Thèse soutenue

Analyse en ondelettes et par paquets d'ondelettes de processus aléatoires stationnaires, et application à l'estimation non-paramétrique
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Auteur / Autrice : Abdourrahmane Mahamane Atto
Direction : Alain Hillion
Type : Thèse de doctorat
Discipline(s) : Mathématiques et applications
Date : Soutenance en 2008
Etablissement(s) : Télécom Bretagne

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Résumé

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Wavelets and wavelet packet transforms are widely used for analyzing and solving problems related to science and engineering techniques. This growth is mainly due to two specific properties that result from decompositions on wavelet bases: the sparse representation of regular and piecewise regular signals, the tendency to transform a stationary random process into quasi decorrelated and Gaussian sequences. Both properties are observed experimentally and justify many treatments carried out on the coefficients obtained by projection on wavelet bases. However, the theoretical results describing the asymptotic autocorrelations and distributions associated with wavelet packet coefficients are not always of the same nature. The first part of this thesis provides these results. The results obtained thus lead to more realistic schemes for the asymptotic autocorrelation functions and distributions of wavelet packet coefficients and are supported by numerous experimental results. The second part of the thesis analyses sparsity properties of wavelet bases and proposes the use of sparse-descriptive based thresholds. These thresholds establish a link between non-parametric detection and estimation in the sense that good detection leads to better estimation. The detection thresholds obtained also unify the minimax and universal thresholds in the sense that these thresholds correspond to detection thresholds associated with different sparsity degrees. On the other hand, this thesis also unifies basic soft and hard thresholding functions by introducing a new family of smooth sigmoid based shrinkage functions: the SSBS functions. In fact, the hard and soft thresholding functions are degenerated SSBS functions, in contrast to non-degenerated SSBS functions which are smooth with very flexible attenuation degrees. In addition, the performance of the non-parametric estimation using detection thresholds and SSBS functions is analyzed for natural image denoising, via the mean square error and visual quality assessment.