867 research outputs found
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
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
SILC: a new Planck Internal Linear Combination CMB temperature map using directional wavelets
We present new clean maps of the CMB temperature anisotropies (as measured by
Planck) constructed with a novel internal linear combination (ILC) algorithm
using directional, scale-discretised wavelets --- Scale-discretised,
directional wavelet ILC or SILC. Directional wavelets, when convolved with
signals on the sphere, can separate the anisotropic filamentary structures
which are characteristic of both the CMB and foregrounds. Extending previous
component separation methods, which use the frequency, spatial and harmonic
signatures of foregrounds to separate them from the cosmological background
signal, SILC can additionally use morphological information in the foregrounds
and CMB to better localise the cleaning algorithm. We test the method on Planck
data and simulations, demonstrating consistency with existing component
separation algorithms, and discuss how to optimise the use of morphological
information by varying the number of directional wavelets as a function of
spatial scale. We find that combining the use of directional and axisymmetric
wavelets depending on scale could yield higher quality CMB temperature maps.
Our results set the stage for the application of SILC to polarisation
anisotropies through an extension to spin wavelets.Comment: 15 pages, 13 figures. Minor changes to match version published in
MNRAS. Map products available at http://www.silc-cmb.or
Directional spin wavelets on the sphere
We construct a directional spin wavelet framework on the sphere by generalising the scalar scale-discretised wavelet transform to signals of arbitrary spin. The resulting framework is the only wavelet framework defined natively on the sphere that is able to probe the directional intensity of spin signals. Furthermore, directional spin scale-discretised wavelets support the exact synthesis of a signal on the sphere from its wavelet coefficients and satisfy excellent localisation and uncorrelation properties. Consequently, directional spin scale-discretised wavelets are likely to be of use in a wide range of applications and in particular for the analysis of the polarisation of the cosmic microwave background (CMB). We develop new algorithms to compute (scalar and spin) forward and inverse wavelet transforms exactly and efficiently for very large data-sets containing tens of millions of samples on the sphere. By leveraging a novel sampling theorem on the rotation group developed in a companion article, only half as many wavelet coefficients as alternative approaches need be computed, while still capturing the full information content of the signal under analysis. Our implementation of these algorithms is made publicly available
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-SILC: CMB polarisation component separation with spin wavelets
We present Spin-SILC, a new foreground component separation method that
accurately extracts the cosmic microwave background (CMB) polarisation and
modes from raw multifrequency Stokes and measurements of the
microwave sky. Spin-SILC is an internal linear combination method that uses
spin wavelets to analyse the spin-2 polarisation signal . The
wavelets are additionally directional (non-axisymmetric). This allows different
morphologies of signals to be separated and therefore the cleaning algorithm is
localised using an additional domain of information. The advantage of spin
wavelets over standard scalar wavelets is to simultaneously and
self-consistently probe scales and directions in the polarisation signal and in the underlying and modes, therefore providing the ability
to perform component separation and - decomposition concurrently for the
first time. We test Spin-SILC on full-mission Planck simulations and data and
show the capacity to correctly recover the underlying cosmological and
modes. We also demonstrate a strong consistency of our CMB maps with those
derived from existing component separation methods. Spin-SILC can be combined
with the pseudo- and pure - spin wavelet estimators presented in a
companion paper to reliably extract the cosmological signal in the presence of
complicated sky cuts and noise. Therefore, it will provide a
computationally-efficient method to accurately extract the CMB and
modes for future polarisation experiments.Comment: 13 pages, 9 figures. Minor changes to match version published in
MNRAS. Map products available at http://www.silc-cmb.org. Companion paper:
arXiv:1605.01414 "Wavelet reconstruction of pure E and B modes for CMB
polarisation and cosmic shear analyses" (B. Leistedt et al.
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
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