New Type of Gegenbauer-Jacobi-Hermite Monogenic Polynomials and Associated Continuous Clifford Wavelet Transform Some Monogenic Clifford Polynomials and Associated Wavelets

• Published in: Acta applicandae mathematicae Vol. 170; no. 1; pp. 1 - 35
• Main Authors: Arfaoui, Sabrine, Ben Mabrouk, Anouar, Cattani, Carlo
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2020
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Probability Theory and Stochastic Processes; Fourier-Plancherel, monogenic polynomials, EEG/ECG, Brain images; 44A15; Continuous wavelet transform; Clifford analysis; 42B10; 30G35; Clifford Fourier transform
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-020-00322-0

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.