2011/09/01 by J. D. McEwen, Jason D. McEwen, Yves Wiaux +1 · 2 citations
Computer Science · Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Computation #Computer science #Geometry #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #NMR spectroscopy and applications #Nyquist–Shannon sampling theorem #Precomputation #Sampling (signal processing) #Scalar (mathematics) #Seismic Imaging and Inversion Techniques #astro-ph.IM #cs.IT #math.IT
paper · pdf · doi:10.1109/tsp.2011.2166394
published as IEEE Trans. Signal Process. 59 (2011) 5876-5887 · 13 pages, 5 figures, accepted for publication by IEEE Trans. Sig. Proc.; We make our Spin Spherical Harmonic Transform (SSHT) package available publicly from http://www.ssht.org.uk
openalex publication_date 2011/09/01 · arxiv created 2011/10/28 · arxiv updated 2012/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 atL, all sampling theorems on the sphere requireO(L2) 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 asO(L3), 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.