Wavelets on the 2-Sphere: A Group-Theoretical Approach

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and +* for dilations on S2, which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X SO0(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO0(3, 1) acting in the space L2(S2, d ) of finite energy signals on S2. This representation is square integrable over X; thus it yields immediately the wavelets on S2 and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S2 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R . Then the parameter space goes into the similitude group of 2 and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.