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Géométrie différentielle des fibrés vectoriels et algèbres de Clifford appliquées au traitement d'images multicanaux

Abstract : This thesis is devoted to supply applications of Clifford algebras to multichannel image processing. Moreover, we introduce the use of vector bundles framework in image processing. Part 1 is devoted to multichannel image segmentation. We generalize Di Zenzo's approach to edge detection by constructing metric tensors related to the choice of the segmentation. Using the framewok of Clifford algebras bundles, we show that the choice of a segmentation of an image is related to the choice of a connection and a section on such a bundle. Part 2 is devoted to regularization. We make use of heat equations associated to generalized Laplacians on vector bundles. The main result of this part is the following. Considering the heat equation associated to the Hodge operator on the Clifford bundle of a well-chosen Riemannian manifold, we obtain a common framework for anisotropic regularization of images (videos), and related fields such as vector fields and orthonormal frame fields. At last, in Part 3, we deal with spectral analysis via the definition of a Fourier transform of a multichannel image. This definition is related to an abstract theory of Fourier transform based on the notion of group representation. From this point of view, the usual Fourier transform of grey level images is related with irreducible representations of the translations of the plane. We extend this Fourier transform to multichannel images by considering reducible representations of this group.
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https://tel.archives-ouvertes.fr/tel-00684250
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Submitted on : Saturday, March 31, 2012 - 1:03:30 PM
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Thomas Batard. Géométrie différentielle des fibrés vectoriels et algèbres de Clifford appliquées au traitement d'images multicanaux. Géométrie différentielle [math.DG]. Université de La Rochelle, 2009. Français. ⟨tel-00684250⟩

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