Practical wavelet design on the sphere

Guilloux, Frédéric; Fay, Gillesn; Cardoso, J.-F.

doi:10.1016/j.acha.2008.03.003

Cited by 57 publications

(79 citation statements)

References 34 publications

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“…1992). Needlets were introduced as a particular construction of a wavelet family on the sphere by Narcowich et al (2006) and Guilloux et al (2009). Needlets can be interpreted as a set of band-pass filters in harmonic space.…”

Section: Discussionmentioning

confidence: 99%

Extracting H i cosmological signal with generalized needlet internal linear combination

Olivari¹,

Remazeilles²,

Dickinson³

2015

Mon. Not. R. Astron. Soc.

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HI intensity mapping is a new observational technique to map fluctuations in the large-scale structure of matter using the 21 cm emission line of atomic hydrogen (HI). Sensitive HI intensity mapping experiments have the potential to detect Baryon Acoustic Oscillations (BAO) at low redshifts (z 1) in order to constrain the properties of dark energy. Observations of the HI signal will be contaminated by instrumental noise and, more significantly, by astrophysical foregrounds, such as Galactic synchrotron emission, which is at least four orders of magnitude brighter than the HI signal. Foreground cleaning is recognised as one of the key challenges for future radio astronomy surveys. We study the ability of the Generalized Needlet Internal Linear Combination (GNILC) method to subtract radio foregrounds and to recover the cosmological HI signal for a general HI intensity mapping experiment. The GNILC method is a new technique that uses both frequency and spatial information to separate the components of the observed data. Our results show that the method is robust to the complexity of the foregrounds. For simulated radio observations including HI emission, Galactic synchrotron, Galactic free-free, radio sources and 0.05 mK thermal noise, we find that the GNILC method can reconstruct the HI power spectrum for multipoles 30 < < 150 with 6% accuracy on 50% of the sky for a redshift z ∼ 0.25.

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Section: Discussionmentioning

confidence: 99%

Extracting H i cosmological signal with generalized needlet internal linear combination

Olivari¹,

Remazeilles²,

Dickinson³

2015

Mon. Not. R. Astron. Soc.

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“…Other constructions may have similarly desirable properties, but even those are often judged on how closely they satisfy quadratic optimality constraints (e.g. Freeden and Michel 1999;Guilloux et al 2009). We therefore remain faithful to the original approach of Slepian, as transformed into spherical geometry by .…”

Section: The Spherical Slepian Basismentioning

confidence: 99%

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

Slobbe

Simons

Klees

2012

J Geod

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The mean dynamic topography (MDT) can be computed as the difference between the mean sea level (MSL) and a gravimetric geoid. This requires that both data sets are spectrally consistent. In practice, it is quite common that the resolution of the geoid data is less than the resolution of the MSL data, hence, the latter need to be low-pass filtered before the MDT is computed. For this purpose conventional low-pass filters are inadequate, failing in coastal regions where they run into the undefined MSL signal on the continents. In this paper, we consider the use of a bandlimited, spatially concentrated Slepian basis to obtain a low-resolution approximation of the MSL signal. We compute Slepian functions for the oceans and parts of the oceans and compare the performance of calculating the MDT via this approach with other methods, in particular the iterative spherical harmonic approach in combination with Gaussian low-pass filtering, and various modifications. Based on the numerical experiments, we conclude that none of these methods provide a lowresolution MSL approximation at the sub-decimetre level. In particular, we show that Slepian functions are not appropriate basis functions for this problem, and a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. We also show that a meaningful definition of a lowresolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. A low-resolution MSL signal, spectrally consistent with a given geoid model, is obtained by a suitable truncation of the expansions of the MSL signal in terms of these orthogonal basis functions. We compute one of these sets of orthogonal basis functions using the Gram-Schmidt orthogonalization for spherical harmonics. For the oceans, we could construct an orthogonal basis only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution fails due to inherent instabilities. Regularization reduces the instabilities but destroys the orthogonality and, therefore, provides unrealistic low-resolution MSL approximations. More research is needed to solve the instability problem, perhaps by finding a different orthogonal basis that avoids it altogether.

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“…Further details can be found in Guilloux et al (2008), with an extensive discussion on the choice of the spectral window functions.…”

Section: Needletsmentioning

confidence: 99%

“…This can be achieved with a spherical wavelet or needlet analysis of the maps (using the tools developed in Marinucci et al 2008 andGuilloux et al 2008), which is our approach in the present work.…”

Section: Full Sky?mentioning

confidence: 99%

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A full sky, low foreground, high resolution CMB map from WMAP

Delabrouille¹,

Cardoso

Jeune³

et al. 2008

A&A

263

317

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Context. The WMAP satellite has made available high quality maps of the sky in five frequency bands ranging from 22 to 94 GHz, with the main scientific objective of studying the anisotropies of the Cosmic Microwave Background (CMB). These maps, however, contain a mixture of emission from various astrophysical origin, superimposed on CMB emission. Aims. The objective of the present work is to make a high resolution CMB map in which contamination by such galactic and extragalactic foreground emissions, as well as by instrumental noise, is as low as possible. Methods. The method used is an implementation of a constrained linear combination of the channels with minimum error variance, and of Wiener filtering, on a frame of spherical wavelets called needlets, allowing localised filtering in both pixel space and harmonic space. Results. We obtain a low contamination low noise CMB map at the resolution of the WMAP W channel, which can be used for a range of scientific studies. We obtain also a Wiener-filtered version with minimal integrated error. Conclusions. The resulting CMB maps offer significantly better rejection of galactic foreground than previous CMB maps from WMAP data. They can be considered as the most precise full-sky CMB temperature maps to date.

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Practical wavelet design on the sphere

Cited by 57 publications

References 34 publications

Extracting H i cosmological signal with generalized needlet internal linear combination

Extracting H i cosmological signal with generalized needlet internal linear combination

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

A full sky, low foreground, high resolution CMB map from WMAP

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