primarily to provide a fast and flexible implementation of the scale-discretised transform with exact reconstruction on the sphere using the sampling theorem of McEwen & Wiaux (2011), although it has also been extended to support some of the features of these other codes. Furthermore, particular attention has been paid in the development of S2LET to prove a user-friendly code, supporting multiple programming languages, and which is extensively documented.…”

Spin-SILC: CMB polarization component separation with spin wavelets

“…The wavelet transforms in Spin-SILC are carried out using the latest version of the S2LET 7 code (Leistedt et al 2013;McEwen et al 2015), written in C with Python wrappers. This employs SSHT 8 (McEwen & Wiaux 2011) and SO3 9 to compute spin spherical harmonics and Wigner transforms exactly and efficiently. Spin-SILC is developed from the scalar SILC 10 code (Rogers et al 2016) (which performs component separation on the temperature anisotropies of the CMB).…”