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

Abstract

We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the 2sphere 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 SO o (3; 1) via the Iwasawa decomposition, so that X ' SO o (3; 1)=N, where N ' C . We select an appropriate unitary representation of SO o (3; 1) acting in the space L 2 (S 2 ; dd) of nite 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 nd 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 ! 1. 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.