Directional spin wavelets on the sphere

TL;DR

This paper introduces a novel directional spin wavelet framework on the sphere that enables precise analysis of spin signals, with efficient algorithms suitable for large datasets, and has potential applications in cosmic microwave background polarization studies.

Contribution

It generalizes scalar wavelets to arbitrary spin signals on the sphere, supporting exact synthesis, and develops efficient algorithms leveraging a new sampling theorem.

Findings

Supports exact synthesis of signals from wavelet coefficients

Achieves efficient computation with fewer coefficients needed

Demonstrates applicability to large-scale data analysis

Abstract

We construct a directional spin wavelet framework on the sphere by generalising the scalar scale-discretised wavelet transform to signals of arbitrary spin. The resulting framework is the only wavelet framework defined natively on the sphere that is able to probe the directional intensity of spin signals. Furthermore, directional spin scale-discretised wavelets support the exact synthesis of a signal on the sphere from its wavelet coefficients and satisfy excellent localisation and uncorrelation properties. Consequently, directional spin scale-discretised wavelets are likely to be of use in a wide range of applications and in particular for the analysis of the polarisation of the cosmic microwave background (CMB). We develop new algorithms to compute (scalar and spin) forward and inverse wavelet transforms exactly and efficiently for very large data-sets containing tens of millions of samples on the sphere. By leveraging a novel sampling theorem on the rotation group developed in a companion article, only half as many wavelet coefficients as alternative approaches need be computed, while still capturing the full information content of the signal under analysis. Our implementation of these algorithms is made publicly available.