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Analyses multirésolutions et problèmes de bords: applications au traitement d'images et à la résolution numérique d'équations aux dérivées partielles

Abstract : This work is devoted to the construction of new numerical wavelet-based methods for the resolution of Partial Differential Equations and for image compression. In the first part, we define and analyse a numerical algorithm that couples wavelet approximations with fictitious domain approach for the approximation of parabolic equations on a general 2D domain. We provide a mathematical analysis that proves the efficiency of this approach in terms of quality of results (error bound, local refinement), numerical efficiency (condition number, simple diagonal preconditioning) and tractability (fast and efficient computation). Two applications to the resolution of the heat equation defined on non-polygonal domains or evolving-in-time domains are presented. The second part deals with the construction of a new compression algorithm adapted to the geometry of the image. We start by introducing 1D multi-scale analyses of Harten's type depending on a family of points. These analyses lead to efficient multi-scale decompositions for discontinuous signals. This approach is then generalized to the 2D case and a compression algorithm depending on the edges of the image is introduced. It uses a map of edges previously obtained. Several comparisons between this new approach and other approaches are then presented.
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https://tel.archives-ouvertes.fr/tel-00008618
Contributor : Jean Baccou <>
Submitted on : Tuesday, March 1, 2005 - 5:06:36 PM
Last modification on : Wednesday, October 10, 2018 - 1:26:40 AM
Long-term archiving on: : Friday, September 14, 2012 - 11:35:38 AM

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  • HAL Id : tel-00008618, version 1

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Jean Baccou. Analyses multirésolutions et problèmes de bords: applications au traitement d'images et à la résolution numérique d'équations aux dérivées partielles. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2004. Français. ⟨tel-00008618⟩

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