13 research outputs found
Subsampling needlet coefficients on the sphere
In a recent paper, we analyzed the properties of a new kind of spherical
wavelets (called needlets) for statistical inference procedures on spherical
random fields; the investigation was mainly motivated by applications to
cosmological data. In the present work, we exploit the asymptotic uncorrelation
of random needlet coefficients at fixed angular distances to construct
subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate
how such statistics can be used for isotropy tests and for bootstrap estimation
of nuisance parameters, even when a single realization of the spherical random
field is observed. The asymptotic theory is developed in detail in the high
resolution sense.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ164 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Scale-discretised ridgelet transform on the sphere
We revisit the spherical Radon transform, also called the Funk-Radon
transform, viewing it as an axisymmetric convolution on the sphere. Viewing the
spherical Radon transform in this manner leads to a straightforward derivation
of its spherical harmonic representation, from which we show the spherical
Radon transform can be inverted exactly for signals exhibiting antipodal
symmetry. We then construct a spherical ridgelet transform by composing the
spherical Radon and scale-discretised wavelet transforms on the sphere. The
resulting spherical ridgelet transform also admits exact inversion for
antipodal signals. The restriction to antipodal signals is expected since the
spherical Radon and ridgelet transforms themselves result in signals that
exhibit antipodal symmetry. Our ridgelet transform is defined natively on the
sphere, probes signal content globally along great circles, does not exhibit
blocking artefacts, supports spin signals and exhibits an exact and explicit
inverse transform. No alternative ridgelet construction on the sphere satisfies
all of these properties. Our implementation of the spherical Radon and ridgelet
transforms is made publicly available. Finally, we illustrate the effectiveness
of spherical ridgelets for diffusion magnetic resonance imaging of white matter
fibers in the brain.Comment: 5 pages, 4 figures, matches version accepted by EUSIPCO, code
available at http://www.s2let.or
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.
Sparse image reconstruction on the sphere: analysis and synthesis
We develop techniques to solve ill-posed inverse problems on the sphere by
sparse regularisation, exploiting sparsity in both axisymmetric and directional
scale-discretised wavelet space. Denoising, inpainting, and deconvolution
problems, and combinations thereof, are considered as examples. Inverse
problems are solved in both the analysis and synthesis settings, with a number
of different sampling schemes. The most effective approach is that with the
most restricted solution-space, which depends on the interplay between the
adopted sampling scheme, the selection of the analysis/synthesis problem, and
any weighting of the l1 norm appearing in the regularisation problem. More
efficient sampling schemes on the sphere improve reconstruction fidelity by
restricting the solution-space and also by improving sparsity in wavelet space.
We apply the technique to denoise Planck 353 GHz observations, improving the
ability to extract the structure of Galactic dust emission, which is important
for studying Galactic magnetism.Comment: 11 pages, 6 Figure
Localisation of directional scale-discretised wavelets on the sphere
Scale-discretised wavelets yield a directional wavelet framework on the
sphere where a signal can be probed not only in scale and position but also in
orientation. Furthermore, a signal can be synthesised from its wavelet
coefficients exactly, in theory and practice (to machine precision).
Scale-discretised wavelets are closely related to spherical needlets (both were
developed independently at about the same time) but relax the axisymmetric
property of needlets so that directional signal content can be probed. Needlets
have been shown to satisfy important quasi-exponential localisation and
asymptotic uncorrelation properties. We show that these properties also hold
for directional scale-discretised wavelets on the sphere and derive similar
localisation and uncorrelation bounds in both the scalar and spin settings.
Scale-discretised wavelets can thus be considered as directional needlets.Comment: 28 pages, 8 figures, minor changes to match version accepted for
publication by ACH
Matrix Filters for the Detection of Extragalactic Point Sources in Cosmic Microwave Background Images
In this paper we introduce a new linear filtering technique, the so-called
matrix filters, that maximizes the signal-to-interference ratio of compact
sources of unknown intensity embedded in a set of images by taking into account
the cross-correlations between the different channels. By construction, the new
filtering technique outperforms (or at least equals) the standard matched
filter applied on individual images. An immediate application is the detection
of extragalactic point sources in Cosmic Microwave Background images obtained
at different wavelengths. We test the new technique in two simulated cases: a
simple two-channel case with ideal correlated color noise and more realistic
simulations of the sky as it will be observed by the LFI instrument of the
upcoming ESA's Planck mission. In both cases we observe an improvement with
respect to the standard matched filter in terms of signal-to-noise
interference, number of detections and number of false alarms.Comment: 9 pages, 7 figures, submitted to IEEE Journal of Selected Topics in
Signal Processin
New approaches to probing Minkowski functionals
We generalize the concept of the ordinary skew-spectrum to probe the effect of non-Gaussianity
on the morphology of cosmic microwave background (CMB) maps in several domains: in
real space (where they are commonly known as cumulant-correlators), and in harmonic and
needlet bases. The essential aim is to retain more information than normally contained in these
statistics, in order to assist in determining the source of any measured non-Gaussianity, in the
same spirit as Munshi & Heavens skew-spectra were used to identify foreground contaminants
to the CMB bispectrum in Planck data. Using a perturbative series to construct the Minkowski
functionals (MFs), we provide a pseudo-C based approach in both harmonic and needlet
representations to estimate these spectra in the presence of a mask and inhomogeneous noise.
Assuming homogeneous noise, we present approximate expressions for error covariance for
the purpose of joint estimation of these spectra. We present specific results for four different
models of primordial non-Gaussianity local, equilateral, orthogonal and enfolded models, as
well as non-Gaussianity caused by unsubtracted point sources. Closed form results of nextorder
corrections to MFs too are obtained in terms of a quadruplet of kurt-spectra. We also
use the method of modal decomposition of the bispectrum and trispectrum to reconstruct the
MFs as an alternative method of reconstruction of morphological properties of CMB maps.
Finally, we introduce the odd-parity skew-spectra to probe the odd-parity bispectrum and its
impact on the morphology of the CMB sky. Although developed for the CMB, the generic
results obtained here can be useful in other areas of cosmology