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Metric Dependent Clifford Analysis with Applications to Wavelet Analysis

  • Fred Brackx
  • Nele De Schepper
  • Frank Sommen
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 167)

Abstract

In earlier research multi-dimensional wavelets have been constructed in the framework of Clifford analysis. Clifford analysis, centered around the notion of monogenic functions, may be regarded as a direct and elegant generalization to higher dimension of the theory of the holomorphic functions in the complex plane. This Clifford wavelet theory might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one.

In this paper we develop the idea of a metric dependent Clifford analysis leading to a so-called anisotropic Clifford wavelet theory featuring wavelet functions which are adaptable to preferential, not necessarily orthogonal, directions in the signals or textures to be analyzed.

Keywords

Continuous Wavelet Transform Clifford analysis Hermite polynomials 

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References

  1. [1]
    J.-P. Antoine, R. Murenzi, P. Vandergheynst and Syed Twareque Ali, Two-Dimensional Wavelets and their Relatives, Cambridge University Press, Cambridge, 2004.zbMATHGoogle Scholar
  2. [2]
    F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Publishers (Boston-London-Melbourne, 1982).zbMATHGoogle Scholar
  3. [3]
    F. Brackx, R. Delanghe and F. Sommen, Differential Forms and/or Multi-vector Functions, CUBO A Mathematical Journal 7 (2005), no. 2, 139–169.MathSciNetzbMATHGoogle Scholar
  4. [4]
    F. Brackx, N. De Schepper and F. Sommen, The Bi-axial Clifford-Hermite Continuous Wavelet Transform, Journal of Natural Geometry 24 (2003), 81–100.zbMATHGoogle Scholar
  5. [5]
    F. Brackx, N. De Schepper and F. Sommen, The Clifford-Gegenbauer Polynomials and the Associated Continuous Wavelet Transform, Integral Transform. Spec. Funct. 15 (2004), no. 5, 387–404.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    F. Brackx, N. De Schepper and F. Sommen, The Clifford-Laguerre Continuous Wavelet Transform, Bull. Belg. Math. Soc. — Simon Stevin 11(2), 2004, 201–215.zbMATHMathSciNetGoogle Scholar
  7. [7]
    F. Brackx, N. De Schepper and F. Sommen, Clifford-Jacobi Polynomials and the Associated Continuous Wavelet Transform in Euclidean Space (accepted for publication in the Proceedings of the 4th International Conference on Wavelet Analysis and Its Applications, University of Macau, China, 2005).Google Scholar
  8. [8]
    F. Brackx, N. De Schepper and F. Sommen, New multivariable polynomials and their associated Continuous Wavelet Transform in the framework of Clifford Analysis (submitted for publication in the Proceedings of the International Conference on Recent trends of Applied Mathematics based on partial differential equations and complex analysis, Hanoi, 2004).Google Scholar
  9. [9]
    F. Brackx and F. Sommen, Clifford-Hermite Wavelets in Euclidean Space, Journal of Fourier Analysis and Applications 6, no. 3 (2000), 299–310.zbMATHMathSciNetGoogle Scholar
  10. [10]
    F. Brackx and F. Sommen, The Generalized Clifford-Hermite Continuous Wavelet Transform, Advances in Applied Clifford Algebras 11(S1), 2001, 219–231.MathSciNetCrossRefGoogle Scholar
  11. [11]
    P. Calderbank, Clifford analysis for Dirac operators on manifolds-with-boundary, Max Planck-Institut für Mathematik (Bonn, 1996).Google Scholar
  12. [12]
    C.K. Chui, An Introduction to Wavelets, Academic Press, Inc., San Diego, 1992.zbMATHGoogle Scholar
  13. [13]
    J. Cnops, An introduction to Dirac operators on manifolds, Birkhäuser Verlag (Basel, 2002).zbMATHGoogle Scholar
  14. [14]
    I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.zbMATHGoogle Scholar
  15. [15]
    R. Delanghe, F. Sommen and V. Souícek, Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publishers (Dordrecht, 1992).zbMATHGoogle Scholar
  16. [16]
    D. Eelbode and F. Sommen, Differential Forms in Clifford Analysis (accepted for publication in the Proceedings of the International Conference on Recent trends of Applied Mathematics based on partial differential equations and complex analysis, Hanoi, 2004).Google Scholar
  17. [17]
    M. Felsberg, Low-Level Image Processing with the Structure Multivector, PhD-thesis, Christian-Albrechts-Universität, Kiel, 2002.zbMATHGoogle Scholar
  18. [18]
    J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press (Cambridge, 1991).zbMATHGoogle Scholar
  19. [19]
    K. Gürlebeck and W. Sprössig, Quaternionic analysis and elliptic boundary value problems, Birkhäuser Verlag (Basel, 1990).zbMATHGoogle Scholar
  20. [20]
    K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley & Sons (Chichester etc., 1997).zbMATHGoogle Scholar
  21. [21]
    G. Kaiser, A Friendly Guide to Wavelets, Birkhäuser Verlag (Boston, 1994).zbMATHGoogle Scholar
  22. [22]
    G. Kaiser, private communication.Google Scholar
  23. [23]
    N. Marchuk, The Dirac Type Tensor Equation in Riemannian Spaces, In: F. Brackx, J.S.R. Chisholm and V. Souček (eds.), Clifford Analysis and Its Applications, Kluwer Academic Publishers (Dordrecht-Boston-London, 2001).Google Scholar
  24. [24]
    M. Mitrea, Clifford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics 1575, Springer-Verlag (Berlin, 1994).zbMATHGoogle Scholar
  25. [25]
    T. Qian, Th. Hempfling, A. McIntosh and F. Sommen (eds.), Advances in Analysis and Geometry: New Developments Using Clifford Algebras, Birkhäuser verlag (Basel-Boston-Berlin, 2004).zbMATHGoogle Scholar
  26. [26]
    J. Ryan and D. Struppa (eds.), Dirac operators in analysis, AddisonWesley Longman Ltd, (Harlow, 1998).zbMATHGoogle Scholar
  27. [27]
    F. Sommen, Special Functions in Clifford analysis and Axial Symmetry. Journal of Math. Analysis and Applications 130 (1988), no. 1, 110–133.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Fred Brackx
    • 1
  • Nele De Schepper
    • 1
  • Frank Sommen
    • 1
  1. 1.Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering — Faculty of SciencesGhent UniversityGentBelgium

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