Real Clifford Algebra Cl(n,0), n=2,3(mod 4) Wavelet Transform

TL;DR

This paper constructs real Clifford algebra-valued wavelets for specific dimensions using the similitude group, emphasizing geometric interpretation and deriving properties like covariance, inversion, and an uncertainty principle.

Contribution

It introduces a method to build Clifford algebra wavelets for n=2,3 mod 4, replacing complex units with geometric blades, and explores their properties and applications.

Findings

Constructed Clifford wavelets using $SIM(n)$ group.

Derived admissibility and covariance properties.

Introduced a generalized Clifford wavelet uncertainty principle.

Abstract

We show how for $n=2,3 (\mod 4)$ continuous Clifford (geometric) algebra (GA) $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \in \C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a $Cl_{n}$ Clifford Fourier Transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle.