Clifford Wavelet Transform and the Uncertainty Principle
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Advances in Applied Clifford Algebras
Clifford Wavelet Transform and the Uncertainty Principle Hicham Banouh, Anouar Ben Mabrouk∗
and Mhamed Kesri
Communicated by Uwe Kaehler Abstract. The present paper lies in the whole topic of wavelet harmonic analysis on Clifford algebras. In which we derive a Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform. A brief review of Clifford algebra/analysis, wavelet transform on R and Clifford Fourier transform and their properties is conducted. Next, such concepts are applied to develop an uncertainty principle based on Clifford wavelets. Mathematics Subject Classification. 30G35, 42C40, 42B10, 15A66. Keywords. Harmonic analysis, Clifford algebra, Clifford analysis, Continuous wavelet transform, Clifford Fourier transform, Clifford wavelet transform, Uncertainty principle.
1. Introduction Transformations such as Fourier and wavelet types are powerful methods for signal/image processing. In Fourier analysis, the signals are transformed from the original domain to the spectral or frequency one. In the frequency domain many characteristics of a signal are seen more clearly. In a counterpart, wavelet bases functions are localized in both spatial and frequency domains and thus yield very sparse and well-structured representations of signals which are important facts in the processing. The first work on wavelet analysis has been developed by Morlet in Ref. [62] to study seismic waves. He also, with Grossman, developed a mathematical study of a continuous wavelet transform (see [33]). In Ref. [61] Meyer recognised the link between harmonic analysis and Morlet’s theory and gave a mathematical foundation to the continuous wavelet theory and thus gave rise to wavelet analysis. The continuous wavelet analysis of a square integrable function f begins by a ∗ Corresponding
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convolution with copies of a given “mother wavelet” ψ translated and dilated respectively by b ∈ R and a > 0. Such a function ψ has to fulfil an admissibility condition such as
Aψ =
2 ψ(ξ) R
|ξ|
dξ < +∞
where ψ is the classical Fourier transform of ψ. More information on real wavelets can be found in Refs. [20] and [32]. Wavelet theory has a great success especially in the applied fields such as image/signal processing, statsitics, finance and engineering. This success encourages researchers to develop new and more efficient tools in this theory to be adapted to understand different problems. Recall that nowadays 3D image processing for example is a challenging topic in many fields such as medicine, arts, physics, informatics, biology and thus needs more theoretical developments. Wavelets and Clifford algebras/analysis are ones of the powerful tools in this subject. The authors in Ref. [15] applied Clifford algebra concepts for the embedding of color information in images. Similarly, in Refs. [25] and [70] Clifford algebra has been used for image compression. In Ref. [69] a quaternionic wavelet met
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