Overcomplete Multiscale Dictionary of Slepian Functions for HEALPix on the Sphere

We present a framework for exact analytical computation of bandlimited Slepian functions for Hierarchical Equal Area iso-Latitude Pixelization (HEALPix) scheme on the sphere. Slepian functions are bandlimited eigenfunctions obtained by solving the spatial-spectral concentration problem on the sphere. Utilizing rotational symmetries between the HEALPix pixels, we employ Wigner-
$D$
functions to efficiently compute the bandlimited Slepian functions at different resolutions of the HEALPix partitioning scheme. We present convergence criteria for the infinite series expansions involved in the analytical expressions, analyze the complexity of computing bandlimited Slepian functions and construct an overcomplete multiscale dictionary of bandlimited Slepian functions on the sphere. We show that the dictionary spans the space of bandlimited functions, which are well-optimally (energy) concentrated within a region on the sphere, and that its elements exhibit small mutual coherence. As a demonstration of the utility of the dictionary, we present localized representation of a bandlimited Earth topography map over the South American continent, and conduct performance comparison with the representations obtained from the Slepian functions for a rotationally symmetric region, centered on South America, and for the South American coastal region, extracted from the coastlines data available in
MATLAB
.