Wavelets on non-Euclidean manifolds

This 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.