919 research outputs found

    Compactly Supported Wavelets Derived From Legendre Polynomials: Spherical Harmonic Wavelets

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    A new family of wavelets is introduced, which is associated with Legendre polynomials. These wavelets, termed spherical harmonic or Legendre wavelets, possess compact support. The method for the wavelet construction is derived from the association of ordinary second order differential equations with multiresolution filters. The low-pass filter associated with Legendre multiresolution analysis is a linear phase finite impulse response filter (FIR).Comment: 6 pages, 6 figures, 1 table In: Computational Methods in Circuits and Systems Applications, WSEAS press, pp.211-215, 2003. ISBN: 960-8052-88-

    S2LET: A code to perform fast wavelet analysis on the sphere

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    We describe S2LET, a fast and robust implementation of the scale-discretised wavelet transform on the sphere. Wavelets are constructed through a tiling of the harmonic line and can be used to probe spatially localised, scale-depended features of signals on the sphere. The scale-discretised wavelet transform was developed previously and reduces to the needlet transform in the axisymmetric case. The reconstruction of a signal from its wavelets coefficients is made exact here through the use of a sampling theorem on the sphere. Moreover, a multiresolution algorithm is presented to capture all information of each wavelet scale in the minimal number of samples on the sphere. In addition S2LET supports the HEALPix pixelisation scheme, in which case the transform is not exact but nevertheless achieves good numerical accuracy. The core routines of S2LET are written in C and have interfaces in Matlab, IDL and Java. Real signals can be written to and read from FITS files and plotted as Mollweide projections. The S2LET code is made publicly available, is extensively documented, and ships with several examples in the four languages supported. At present the code is restricted to axisymmetric wavelets but will be extended to directional, steerable wavelets in a future release.Comment: 8 pages, 6 figures, version accepted for publication in A&A. Code is publicly available from http://www.s2let.or

    3D weak lensing with spin wavelets on the ball

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    We construct the spin flaglet transform, a wavelet transform to analyze spin signals in three dimensions. Spin flaglets can probe signal content localized simultaneously in space and frequency and, moreover, are separable so that their angular and radial properties can be controlled independently. They are particularly suited to analyzing of cosmological observations such as the weak gravitational lensing of galaxies. Such observations have a unique 3D geometrical setting since they are natively made on the sky, have spin angular symmetries, and are extended in the radial direction by additional distance or redshift information. Flaglets are constructed in the harmonic space defined by the Fourier-Laguerre transform, previously defined for scalar functions and extended here to signals with spin symmetries. Thanks to various sampling theorems, both the Fourier-Laguerre and flaglet transforms are theoretically exact when applied to bandlimited signals. In other words, in numerical computations the only loss of information is due to the finite representation of floating point numbers. We develop a 3D framework relating the weak lensing power spectrum to covariances of flaglet coefficients. We suggest that the resulting novel flaglet weak lensing estimator offers a powerful alternative to common 2D and 3D approaches to accurately capture cosmological information. While standard weak lensing analyses focus on either real or harmonic space representations (i.e., correlation functions or Fourier-Bessel power spectra, respectively), a wavelet approach inherits the advantages of both techniques, where both complicated sky coverage and uncertainties associated with the physical modeling of small scales can be handled effectively. Our codes to compute the Fourier-Laguerre and flaglet transforms are made publicly available.Comment: 24 pages, 4 figures, version accepted for publication in PR

    Simulating full-sky interferometric observations

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    Aperture array interferometers, such as that proposed for the Square Kilometre Array (SKA), will see the entire sky, hence the standard approach to simulating visibilities will not be applicable since it relies on a tangent plane approximation that is valid only for small fields of view. We derive interferometric formulations in real, spherical harmonic and wavelet space that include contributions over the entire sky and do not rely on any tangent plane approximations. A fast wavelet method is developed to simulate the visibilities observed by an interferometer in the full-sky setting. Computing visibilities using the fast wavelet method adapts to the sparse representation of the primary beam and sky intensity in the wavelet basis. Consequently, the fast wavelet method exhibits superior computational complexity to the real and spherical harmonic space methods and may be performed at substantially lower computational cost, while introducing only negligible error to simulated visibilities. Low-resolution interferometric observations are simulated using all of the methods to compare their performance, demonstrating that the fast wavelet method is approximately three times faster that the other methods for these low-resolution simulations. The computational burden of the real and spherical harmonic space methods renders these techniques computationally infeasible for higher resolution simulations. High-resolution interferometric observations are simulated using the fast wavelet method only, demonstrating and validating the application of this method to realistic simulations. The fast wavelet method is estimated to provide a greater than ten-fold reduction in execution time compared to the other methods for these high-resolution simulations.Comment: 16 pages, 9 figures, replaced to match version accepted by MNRAS (major additions to previous version including new fast wavelet method

    Splines and Wavelets on Geophysically Relevant Manifolds

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    Analysis on the unit sphere S2\mathbb{S}^{2} found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres S2\mathbb{S}^{2},   S3\>\>\mathbb{S}^{3} and the rotation group SO(3)SO(3). Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on Sd\mathbb{S}^{d} and SO(3)SO(3).Comment: The final publication is available at http://www.springerlink.co

    True CMB Power Spectrum Estimation

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    The cosmic microwave background (CMB) power spectrum is a powerful cosmological probe as it entails almost all the statistical information of the CMB perturbations. Having access to only one sky, the CMB power spectrum measured by our experiments is only a realization of the true underlying angular power spectrum. In this paper we aim to recover the true underlying CMB power spectrum from the one realization that we have without a need to know the cosmological parameters. The sparsity of the CMB power spectrum is first investigated in two dictionaries; Discrete Cosine Transform (DCT) and Wavelet Transform (WT). The CMB power spectrum can be recovered with only a few percentage of the coefficients in both of these dictionaries and hence is very compressible in these dictionaries. We study the performance of these dictionaries in smoothing a set of simulated power spectra. Based on this, we develop a technique that estimates the true underlying CMB power spectrum from data, i.e. without a need to know the cosmological parameters. This smooth estimated spectrum can be used to simulate CMB maps with similar properties to the true CMB simulations with the correct cosmological parameters. This allows us to make Monte Carlo simulations in a given project, without having to know the cosmological parameters. The developed IDL code, TOUSI, for Theoretical pOwer spectrUm using Sparse estImation, will be released with the next version of ISAP
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