New Type of Gegenbauer-Jacobi-Hermite Monogenic Polynomials and Associated Continuous Clifford Wavelet Transform
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New Type of Gegenbauer-Jacobi-Hermite Monogenic Polynomials and Associated Continuous Clifford Wavelet Transform Some Monogenic Clifford Polynomials and Associated Wavelets Sabrine Arfaoui1,2 · Anouar Ben Mabrouk1,2,3 · Carlo Cattani4 Received: 4 September 2018 / Accepted: 6 March 2020 © Springer Nature B.V. 2020
Abstract Recently 3D image processing has interested researchers in both theoretical and applied fields and thus has constituted a challenging subject. Theoretically, this needs suitable functional bases that are easy to implement by the next. It holds that Clifford wavelets are main tools to achieve this necessity. In the present paper we intend to develop some new classes of Clifford wavelet functions. Some classes of new monogenic polynomials are developed firstly from monogenic extensions of 2-parameters Clifford weights. Such classes englobe the well known Jacobi, Gegenbauer and Hermite ones. The constructed polynomials are next applied to develop new Clifford wavelets. Reconstruction and Fourier-Plancherel formulae have been proved. Finally, computational examples are developed provided with graphical illustrations of the Clifford mother wavelets in some cases. Some graphical illustrations of the constructed wavelets have been provided and finally concrete applications in biofields have been developed dealing with EEG/ECG and Brain image processing. Keywords Continuous wavelet transform · Clifford analysis · Clifford Fourier transform · Fourier-Plancherel, monogenic polynomials, EEG/ECG, Brain images Mathematics Subject Classification 42B10 · 44A15 · 30G35
B S. Arfaoui
[email protected] A. Ben Mabrouk [email protected] C. Cattani [email protected]
1
Algebra, Number Theory and Nonlinear Analysis Laboratory LR18ES15, Department of Mathematics, Faculty of Sciences, University of Monastir, 5000 Monastir, Tunisia
2
Department of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk, Saudi Arabia
3
Department of Mathematics, Higher Institute of Applied Mathematics and Computer Science, University of Kairouan, 3100 Kairouan, Tunisia
4
Engineering School (DEIM), Tuscia University, Tuscia, Italy
S. Arfaoui et al.
1 Introduction and Motivations Wavelets since their discovery have been proved to be powerful tools that are able to resolve and/or to describe many complex phenomena that are miss-understood with classical tools. Nowadays, wavelets appear in quasi all scientific fields from pure mathematics to applied mathematics, physics, engineering, computer sciences, statistics and especially bio-fields. These facts have impressed researchers especially in pure mathematics to develop more and more its theory and to provide for scientific community more and more wavelet bases to overcome problems that remain needing more sophisticated functional bases to be understood. It seems that Clifford algebras which are already known in image processing especially, are important frameworks to develop new wavelet functions which by the next will be addressed to describe many probl
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