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Développement de méthodes d'estimation modale de signaux multidimensionnels. Application à la spectroscopie RMN multidimensionnelle

Abstract : This thesis aims at the developpement of modal analysis algorithms for multidimensional signals (R-D) presenting resolution and numerical complexity problems. A multidimensional signal of dimension R is the superimposition of products of R monodimensional sinusoids. The intended application is NMR spectroscopy. Firstly, after a state-of-the-art on the so-called ''algebraic'' estimation methods, we propose a parametric method based on tensors. It uses the multidimensional tensor lattice of the R-D modal signal and exploits the eigenvectors structure of the signal subspace obtained using a higher-order singular value decomposition (HOSVD). Unlike most tensor-based eigenvalue approaches, modes estimated by the proposed method are automatically paired, thus it avoids a separate pairing step and joint diagonalization. Secondly, the multidimensional modal estimation problem is formulated as a sparse approximation problem in which the dictionary is obtained by the discretization of complex exponential functions. To achieve good spectral resolution, it is necessary to choose a very fine grid, which leads to handling a large dictionary with all the underlying computational problems. Hence, we propose a novel method that consists in combining a sparse approximation and a multigrid approach on several levels of resolution. The approach is demonstrated using several 1-D and 2-D examples. In addition, the influence of the initial dictionary on the algorithm convergence is also studied. The developed methods are then applied to estimate 1-D and 2-D NMR signal parameters. To reduce the computation cost in the case of large bidimensional signals, we also propose an approach exploiting the simultaneous sparsity principle to estimate the coordinates of the modes on each dimension. The procedure involves two 1-D sparse approximations followed by a 2-D modes painring step.
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Souleymen Sahnoun. Développement de méthodes d'estimation modale de signaux multidimensionnels. Application à la spectroscopie RMN multidimensionnelle. Autre. Université de Lorraine, 2012. Français. ⟨NNT : 2012LORR0169⟩. ⟨tel-01749354⟩

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