A directional continuous wavelet transform on the sphere

TL;DR

This paper introduces a novel directional continuous wavelet transform on the sphere, enabling analysis of oriented structures directly on spherical data, with a flexible dilation framework extending previous harmonic scaling methods.

Contribution

It extends harmonic scaling-based spherical wavelet analysis to a more general directional framework, allowing for oriented feature detection directly on the sphere.

Findings

Wavelet construction demonstrated with an explicit example.

Framework allows analysis of oriented structures on spherical data.

All functions and operators are defined directly on the sphere.

Abstract

A new construction of a directional continuous wavelet analysis on the sphere is derived herein. We adopt the harmonic scaling idea for the spherical dilation operator recently proposed by Sanz et al. but extend the analysis to a more general directional framework. Directional wavelets are a powerful extension that allow one to also probe oriented structure in the analysed function. Our spherical wavelet methodology has the advantage that all functions and operators are defined directly on the sphere. The construction of wavelets in our framework is demonstrated with an example.