Construction of Overcomplete Multiscale Dictionary of Slepian Functions on the Sphere

Abstract

We construct an overcomplete and multiscale dictionary of bandlimited Slepian functions on the sphere. Slepian functions are the bandlimited eigenfunctions obtained by solving spatial-spectral concentration problem on the sphere. To this end, we develop the hierarchical equal area iso-latitude iso-longitude pixelization (HEALLPix) scheme for hierarchical partitioning of the sphere into equal area sub-regions called pixels and present its quaternary tree structure. We then solve the concentration problem of finding bandlimited functions with maximal energy concentration in the given spatial region for each pixel and use these spatially concentrated bandlimited functions as dictionary elements. We analyze the span of the dictionary elements and their mutual coherence and show that the dictionary spans the space of bandlimited functions which are optimally (energy) concentrated within a pixel on the sphere with most of its elements exhibiting negligibly small mutual coherence. Hence, the proposed dictionary is a significant tool for use in multi-resolution analysis and sparse reconstruction of signals on the sphere.