An Antipodally Symmetric Optimal Dimensionality Sampling on the Sphere

Abstract

We propose an antipodally symmetric sampling scheme of optimal dimensionality for the sampling of band-limited signals. The proposed scheme takes ~L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> number of samples for the sampling of spherical signal of band-limit L and the accurate computation of its spherical harmonic transform (SHT). Since the number of samples are asymptotically equal to the degrees of freedom of the signal in harmonic space, the proposed scheme attains optimal spatial dimensionality. We also formulate the SHT associated with proposed sampling scheme. We employ the antipodal symmetry of the sampling points that is exploited to separate the signal into antipodally symmetric and asymmetric signals due to which the signal splits in harmonic space into the signals of even and odd spherical harmonic degrees. The exploitation of this splitting in the formulation of the SHT makes our method computationally efficient by a factor of four in comparison with the existing methods developed for sampling schemes that attain optimal spatial dimensionality. We also analyse the numerical accuracy of the proposed SHT by conducting numerical experiments and show that the proposed sampling and its associated SHT enable accurate signal reconstruction for band-limits in the range 15 ≤ L ≤ 127.