An algebraic approach to discrete dilations. Application to discrete wavelet transforms |
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| Authors: | J. -P. Antoine Y. B. Kouagou D. Lambert B. Torrésani |
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| Affiliation: | (1) Institut de Physique Théorique, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium;(2) Present address: Institut de Mathématiques et de Sciences Physiques, BP-613, Porto-Novo, Bénin, West Africa;(3) Faculté des Sciences, FUNDP, B-5000 Namur, Belgium;(4) Laboratoire d’Analyse, Topologie et Probabilités, CMI, Université de Provence, 39 Avenue F. Joliot-Curie, 13453 Marseille, France;(5) Present address: Centre de Physique Théorique, CNRS Luminy, Case 907, F-13288 Marseille Cedex 09 |
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| Abstract: | We investigate the connections between continuous and discrete wavelet transforms on the basis of algebraic arguments. The discrete approach is formulated abstractly in terms of the action of a semidirect product A×Γ on ℓ2(Γ), with Γ a lattice and A an abelian semigroup acting of Γ. We show that several such actions may be considered, and investigate those which may be written as deformations of the canonical one. The corresponding deformed dilations (the pseudodilations) turn out to be characterized by compatibility relations of a cohomological nature. The connection with multiresolution wavelet analysis is based on families of pseudodilations of a different type. |
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| Keywords: | 42C15 43A30 20J05 |
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