Different wavelet-based approaches for the separation of noisy and blurred mixtures of components. Application to astrophysical data.
Plusieurs approches en ondelettes pour la séparation et déconvolection de composantes. Application à des données astrophysiques.
Résumé
This thesis addresses the problem of separating image components that have different structure, when different observations of blurred mixtures of these components are available. When only a single component is present and has to be extracted from a single observation, this reduces to the deblurring and denoising of one image, a problem well described in the image processing literature. On the other hand, the separation problem has been mainly studied in the simple case of linear mixtures (i.e. without blurring). In this thesis, the full problem is addressed globally, the separation being done simultaneously with the denoising and deblurring of the data at hand. One natural way to tackle the multi-components/multi-observations problem in the blurred context is to generalize methods that exist for the enhancement of a single image. The first result presented in this thesis is a mathematical analysis of a heuristic iterative algorithm for the enhancement of a single image. This algorithm is proved to be convergent but not regularizing; a modification is introduced restores this property. The main object of this thesis is to develop and compare two methods for the multi-components/multi-observations problem: the first method uses functional spaces to describe the signals; the second method models the local statistical properties of the signals. Both methods use wavelet frames to simplify the description of the data. In addition, the functional method uses different frames to characterize different components. The performances of both algorithms are evaluated with regards to a particular astrophysical problem: the reconstruction of clusters of galaxies by the extraction of their Sunyaev-Zel'dovich effect in multifrequency measurements of the Cosmic Microwave Background anisotropies. Realistic simulations are studied, that correspond to different experiments, future or underway. It is shown that both methods yield clusters maps of sufficient quality for subsequent cosmological studies when the resolution of the observations is high and the level of noise moderate, that the noise level is a limiting factor for observations at lower resolution, and that the statistical algorithm is robust to the presence of point sources at higher frequencies.
Cette thèse est consacree au problµeme de separation de composantes lorsque celles ci sont des images de structure differente et que l'on en observe un ou plusieurs melange(s) ou(s) et bruite(s). Les problµemes de deconvolution et de separation, traditionnellement etudies separement, sont ici traites simultanément. Une façon naturelle d'aborder le problµeme multicomposants/multiobservations est de generaliser les techniques de deconvolution d'une image unique. Le premier resultat presente est une etude mathematique d'un tel algorithme. Preuve est faite que celuici est convergent mais pas regularisant et une modification restaurant cette propriete est proposee. Le sujet principal est le developpement et la comparaison de deux methodes pour traiter la deconvolution et separation simultanees de composantes. La premiµere methode est basee sur les propriétes statistiques locales des composantes tandis que dans la seconde, ces signaux sont decrits par des espaces fonctionnels. Les deux methodes utilisent des transformees en ondelettes redondantes pour simplifier les donnees. Les performances des deux algorithmes sont evaluees et comparees dans le cadre d'un problµeme astrophysique : celui de l'extraction des amas de galaxies par l'effet Sunyaev-Zel'dovich dans les images multispectrales des anisotropies du fond cosmique. Des simulations realistes sont etudiees. On montre qu'µa haute resolution et niveau de bruit modere, les deux methodes permettent d'extraire des cartes d'amas de galaxies de qualite suffisante pour des etudes cosmologiques. Le niveau de bruit est un facteur limitant µa basse resolution et la methode statistique est robuste µa la presence de points sources.
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