Comparative analysis of geometrical properties of sampling schemes on the sphere

Abstract

In this work, we carry out the comparative analysis of the geometrical properties of the sampling schemes on the sphere. Among the sampling schemes devised on the sphere, we focus on equiangular sampling, Gauss-Legendre (GL) quadrature based sampling, optimal-dimensionality sampling, sampling points of extremal systems and spherical design as these schemes support the accurate representation of the band-limited signals. We analyse sampling efficiency, minimum geodesic distance, mesh norm, mesh ratio and Riesz s-energy for these sampling schemes. Since these sampling schemes require different number of samples for the accurate representation of a band-limited signal and therefore have different sampling efficiency, we formulate these geometrical properties to take into account the sampling efficiency for a meaningful comparison. We illustrate that the optimal dimensionality, extremal system and spherical design sampling schemes exhibit desirable geometrical properties. Among these schemes, extremal system sampling scheme has superior geometrical properties. However, the accuracy of the representation of a band-limited signal degrades with the increase in band-limit for extremal system sampling scheme, due to which we propose to use extremal point sampling scheme for small band-limits. We also propose to use optimal dimensional sampling scheme for moderate to large band-limits as it exhibits desirable geometrical properties and has the capability to accurately represent the band-limited signal.