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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