Slepian Spatial-Spectral Concentration Problem on the Sphere: Analytical Formulation for Limited Colatitude-Longitude Spatial Region

TL;DR

This paper develops an analytical approach to the Slepian spatial-spectral concentration problem on the sphere for limited colatitude-longitude regions, enabling efficient localized spectral analysis and sparse signal representation.

Contribution

It introduces an analytical formulation for the Slepian problem on the sphere for specific regions and extends it to unions of rotated subregions, improving computational efficiency.

Findings

The formulation is computationally efficient due to symmetry exploitation.

Numerical examples demonstrate sparse representation of localized signals.

The method supports localized spectral analysis on the sphere.

Abstract

In this paper, we develop an analytical formulation for the Slepian spatial-spectral concentration problem on the sphere for a limited colatitude-longitude spatial region on the sphere, defined as the Cartesian product of a range of positive colatitudes and longitudes. The solution of the Slepian problem is a set of functions that are optimally concentrated and orthogonal within a spatial or spectral region. These properties make them useful for applications where measurements are taken within a spatially limited region of the sphere and/or a signal is only to be analyzed within a region of the sphere. To support localized spectral/spatial analysis, and estimation and sparse representation of localized data in these applications, we exploit the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the Slepian concentration problem for a limited colatitude-longitude spatial region. We also extend the analytical formulation for spatial regions that are comprised of a union of rotated limited colatitude-longitude subregions. By exploiting various symmetries of the proposed formulation, we design a computationally efficient algorithm for the implementation of the proposed analytical formulation. Such a reduction in computation time is demonstrated through numerical experiments. We present illustrations of our results with the help of numerical examples and show that the representation of a spatially concentrated signal is indeed sparse in the Slepian basis.