Wavelets on the 2-Hyperboloid

The continuous wavelet transform is already a well established procedure for analysing data. Its main advantages over the classical Fourier transform are its local and multiresolution nature, which provide the interesting properties of a mathematical microscope. Its theory is well known in the case of the line and plane. However, the representation and analysis of signals in non-Euclidean geometry is a recurrent problem in many scientific domains. Not only certain data are constrained by nature on curved surfaces, but a lot of detectors collect information via interfaces which are geometrically complicated. Because of these demands, spherical wavelets [Antoine and Vandergheynst, 1999] were recently developed and applied, for example in Cosmology [Martinez-Gonzalez et al., 2002, Cayon et al., 2003]. Although the sphere is a manifold most desirable for applications, the mathematical analysis made so far invites us to consider other manifolds with similar geometrical properties, and first of all, other Riemmanian symmetric spaces of constant curvature. Among them, the two-sheeted hyperboloid stands as a very interesting case. For instance, in quantum mechanics such a manifold may be a particular example of a phase space [Gazeau et al., 1989]. Other examples come from physical systems constrained on a hyperbolic manifold, for instance, an open expanding model of the universe. A completely different example of application is provided by the emerging field of catadioptric image processing [Makadia and Daniilidis, 2003, Daniilidis et al.,