Spherical continuous wavelet transforms arising from sections of the Lorentz group

Abstract

We consider the conformal group of the unit sphere S n−1 , the so-called proper Lorentz group Spin + (1, n), for the study of spherical continuous wavelet transforms. Our approach is based on the method for construction of general coherent states associated to square integrable group representations over homogeneous spaces. The underlying homogeneous space is an extension to the whole of the group Spin + (1, n) of the factorization of the gyrogroup of the unit ball by an appropriate gyro-subgroup. Sections on it are constituted by rotations of the subgroup Spin(n) and Möbius transformations of the type ϕ a (x) = (x − a)(1 + ax) −1 , where a belongs to a given section on a quotient space of the unit ball. This extends in a natural way the work of Antoine and Vandergheynst to anisotropic conformal dilations on the unit sphere.

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

“….V; Q// Â .V; Q/ because .V; Q/ is invariant under the Clifford conjugation. Note also that if v is a vector in V, then, by (15),…”

“…For links between gyrogroups and loops, see [34-36, 41, 44, 45]. For applications of gyrogroup theory in analysis and signal processing, see [15,17].…”

Wavelets onS3andSO(3)-Their construction, relation to each other and Radon transform of wavelets onSO(3)

“…In the paper of M. Ferreira it can be seen that AK B n , where B n denotes the unit ball. The decomposition of an element with the help of the weaker Cartan decomposition is not unique [9].…”

“…All possible sections are worked out in [9]. The dilation and translation corresponds to what is obtained by the stereographic projection of the dilation and translation from the Tangent space at the north pole onto the sphere [10].…”

Wavelets invariant under finite reflection groups

“…Among several approaches two appear as the most promising, one initiated by Antoine and Vandergheynst [1,2] (see also [3,4]) where the authors use representations over homogeneous spaces and the approach via approximate identities and singular kernels (see Freeden, et al [5]). These wavelets have been used in a variety of applications.…”