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

✍ Scribed by J.-P Antoine; P Vandergheynst

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 421 KB
• Volume: 7
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.1999.0272

✦ Synopsis

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 SO 0 (3, 1) via the Iwasawa decomposition, so that X SO 0 (3, 1)/N, where N C. We select an appropriate unitary representation of SO 0 (3, 1) acting in the space L 2 (S 2 , dµ) 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 → ∞. 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.