E. Hitzer: Real Cliff. Alg. Cl(n,0), n = 2,3(mod 4) Wavelet Transf.


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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