Báo cáo hóa học: Blind Component Separation in Wavelet Space: Application to CMB Analysis

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Báo cáo hóa học: " Blind Component Separation in Wavelet Space: Application to CMB Analysis"EURASIP Journal on Applied Signal Processing 2005:15, 2437–2454 c 2005 Hindawi Publishing CorporationBlind Component Separation in Wavelet Space:Application to CMB Analysis Y. Moudden DAPNIA/SEDI-SAP, CEA/Saclay, 91191 Gif-sur-Yvette, France Email: yassir.moudden@cea.fr J.-F. Cardoso ´ ´ CNRS, Ecole National Superieure des T´l´communications, 46 rue Barrault, 75634 Paris, France ee Email: cardoso@tsi.enst.fr J.-L. Starck DAPNIA/SEDI-SAP, CEA/Saclay, 91191 Gif-sur-Yvette, France Email: jstarck@cea.fr J. Delabrouille CNRS/PCC, Coll`ge de France, 11 place Marcelin Berthelot, 75231 Paris, France e Email: delabrouille@cdf.in2p3.fr 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 wavelet- based statistics for dealing with gaps in the data. Keywords and phrases: blind source separation, cosmic microwave background, wavelets, data analysis, missing data.1. INTRODUCTION constrain these models as well as to measure the cosmologi- cal parameters describing the matter content, the geometry,The detection of cosmic microwave background (CMB) and the evolution of our universe [6].anisotropies on the sky has been over the past three decades a Accessing this information, however, requires disentan-subject of intense activity in the cosmology community. The gling in the data the contributions of several distinct astro-CMB, discovered in 1965 by Penzias and Wilson, is a relic ra- physical sources, all of which emit radiation in the frequencydiation emitted some 13 billion years ago, when the universe range used for CMB observations [7]. This problem of com-was about 370 000 years old. Small fluctuations of this emis- ponent separation, in the field of CMB studies, has thus beension, tracing the seeds of the primordial inhomogeneities the object of many dedicated studies in the past.which gave rise to present large scale structures as galaxies To first order, the total sky emission can be modeled asand clusters of galaxies, were first discovered in the observa- a linear superposition of a few independent processes. Thetions made by COBE [1] and further investigated by a num- observation of the sky in direction (θ , ϕ) with detector d isber of experiments among which Archeops [2], boomerang then a noisy linear mixture of Nc components:[3], maxima [4], and WMAP [5]. Nc The precise measurement of these fluctuations is of ut- xd (ϑ, ϕ) = Adj s j (ϑ, ϕ) + nd (ϑ, ϕ), (1)most importance to cosmology. Their statistical properties j =1(spatial power spectrum, Gaussianity) strongly depend onthe cosmological scenarios describing the properties and where s j is the emission template for the j th astrophysi-evolution of our universe as a whole, and thus permit to cal process, herein referred to as a source or a component.2438 EURASIP Journal on Applied Signal ProcessingThe coefficients Adj reflect emission laws while nd accounts Blind component separation (and in particular estima-for noise. When Nd detectors provide independent observa- tion of the mixing matrix), as discussed by Cardoso [17], can be achieved in several different ways. The first of these ex-tions, this equation can be put in vector-matrix form: ploits non-Gaussianity of all, but possibly one, components. The component separation method of Baccigalupi [11] and X (ϑ, ϕ) = AS(ϑ, ϕ) + N (ϑ, ϕ), (2) ...