35 research outputs found
Mixed Needlets
The construction of needlet-type wavelets on sections of the spin line
bundles over the sphere has been recently addressed in Geller and Marinucci
(2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal
for needlets on this spin line bundle, in which needlet coefficients arise from
the usual, rather than the spin, spherical harmonics, as in the previous
constructions. We label this system mixed needlets and investigate in full
their properties, including localization, the exact tight frame
characterization, reconstruction formula, decomposition of functional spaces,
and asymptotic uncorrelation in the stochastic case. We outline astrophysical
applications.Comment: 26 page
Adaptive Nonparametric Regression on Spin Fiber Bundles
The construction of adaptive nonparametric procedures by means of wavelet
thresholding techniques is now a classical topic in modern mathematical
statistics. In this paper, we extend this framework to the analysis of
nonparametric regression on sections of spin fiber bundles defined on the
sphere. This can be viewed as a regression problem where the function to be
estimated takes as its values algebraic curves (for instance, ellipses) rather
than scalars, as usual. The problem is motivated by many important
astrophysical applications, concerning for instance the analysis of the weak
gravitational lensing effect, i.e. the distortion effect of gravity on the
images of distant galaxies. We propose a thresholding procedure based upon the
(mixed) spin needlets construction recently advocated by Geller and Marinucci
(2008,2010) and Geller et al. (2008,2009), and we investigate their rates of
convergence and their adaptive properties over spin Besov balls.Comment: 40 page
Spin Needlets for Cosmic Microwave Background Polarization Data Analysis
Scalar wavelets have been used extensively in the analysis of Cosmic
Microwave Background (CMB) temperature maps. Spin needlets are a new form of
(spin) wavelets which were introduced in the mathematical literature by Geller
and Marinucci (2008) as a tool for the analysis of spin random fields. Here we
adopt the spin needlet approach for the analysis of CMB polarization
measurements. The outcome of experiments measuring the polarization of the CMB
are maps of the Stokes Q and U parameters which are spin 2 quantities. Here we
discuss how to transform these spin 2 maps into spin 2 needlet coefficients and
outline briefly how these coefficients can be used in the analysis of CMB
polarization data. We review the most important properties of spin needlets,
such as localization in pixel and harmonic space and asymptotic uncorrelation.
We discuss several statistical applications, including the relation of angular
power spectra to the needlet coefficients, testing for non-Gaussianity on
polarization data, and reconstruction of the E and B scalar maps.Comment: Accepted for publication in Phys. Rev.