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Friday, November 09, 2012

FLAGLET: Exact wavelets on the ball - implementation -

After S2LET, here is a related toolbox FLAGLET producing exact wavelets on the ball. From the paper:

Exact Wavelets on the Ball by B. Leistedt, J. D. McEwen. The abstract reads:
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.

The code is located at: The introduction reads:


The FLAGLET code provides high-performance routines for fast wavelet analysis of signals on the ball using theFlaglet transform (ArXiv | DOI). It exploits S2LETFLAG and SSHT codes. The flaglet transform is theoretically exact, i.e. the original signal can be synthesises from its wavelet coefficients exactly since the wavelet coefficients capture all the information of band-limited signals.
This page outlines the main features of FLAGLET, installation details as well as the core functionalities and interfaces. References, version, and license information then follows. FLAGLET requires the FLAGS2LETSSHT andFFTW libraries.

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