An Optimal-Dimensionality Sampling Scheme on the Sphere with Fast Spherical Harmonic Transforms

Abstract

We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band-limited at $L$ using only $L^2$ samples. We obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space. The number of samples required in our scheme is a factor of two or four fewer than existing techniques, which require either $2L^2$ or $4L^2$ samples. We note, however, that we do not recover a sampling theorem on the sphere, where spherical harmonic transforms are theoretically exact. Nevertheless, we achieve high accuracy even for very large band-limits. For our optimal-dimensionality sampling scheme, we develop a fast and accurate algorithm to compute the spherical harmonic transform (and inverse), with computational complexity comparable with existing schemes in practice. We conduct numerical experiments to study in detail the stability, accuracy and computational complexity of the proposed transforms. We also highlight the advantages of the proposed sampling scheme and associated transforms in the context of potential applications.

14 pages, 11 figures

References in corpus (7)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Spherical Needlets for CMB Data Analysis
• Minimum-variance multitaper spectral estimation on the sphere
• Exact reconstruction with directional wavelets on the sphere
• Libsharp - spherical harmonic transforms revisited
• Fast and Exact Spin-s Spherical Harmonic Transforms
• Sparse image reconstruction on the sphere: implications of a new sampling theorem

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