E.S.M. Hitzer
Clifford (Geometric) Algebra Wavelet Transform
accepted for: V. Skala and D. Hildenbrand (eds.), Proceedings of GraVisMa 2009, 02-04 Sep. 2009, Plzen, Czech Republic, pp. 94-101 (2009). Revised: 22 January 2010.

Abstract: 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 element of C (complex numbers) 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 CliRord 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.

AMS Subj. Class. 15A66, 42C40, 94A12.

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


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