On the computation of directional scale-discretized wavelet transforms on the sphere

TL;DR

This paper reviews and improves algorithms for computing directional scale-discretized wavelet transforms on the sphere, enabling efficient and exact analysis of spherical data with oriented structures.

Contribution

It introduces optimized, parallelized algorithms for exact spherical wavelet transforms and discusses future improvements using a new sampling theorem.

Findings

Algorithms are exact and efficient for band-limited signals

The S2DW code is now parallelized with additional optimizations

Wavelets serve as a directional generalization of needlets

Abstract

We review scale-discretized wavelets on the sphere, which are directional and allow one to probe oriented structure in data defined on the sphere. Furthermore, scale-discretized wavelets allow in practice the exact synthesis of a signal from its wavelet coefficients. We present exact and efficient algorithms to compute the scale-discretized wavelet transform of band-limited signals on the sphere. These algorithms are implemented in the publicly available S2DW code. We release a new version of S2DW that is parallelized and contains additional code optimizations. Note that scale-discretized wavelets can be viewed as a directional generalization of needlets. Finally, we outline future improvements to the algorithms presented, which can be achieved by exploiting a new sampling theorem on the sphere developed recently by some of the authors.