Clifford Algebra-Valued Wavelet Transform on Multivector Fields

Abstract

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cln,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \({\mathbb{R}^{n}}\) . We express the admissibility condition in terms of the Cln,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cln,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.