Exact Wavelets on the Ball

TL;DR

This paper introduces an exact wavelet transform on the three-dimensional ball, called the flaglet transform, enabling precise multiscale analysis of functions on the solid sphere with applications demonstrated in denoising.

Contribution

The authors develop the first exact wavelet transform on the 3D ball using Fourier-Laguerre basis, including a fast multiresolution algorithm and publicly available implementation.

Findings

Achieves floating-point precision in wavelet transforms on the ball.

Provides a fast, multiresolution algorithm for the flaglet transform.

Demonstrates effectiveness in a denoising example.

Abstract

We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial half-line using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel three-dimensional decomposition which we call the Fourier-Laguerre transform. We relate this new transform to the well-known Fourier-Bessel decomposition and show that band-limitedness in the Fourier-Laguerre basis is a sufficient condition to compute the Fourier-Bessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the Fourier-Laguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and Fourier-Laguerre spaces and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floating-point precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example.