1,169 research outputs found
Fast spin +-2 spherical harmonics transforms and application in cosmology
A fast and exact algorithm is developed for the spin +-2 spherical harmonics
transforms on equi-angular pixelizations on the sphere. It is based on the
Driscoll and Healy fast scalar spherical harmonics transform. The theoretical
exactness of the transform relies on a sampling theorem. The associated
asymptotic complexity is of order O(L^2 log^2_2(L)), where 2L stands for the
square-root of the number of sampling points on the sphere, also setting a band
limit L for the spin +-2 functions considered. The algorithm is presented as an
alternative to existing fast algorithms with an asymptotic complexity of order
O(L^3) on other pixelizations. We also illustrate these generic developments
through their application in cosmology, for the analysis of the cosmic
microwave background (CMB) polarization data.Comment: 20 pages, 2 figures. Version accepted for publication in J. Comput.
Phys
Fast directional correlation on the sphere with steerable filters
A fast algorithm is developed for the directional correlation of scalar
band-limited signals and band-limited steerable filters on the sphere. The
asymptotic complexity associated to it through simple quadrature is of order
O(L^5), where 2L stands for the square-root of the number of sampling points on
the sphere, also setting a band limit L for the signals and filters considered.
The filter steerability allows to compute the directional correlation uniquely
in terms of direct and inverse scalar spherical harmonics transforms, which
drive the overall asymptotic complexity. The separation of variables technique
for the scalar spherical harmonics transform produces an O(L^3) algorithm
independently of the pixelization. On equi-angular pixelizations, a sampling
theorem introduced by Driscoll and Healy implies the exactness of the
algorithm. The equi-angular and HEALPix implementations are compared in terms
of memory requirements, computation times, and numerical stability. The
computation times for the scalar transform, and hence for the directional
correlation, of maps of several megapixels on the sphere (L~10^3) are reduced
from years to tens of seconds in both implementations on a single standard
computer. These generic results for the scale-space signal processing on the
sphere are specifically developed in the perspective of the wavelet analysis of
the cosmic microwave background (CMB) temperature (T) and polarization (E and
B) maps of the WMAP and Planck experiments. As an illustration, we consider the
computation of the wavelet coefficients of a simulated temperature map of
several megapixels with the second Gaussian derivative wavelet.Comment: Version accepted in APJ. 14 pages, 2 figures, Revtex4 (emulateapj).
Changes include (a) a presentation of the algorithm as directly built on
blocks of standard spherical harmonics transforms, (b) a comparison between
the HEALPix and equi-angular implementation
An Optimal-Dimensionality Sampling for Spin- Functions on the Sphere
For the representation of spin- band-limited functions on the sphere, we
propose a sampling scheme with optimal number of samples equal to the number of
degrees of freedom of the function in harmonic space. In comparison to the
existing sampling designs, which require samples for the
representation of spin- functions band-limited at , the proposed scheme
requires samples for the accurate computation of the spin-
spherical harmonic transform~(-SHT). For the proposed sampling scheme, we
also develop a method to compute the -SHT. We place the samples in our
design scheme such that the matrices involved in the computation of -SHT are
well-conditioned. We also present a multi-pass -SHT to improve the accuracy
of the transform. We also show the proposed sampling design exhibits superior
geometrical properties compared to existing equiangular and Gauss-Legendre
sampling schemes, and enables accurate computation of the -SHT corroborated
through numerical experiments.Comment: 5 pages, 2 figure
A novel sampling theorem on the rotation group
We develop a novel sampling theorem for functions defined on the
three-dimensional rotation group SO(3) by connecting the rotation group to the
three-torus through a periodic extension. Our sampling theorem requires
samples to capture all of the information content of a signal band-limited at
, reducing the number of required samples by a factor of two compared to
other equiangular sampling theorems. We present fast algorithms to compute the
associated Fourier transform on the rotation group, the so-called Wigner
transform, which scale as , compared to the naive scaling of .
For the common case of a low directional band-limit , complexity is reduced
to . Our fast algorithms will be of direct use in speeding up the
computation of directional wavelet transforms on the sphere. We make our SO3
code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for
publication. Code available at http://www.sothree.or
Polarized wavelets and curvelets on the sphere
The statistics of the temperature anisotropies in the primordial cosmic
microwave background radiation field provide a wealth of information for
cosmology and for estimating cosmological parameters. An even more acute
inference should stem from the study of maps of the polarization state of the
CMB radiation. Measuring the extremely weak CMB polarization signal requires
very sensitive instruments. The full-sky maps of both temperature and
polarization anisotropies of the CMB to be delivered by the upcoming Planck
Surveyor satellite experiment are hence being awaited with excitement.
Multiscale methods, such as isotropic wavelets, steerable wavelets, or
curvelets, have been proposed in the past to analyze the CMB temperature map.
In this paper, we contribute to enlarging the set of available transforms for
polarized data on the sphere. We describe a set of new multiscale
decompositions for polarized data on the sphere, including decimated and
undecimated Q-U or E-B wavelet transforms and Q-U or E-B curvelets. The
proposed transforms are invertible and so allow for applications in data
restoration and denoising.Comment: Accepted. Full paper will figures available at
http://jstarck.free.fr/aa08_pola.pd
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
Fast and Exact Spin-s Spherical Harmonic Transforms
We demonstrate a fast spin-s spherical harmonic transform algorithm, which is
flexible and exact for band-limited functions. In contrast to previous work,
where spin transforms are computed independently, our algorithm permits the
computation of several distinct spin transforms simultaneously. Specifically,
only one set of special functions is computed for transforms of quantities with
any spin, namely the Wigner d-matrices evaluated at {\pi}/2, which may be
computed with efficient recursions. For any spin the computation scales as
O(L^3) where L is the band-limit of the function. Our publicly available
numerical implementation permits very high accuracy at modest computational
cost. We discuss applications to the Cosmic Microwave Background (CMB) and
gravitational lensing.Comment: 22 pages, preprint format, 5 figure
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
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