Stereographic Wavelet Frames on the Sphere

In this paper we exploit the Continuous Wavelet Transform (CWT) on the two dimensional sphere, introduced previously by two of us, to build associated Discrete Wavelet Frames. We first explore half-continuous frames, i.e., frames where the position remains a continuous variable, and then move on to a fully discrete theory. We introduce the notion of controlled frames, which reflects the particular nature of the underlying theory, in particular the apparent conflict between dilation and the compactness of the $S^2$ manifold. We also highlight some implementation issues and provide numerical illustrations.