Blind Component Separation in Wavelet Space: Application to CMB Analysis
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Blind Component Separation in Wavelet Space: Application to CMB Analysis Y. Moudden DAPNIA/SEDI-SAP, CEA/Saclay, 91191 Gif-sur-Yvette, France Email: [email protected]
J.-F. Cardoso ´ CNRS, Ecole National Sup´erieure des T´el´ecommunications, 46 rue Barrault, 75634 Paris, France Email: [email protected]
J.-L. Starck DAPNIA/SEDI-SAP, CEA/Saclay, 91191 Gif-sur-Yvette, France Email: [email protected]
J. Delabrouille CNRS/PCC, Coll`ege de France, 11 place Marcelin Berthelot, 75231 Paris, France Email: [email protected] Received 30 June 2004; Revised 22 November 2004 It is a recurrent issue in astronomical data analysis that observations are incomplete maps with missing patches or intentionally masked parts. In addition, many astrophysical emissions are nonstationary processes over the sky. All these effects impair data processing techniques which work in the Fourier domain. Spectral matching ICA (SMICA) is a source separation method based on spectral matching in Fourier space designed for the separation of diffuse astrophysical emissions in cosmic microwave background observations. This paper proposes an extension of SMICA to the wavelet domain and demonstrates the effectiveness of waveletbased statistics for dealing with gaps in the data. Keywords and phrases: blind source separation, cosmic microwave background, wavelets, data analysis, missing data.
1.
INTRODUCTION
The detection of cosmic microwave background (CMB) anisotropies on the sky has been over the past three decades a subject of intense activity in the cosmology community. The CMB, discovered in 1965 by Penzias and Wilson, is a relic radiation emitted some 13 billion years ago, when the universe was about 370 000 years old. Small fluctuations of this emission, tracing the seeds of the primordial inhomogeneities which gave rise to present large scale structures as galaxies and clusters of galaxies, were first discovered in the observations made by COBE [1] and further investigated by a number of experiments among which Archeops [2], boomerang [3], maxima [4], and WMAP [5]. The precise measurement of these fluctuations is of utmost importance to cosmology. Their statistical properties (spatial power spectrum, Gaussianity) strongly depend on the cosmological scenarios describing the properties and evolution of our universe as a whole, and thus permit to
constrain these models as well as to measure the cosmological parameters describing the matter content, the geometry, and the evolution of our universe [6]. Accessing this information, however, requires disentangling in the data the contributions of several distinct astrophysical sources, all of which emit radiation in the frequency range used for CMB observations [7]. This problem of component separation, in the field of CMB studies, has thus been the object of many dedicated studies in the past. To first order, the total sky emission can be modeled as a linear superposition of a few independent processes. The observation of the sky in direction (θ, ϕ) with detector d is then a noisy linear mixture
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