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000056050 001__ 56050 000056050 005__ 20190509132101.0 000056050 0247_ $$2doi$$a10.5075/epfl-thesis-3409 000056050 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis3409-9 000056050 02471 $$2nebis$$a5051236 000056050 037__ $$aTHESIS 000056050 041__ $$aeng 000056050 088__ $$a3409 000056050 245__ $$aWavelets on non-Euclidean manifolds 000056050 269__ $$a2006 000056050 260__ $$bEPFL$$c2006$$aLausanne 000056050 300__ $$a145 000056050 336__ $$aTheses 000056050 502__ $$aTwareque Ali, Jean-Pierre Antoine, Murat Kunt, Jean-Philippe Thiran 000056050 520__ $$aThis dissertation investigates wavelets as a multiscale tool on non-Euclidean manifolds. The growing importance of using non-Euclidean manifolds as a geometric model for data comes from the diversity of the data collected. In this work we mostly deal with the sphere and the hyperboloid. First, given the recent success of the continuous wavelet transform on the sphere a natural extension is to build discrete frames. Then, from a more theoretical perspective, having already wavelets on the sphere, which is a non-Euclidean manifold of constant positive curvature, it is interesting and even challenging to build and prove the existence of wavelets on its dual manifold-the hyperboloid as non-Euclidean manifold of constant negative curvature. This dissertation starts with detailing the construction of one- and two-dimensional Euclidean wavelets in both continuous and discrete versions. Next, it continues with details on the construction of wavelets on the sphere. In the three cases (line, plane and sphere) the group theoretical approach for constructing wavelets is used. We develop discrete wavelet frames on the sphere by discretizing the existing spherical continuous wavelet transform. First, half-continuous wavelet frames are derived. Second, we show that a controlled frame may be constructed in order to get an easy reconstruction of functions from their decomposition coefficients. Finally we completely discretize the continuous wavelet transform on the sphere and give examples of frame decomposition of spherical data. As a close parent of the wavelet transform we also implement the Laplacian Pyramid on the sphere. Another important part of this dissertation is dedicated to the hyperboloid. We build a total family of functions, in the space of square-integrable functions on the hyperboloid, by picking a probe with suitable localization properties, applying on it hyperbolic motions and supplemented by appropriate dilations. Based on a minimal set of axioms, we define appropriate dilations for the hyperbolic geometry. Then, the continuous wavelet transform on the hyperboloid is obtained by convolution of the scaled wavelets with the signal. This transform is proved to be a well-defined invertible map, provided the wavelets satisfy an admissibility condition. As a final part in this dissertation, we discuss one possible application of non-Euclidean wavelets – the processing of non-Euclidean images. This leads to implementing some other basic non-Euclidean image processing techniques, for example scale-space analysis and active contour, that we apply to catadioptric image processing. 000056050 6531_ $$ahyperboloid 000056050 6531_ $$anon-commutative harmonic analysis 000056050 6531_ $$aomni-directional images 000056050 6531_ $$aparaboloid 000056050 6531_ $$asphere 000056050 6531_ $$awavelets 000056050 700__ $$0241302$$g128491$$aBogdanova, Iva 000056050 720_2 $$aVandergheynst, Pierre$$edir.$$g120906$$0240428 000056050 8564_ $$uhttps://infoscience.epfl.ch/record/56050/files/EPFL_TH3409.pdf$$zTexte intégral / Full text$$s12863245$$yTexte intégral / Full text 000056050 909C0 $$xU10380$$0252392$$pLTS2 000056050 909CO $$pthesis$$pthesis-bn2018$$pDOI$$ooai:infoscience.tind.io:56050$$qDOI2$$qGLOBAL_SET$$pSTI 000056050 918__ $$bSTI-SEL$$cITS$$aSTI 000056050 919__ $$aLTS2 000056050 920__ $$b2005$$a2005-12-21 000056050 970__ $$a3409/THESES 000056050 973__ $$sPUBLISHED$$aEPFL 000056050 980__ $$aTHESIS