18,549 research outputs found
Tomograms and other transforms. A unified view
A general framework is presented which unifies the treatment of wavelet-like,
quasidistribution, and tomographic transforms. Explicit formulas relating the
three types of transforms are obtained. The case of transforms associated to
the symplectic and affine groups is treated in some detail. Special emphasis is
given to the properties of the scale-time and scale-frequency tomograms.
Tomograms are interpreted as a tool to sample the signal space by a family of
curves or as the matrix element of a projector.Comment: 19 pages latex, submitted to J. Phys. A: Math and Ge
Complex data processing: fast wavelet analysis on the sphere
In the general context of complex data processing, this paper reviews a
recent practical approach to the continuous wavelet formalism on the sphere.
This formalism notably yields a correspondence principle which relates wavelets
on the plane and on the sphere. Two fast algorithms are also presented for the
analysis of signals on the sphere with steerable wavelets.Comment: 20 pages, 5 figures, JFAA style, paper invited to J. Fourier Anal.
and Appli
Fast directional continuous spherical wavelet transform algorithms
We describe the construction of a spherical wavelet analysis through the
inverse stereographic projection of the Euclidean planar wavelet framework,
introduced originally by Antoine and Vandergheynst and developed further by
Wiaux et al. Fast algorithms for performing the directional continuous wavelet
analysis on the unit sphere are presented. The fast directional algorithm,
based on the fast spherical convolution algorithm developed by Wandelt and
Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature
implementation for Npix pixels on the sphere, and allows one to perform a
directional spherical wavelet analysis of a 10^6 pixel map on a personal
computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE
Trans. Sig. Pro
On The Continuous Steering of the Scale of Tight Wavelet Frames
In analogy with steerable wavelets, we present a general construction of
adaptable tight wavelet frames, with an emphasis on scaling operations. In
particular, the derived wavelets can be "dilated" by a procedure comparable to
the operation of steering steerable wavelets. The fundamental aspects of the
construction are the same: an admissible collection of Fourier multipliers is
used to extend a tight wavelet frame, and the "scale" of the wavelets is
adapted by scaling the multipliers. As an application, the proposed wavelets
can be used to improve the frequency localization. Importantly, the localized
frequency bands specified by this construction can be scaled efficiently using
matrix multiplication
3D weak lensing with spin wavelets on the ball
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
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