LAUSR

To support localized signal analysis on the sphere for applications in geophysics, cosmology, acoustics and beyond, we develop a framework for the analytic evaluation of the integral of spherical signals and for the analytic solution of the Slepian spatial-spectral concentration problem over simple spherical polygons. We propose a polygon right angle triangulation method for the division of a simple spherical polygon into spherical right angle triangles, which allows us to decompose the problem of integrating signals or solving the spatial-spectral concentration problem over the polygon region into sub-problems that require the evaluation of the integral of complex exponential functions over spherical right angle triangles of arbitrary orientation and position. We derive closed-form expressions for the evaluation of such integrals by appropriately choosing the rotations and using Wigner-$D$ functions. The proposed framework enables us to solve the Slepian spatial-spectral concentration problem for simple spherical polygons, resulting in bandlimited, spatially optimally concentrated basis functions for signal representation and reconstruction, localized analysis and signal modeling on the sphere. We also present convergence criterion for the infinite series expansions involved in the evaluation of the integral of complex exponential functions, and establish the validity of the proposed developments by evaluating the integral and computing the Slepian basis functions over the geographical region of Australia and the volcanic plateau of Tharsis, using the Earth and Mars topography maps respectively.