Fast directional spatially localized spherical harmonic transform

TL;DR

The paper introduces a fast, directional spherical harmonic transform that enhances localized, directional analysis of signals on the sphere, with applications demonstrated on synthetic and planetary data.

Contribution

It extends the SLSHT to include asymmetric windows, provides an inversion formula, and develops a fast algorithm for efficient computation.

Findings

Effective directional analysis demonstrated on synthetic data.

Successful application to Mars topographic data.

Fast algorithm with verified numerical accuracy.

Abstract

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.