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FourierLaguerre transform, Convolution and Wavelets
"... Abstract—We review the FourierLaguerre transform, an alternative harmonic analysis on the threedimensional ball to the usual FourierBessel transform. The FourierLaguerre transform exhibits an exact quadrature rule and thus leads to a sampling theorem on the ball. We study the definition of conv ..."
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Abstract—We review the FourierLaguerre transform, an alternative harmonic analysis on the threedimensional ball to the usual FourierBessel transform. The FourierLaguerre transform exhibits an exact quadrature rule and thus leads to a sampling theorem on the ball. We study the definition of convolution on the ball in this context, showing explicitly how translation on the radial line may be viewed as convolution with a shifted Dirac delta function. We review the exact FourierLaguerre wavelet transform on the ball, coined flaglets, and show that flaglets constitute a tight frame. Index Terms—Harmonic analysis, sampling, wavelets, threedimensional ball.
SIGNAL PROCESSING CHALLENGES FOR RADIO ASTRONOMICAL ARRAYS
"... Current and future radio telescopes, in particular the Square Kilometre Array (SKA), are envisaged to produce large images (> 108 pixels) with over 60 dB dynamic range. This poses a number of image reconstruction and technological challenges, which will require novel approaches to image reconst ..."
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Current and future radio telescopes, in particular the Square Kilometre Array (SKA), are envisaged to produce large images (> 108 pixels) with over 60 dB dynamic range. This poses a number of image reconstruction and technological challenges, which will require novel approaches to image reconstruction and design of data processing systems. In this paper, we sketch the limitations of current algorithms by extrapolating their computational requirements to future radio telescopes as well as by discussing their imaging limitations. We discuss a number of potential research directions to cope with these challenges. Index Terms — Array signal processing, radio astronomy, imaging, image reconstruction, computational cost 1.
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"... Preface vii 1 The wavelet transform 1 1.1 Multiscale methods........................ 1 1.1.1 Some perspectives on the wavelet transform...... 2 ..."
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Preface vii 1 The wavelet transform 1 1.1 Multiscale methods........................ 1 1.1.1 Some perspectives on the wavelet transform...... 2
1An OptimalDimensionality Sampling Scheme on the Sphere for Fast Spherical Harmonic Transforms
"... Abstract—We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals bandlimited at L using only L2 samples. We obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space. The ..."
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Abstract—We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals bandlimited at L using only L2 samples. We obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space. The number of samples required in our scheme is a factor of two or four fewer than existing techniques, which require either 2L2 or 4L2 samples. We note, however, that we do not recover a sampling theorem on the sphere, where spherical harmonic transforms are theoretically exact. Nevertheless, we achieve high accuracy even for very large bandlimits. For our optimaldimensionality sampling scheme, we develop a fast and accurate algorithm to compute the spherical harmonic transform (and inverse), with computational complexity comparable with existing schemes in practice. We conduct numerical experiments to study in detail the stability, accuracy and computational complexity of the proposed transforms. We also highlight the advantages of the proposed sampling scheme and associated transforms in the context of potential applications. Index Terms—2sphere (unit sphere), spherical harmonic transform, sampling, harmonic analysis, spherical harmonics.
SLEPIAN SPATIALSPECTRAL CONCENTRATION ON THE BALL
"... Abstract. We formulate and solve the Slepian spatialspectral concentration problem on the threedimensional ball. Both the standard FourierBessel and also the FourierLaguerre spectral domains are considered since the latter exhibits a number of practical advantages (spectral decoupling and exact ..."
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Abstract. We formulate and solve the Slepian spatialspectral concentration problem on the threedimensional ball. Both the standard FourierBessel and also the FourierLaguerre spectral domains are considered since the latter exhibits a number of practical advantages (spectral decoupling and exact computation). The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions. Equivalence between the spatial and spectral problems is shown. The spherical Shannon number on the ball is derived, which acts as the analog of the spacebandwidth product in the Euclidean setting, giving an estimate of the number of concentrated eigenfunctions and thus the dimension of the space of functions that can be concentrated in both the spatial and spectral domains simultaneously. Various symmetries of the spatial region are considered that reduce considerably the computational burden of recovering eigenfunctions, either by decoupling the problem into smaller subproblems or by affording analytic calculations. The family of concentrated eigenfunctions forms a Slepian basis that can be used be represent concentrated signals efficiently. We illustrate our results with numerical examples and show that the Slepian basis indeeds permits a sparse representation of concentrated signals. Key words. Slepian concentration problem, bandlimited function, eigenvalue problem, harmonic analysis, ball
Flaglets for studying the largescale structure of the Universe
"... Pressing questions in cosmology such as the nature of dark matter and dark energy can be addressed using large galaxy surveys, which measure the positions, properties and redshifts of galaxies in order to map the largescale structure of the Universe. We review the FourierLaguerre transform, a nove ..."
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Pressing questions in cosmology such as the nature of dark matter and dark energy can be addressed using large galaxy surveys, which measure the positions, properties and redshifts of galaxies in order to map the largescale structure of the Universe. We review the FourierLaguerre transform, a novel transform in 3D spherical coordinates which is based on spherical harmonics combined with damped Laguerre polynomials and appropriate for analysing galaxy surveys. We also recall the construction of flaglets, 3D wavelets obtained through a tiling of the FourierLaguerre space, which can be used to extract scaledependent, spatially localised features on the ball. We exploit a sampling theorem to obtain exact FourierLaguerre and flaglet transforms, such that bandlimited signals can analysed and reconstructed at floating point accuracy on a finite number of voxels on the ball. We present a potential application of the flaglet transform for finding voids in galaxy surveys and studying the largescale structure of the Universe.
Flaglets: Exact Wavelets on the Ball
"... Abstract—We summarise the construction of exact axisymmetric scalediscretised wavelets on the sphere and on the ball. The wavelet transform on the ball relies on a novel 3D harmonic transform called the FourierLaguerre transform which combines the spherical harmonic transform with damped Laguerre ..."
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Abstract—We summarise the construction of exact axisymmetric scalediscretised wavelets on the sphere and on the ball. The wavelet transform on the ball relies on a novel 3D harmonic transform called the FourierLaguerre transform which combines the spherical harmonic transform with damped Laguerre polynomials on the radial halfline. The resulting wavelets, called flaglets, extract scaledependent, spatially localised features in threedimensions while treating the tangential and radial structures separately. Both the FourierLaguerre and the flaglet transforms are theoretically exact thanks to a novel sampling theorem on the ball. Our implementation of these methods is publicly available [1], [2] and achieves floatingpoint accuracy when applied to bandlimited signals. I.