99 research outputs found
A Note on Spherical Needlets
Compared with the traditional spherical harmonics, the spherical needlets are
a new generation of spherical wavelets that possess several attractive
properties. Their double localization in both spatial and frequency domains
empowers them to easily and sparsely represent functions with small spatial
scale features. This paper is divided into two parts. First, it reviews the
spherical harmonics and discusses their limitations in representing functions
with small spatial scale features. To overcome the limitations, it introduces
the spherical needlets and their attractive properties. In the second part of
the paper, a Matlab package for the spherical needlets is presented. The
properties of the spherical needlets are demonstrated by several examples using
the package.Comment: 12 pages, 7 figures, technical repor
A Note on Global Suprema of Band-Limited Spherical Random Functions
In this note, we investigate the behaviour of suprema for band-limited
spherical random fields. We prove upper and lower bound for the expected values
of these suprema, by means of metric entropy arguments and discrete
approximations; we then exploit the Borell-TIS inequality to establish almost
sure upper and lower bounds for their fluctuations. Band limited functions can
be viewed as restrictions on the sphere of random polynomials with increasing
degrees, and our results show that fluctuations scale as the square root of the
logarithm of these degrees
Asymptotic Uncorrelation for Mexican Needlets
We recall Mexican needlets from [5]. We derive an estimate for certain types
of Legendre series, which we apply to the statistical properties of Mexican
needlets. More precisely, we shall show that, under isotropy and Gaussianity
assumptions, the Mexican needlet coefficients of a random field on the sphere
are asymptotically uncorrelated, as the frequency parameter goes to infinity.
This property is important in the analysis of cosmic microwave background
radiation.Comment: 13 page
On the computation of directional scale-discretized wavelet transforms on the sphere
We review scale-discretized wavelets on the sphere, which are directional and
allow one to probe oriented structure in data defined on the sphere.
Furthermore, scale-discretized wavelets allow in practice the exact synthesis
of a signal from its wavelet coefficients. We present exact and efficient
algorithms to compute the scale-discretized wavelet transform of band-limited
signals on the sphere. These algorithms are implemented in the publicly
available S2DW code. We release a new version of S2DW that is parallelized and
contains additional code optimizations. Note that scale-discretized wavelets
can be viewed as a directional generalization of needlets. Finally, we outline
future improvements to the algorithms presented, which can be achieved by
exploiting a new sampling theorem on the sphere developed recently by some of
the authors.Comment: 13 pages, 3 figures, Proceedings of Wavelets and Sparsity XV, SPIE
Optics and Photonics 2013, Code is publicly available at http://www.s2dw.org
Quantitative central limit theorems for Mexican needlet coefficients on circular Poisson fields
The aim of this paper is to establish rates of convergence to Gaussianity for
wavelet coefficients on circular Poisson random fields. This result is
established by using the Stein-Malliavin techniques introduced by Peccati and
Zheng (2011) and the concentration properties of so-called Mexican needlets on
the circleComment: 26 pages, 4 figure
Bayesian Estimation of Intensity Surfaces on the Sphere via Needlet Shrinkage and Selection
This paper describes an approach for Bayesian modeling in spherical datasets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding, and Bayesian local shrinkage rules. We study the performance of the proposed methodology both on simulated data and on two real data sets: one involving the cosmic microwave background radiation, and one involving the reconstruction of a global news intensity surface inferred from published Reuters articles in August, 1996. The fully Bayesian approach based on robust, sparse shrinkage priors seems to outperform other alternatives.Business Administratio
Asymptotics for spherical needlets
We investigate invariant random fields on the sphere using a new type of
spherical wavelets, called needlets. These are compactly supported in frequency
and enjoy excellent localization properties in real space, with
quasi-exponentially decaying tails. We show that, for random fields on the
sphere, the needlet coefficients are asymptotically uncorrelated for any fixed
angular distance. This property is used to derive CLT and functional CLT
convergence results for polynomial functionals of the needlet coefficients:
here the asymptotic theory is considered in the high-frequency sense. Our
proposals emerge from strong empirical motivations, especially in connection
with the analysis of cosmological data sets.Comment: Published in at http://dx.doi.org/10.1214/08-AOS601 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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