Exact Wavelets on the Ball
Abstract
We develop an exact wavelet transform on the threedimensional 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 halfline 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 threedimensional decomposition which we call the FourierLaguerre transform. We relate this new transform to the wellknown FourierBessel decomposition and show that bandlimitedness in the FourierLaguerre basis is a sufficient condition to compute the FourierBessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the FourierLaguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and FourierLaguerre 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 floatingpoint 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.
 Publication:

IEEE Transactions on Signal Processing
 Pub Date:
 December 2012
 DOI:
 10.1109/TSP.2012.2215030
 arXiv:
 arXiv:1205.0792
 Bibcode:
 2012ITSP...60.6257L
 Keywords:

 Computer Science  Information Theory;
 Astrophysics  Instrumentation and Methods for Astrophysics
 EPrint:
 13 pages, 10 figures, accepted for publication in IEEE Trans. Sig. Proc. The code is publicly available from http://www.flaglets.org