Spherical Continuous Wavelet Transforms arising from sections of the Lorentz group

TL;DR

This paper develops spherical continuous wavelet transforms based on the conformal group of the sphere, extending previous work to include anisotropic conformal dilations using group representations and homogeneous spaces.

Contribution

It introduces a new framework for spherical CWTs using the proper Lorentz group and sections of homogeneous spaces, generalizing prior methods to anisotropic dilations.

Findings

Constructed wavelet transforms from the conformal group of the sphere.

Extended the work of Antoine and Vandergheynst to anisotropic conformal dilations.

Established a method for coherent states associated with square integrable group representations.

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 (CWT). 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 this homogeneous space are constituted by rotations of the subgroup Spin$(n)$ and M\"{o}bius transformations of the type $\phi_a(x)=(x-a)(1+ax)^{-1},$ where $a$ belongs to a given section on a homogeneous space of the unit ball. This extends in a natural way the work of Antoine and Vandergheynst to anisotropic conformal dilations.