Localisation of directional scale-discretised wavelets on the sphere

TL;DR

This paper introduces directional scale-discretised wavelets on the sphere, enabling probing of signals in scale, position, and orientation, with exact synthesis and properties similar to spherical needlets.

Contribution

It demonstrates that directional scale-discretised wavelets share localisation and uncorrelation properties with needlets, extending their applicability to directional signal analysis.

Findings

Wavelets allow probing signals in scale, position, and orientation.

Properties like localisation and uncorrelation are confirmed for these wavelets.

Exact signal synthesis from wavelet coefficients is achievable.

Abstract

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.