Efficient Sampling and Accurate Reconstruction of Order-Limited Spherical Signals

Abstract

In this work, we propose an efficient sampling design on the sphere for sampling order-limited spherical signals. The current sampling schemes on the sphere do not take into account the plausible rotational symmetry that is naturally exhibited by signals in many applications (e.g., climate data and antenna radiation pattern). These nearly rotationally symmetric signals have negligible dependence on the spherical harmonics of higher orders, and therefore can be modeled as order-limited signals. For sampling of spherical signals band-limited at $L$ and order-limited at $M$, we present a sampling design for accurate computation of spherical harmonic transform and accurate signal reconstruction. The proposed sampling scheme achieves optimal dimensionality, requiring $M^{2} + (L-M)(2M+1)$ samples, which match the degrees of freedom in harmonic space.