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

by Eckhard Hitzer

(Department of Applied Physics, University of Fukui, 910-8507 Japan)

edited by T.E. Simos et al., AIP Proceedings of ICNAAM 2009, No. 1168, pp. 781-784 (2009).

**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* of **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.

**Keywords**

Clifford geometric algebra, Clifford wavelet transform, multidimensional wavelets, continuous wavelets, similitude group

**AMS Subj. Class.**

15A66, 42C40, 94A12

*Source:* E. Hitzer, hitzer_at_mech.fukui-u.ac.jp

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