Spherical Wavelet Transform

Abstract.

In this article we present a group-theoretical approach for the continuous wavelet transform (CWT) on the unit sphere S ^n-1 based on its conformal group, the Lorentz group Spin(1, n), which is a double covering of the SO(1, n) group. We introduce transformations on the unit sphere from the decomposition of the conformal group into the maximal compact subgroup of rotations Spin(n) and the set of Möbius transformations of the form φ _a (x)  =  (x - a)(1  +  ax)⁻¹, where a ∈ B ⁿ and B ⁿ denote the open unit ball in ${{\mathbb{R}}}^n$. We present a class of local conformal dilatation/ contraction operators on the unit sphere S ^n-1 that allow us to define a family of spherical continuous wavelet transforms from which the CWT defined by J. P. Antoine and P. Vandergheynst is a particular case (see [1] and [2]).