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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