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 S-2, 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 R-*(+) for dilations on S-2, which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X similar or equal to SO0(3, 1)/N, where N similar or equal to C. We select an appropriate unitary representation of SO0(3, 1) acting in the space L-2(S-2, d mu) of finite energy signals on S-2. This representation is square integrable over X; thus it yields immediately the wavelets on S-2 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 S-2 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R --> infinity. Then the parameter space goes into the similitude group of R-2 and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility. (C) 1999 Academic Press