E. Hitzer: Clifford (Geometric) Algebra Wavelet Transform

Clifford (Geometric) Algebra Wavelet Transform

by Eckhard Hitzer
Department of Applied Physics, University of Fukui, 910-8507 Japan
Submitted (October 10, 2009) to:
Proceedings of GraVisMa 2009, 02-04 September 2009, Plzen, Czech Republic, edited by V. Skala and D. Hildenbrand.

While the Clifford (geometric) algebra Fourier Transform (CFT) is global, we introduce here the local Clifford (geometric) algebra (GA) wavelet concept. We show how for n = 2,3(mod 4) continuous 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 CFT 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 explicit example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle. Extensions of CFTs and Clifford wavelets to Cl_{0,n’}, n’ = 1,2 (mod 4) appear straight forward.

Source: E. Hitzer (hitzer_at_mech.fukui-u.ac.jp)


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