Flaglets: Exact Wavelets on the Ball

TL;DR

This paper introduces flaglets, a new class of exact wavelets on the ball, utilizing a novel Fourier-Laguerre transform for precise 3D analysis of radial and tangential features.

Contribution

It presents the construction of exact axisymmetric wavelets on the ball using a new 3D harmonic transform, with a publicly available implementation achieving high accuracy.

Findings

Exact wavelet transform on the ball achieved

Fourier-Laguerre transform combines spherical harmonic and Laguerre polynomials

Implementation attains floating-point accuracy for band-limited signals

Abstract

We summarise the construction of exact axisymmetric scale-discretised wavelets on the sphere and on the ball. The wavelet transform on the ball relies on a novel 3D harmonic transform called the Fourier-Laguerre transform which combines the spherical harmonic transform with damped Laguerre polynomials on the radial half-line. The resulting wavelets, called flaglets, extract scale-dependent, spatially localised features in three-dimensions while treating the tangential and radial structures separately. Both the Fourier-Laguerre and the flaglet transforms are theoretically exact thanks to a novel sampling theorem on the ball. Our implementation of these methods is publicly available and achieves floating-point accuracy when applied to band-limited signals.