E. Hitzer

**Tutorial
on Fourier Transformations and Wavelet Transformations in Clifford Geometric Algebra**

in K. Tachibana (ed.)
*Lecture notes of the International Workshop for
"Computational Science with Geometric Algebra"
(FCSGA2007),
Nagoya University, Japan, 14-21 Feb. 2007*,
pp. 65-87 (2007).

**Abstract:**
First, the basic concept multivector functions and their vector derivative
in geometric algebra (GA) is introduced. Second, beginning
with the Fourier transform on a scalar function we generalize to a
real Fourier transform on GA multivector-valued functions (f : R^3 -->
Cl(3,0)). Third, we show a set of important properties of the Clifford
Fourier transform (CFT) on Cl(3,0) such as differentiation properties,
and the Plancherel theorem. We round off the treatment of the CFT
(at the end of this tutorial) by applying the Clifford Fourier transform
properties for proving an uncertainty principle for Cl(3,0) multivector
functions.
For wavelets in GA it is shown how continuous Clifford Cl(3,0)-valued
admissible wavelets can be constructed using the similitude
group SIM(3), a subgroup of the affine group of R^3. We express the
admissibility condition in terms of the 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 of multivector functions. We explain (at the end of this tutorial)
a generalized Clifford wavelet uncertainty principle. For scalar
admissibility constant it sets bounds of accuracy in multivector wavelet
signal and image processing. As concrete example we introduce
multivector Clifford Gabor wavelets, and describe important properties
such as the Clifford Gabor transform isometry, a reconstruction
formula, and (at the end of this tutorial) an uncertainty principle for
Clifford Gabor wavelets.

**Keywords:**
Vector derivative, multivector-valued function, Clifford
(geometric) algebra, Clifford Fourier transform, uncertainty principle,
similitude group, geometric algebra wavelet transform, geometric
algebra Gabor wavelets.

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