Wavelet transform on manifolds: Old and new approaches

Given a two-dimensional smooth manifold M and a bijective projection p from M on a fixed plane (or a subset of that plane), we explore systematically how a wavelet transform (WT) on M may be generated from a plane WT by the inverse projection p−1. Examples where the projection maps the whole manifold onto a plane include the two-sphere, the upper sheet of the two-sheeted hyperboloid and the paraboloid. When no such global projection is available, the construction may be performed locally, i.e., around a given point on M. We apply this procedure both to the continuous WT, already treated in the literature, and to the discrete WT. Finally, we discuss the case of a WT on a graph, for instance, the graph defined by linking the elements of a discrete set of points on the manifold.