Skip to main content
SpringerLink
Log in

Wavelets in Clifford Analysis

  • Reference work entry
  • First Online:
Operator Theory

Abstract

Clifford analysis is a higher dimensional functions theory for the Dirac operator and builds a bridge between complex function theory and harmonic analysis. The construction of wavelets is done in three different ways. Firstly, a monogenic mother wavelet is obtained from monogenic extensions (Cauchy–Kovalevskaya extensions) of special functions like Hermite and Laguerre polynomials. Based on the kernel function, Cauchy wavelets are also monogenic but not square integrable in the usual sense. On the other hand, these wavelets and their kernels are connected to the Cauchy–Riemann equations in the upper half space as well as to Bergman and Hardy spaces. Secondly, a group theoretical approach is used to construct wavelets. This approach considers pure dilations and rotations as group actions on the unit sphere. It can be generalized by using the action of the Spin group because the Spin group is a double cover of the rotation group, whereas dilations arise from Möbius transformations. Here, Clifford analysis gives the tools to construct wavelets. Finally, an application to image processing based on monogenic wavelets is considered. Here, the starting point are scalar-valued functions and the resulting Clifford wavelets are boundary values of monogenic functions in the upper half space. One proceeds in two steps. First choose a real- or complex-valued primary wavelet and then construct from that using the Riesz transform = Hilbert transform Clifford wavelets and Clifford wavelet frames.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ahlfors, L.V.: Clifford numbers and Möbius transformations in \(\mathbb{R}^{n}\). In: Chisholm, J.S.R., Commen, A.K. (eds.) Clifford Algebras and Their Applications in Mathematical Physics. NATO ASI Series C, vol. 183, pp. 167–175. D. Reidel, Dordrecht (1986)

    Chapter  Google Scholar 

  2. Antoine, J.-P., Vandergheynst, P.: Wavelets on the n-sphere and other manifolds. J. Math. Phys. 39, 3987–4008 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernstein, S.: Spherical singular integrals, monogenic Kernels and wavelets on the three dimensional sphere. Adv. Appl. Clifford Algebras 19(2), 173–189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernstein, S., Ebert, S.: Kernel based wavelets on S 3. J. Concr. Appl. Math. 8(1), 110–124 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Bernstein, S., Ebert, S.: Wavelets on S 3 and SO(3)—their construction, relation to each other and Radon transform of wavelets on SO(3). Math. Meth. Appl. Sci. 30(16), 1895–1909 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Brackx, F., Sommen, F.: Clifford-Hermite wavelets in euclidean space. J. Fourier Anal. Appl. 6(3), 299–310 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brackx, F., Sommen, F.: The generalized Clifford-Hermite continuous wavelet transform. Adv. Appl. Clifford Algebras 11(S1), 219–231 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brackx, F., De Schepper, N., Sommen, F.: The Clifford-Laguerre continuous wavelet transform. Bull. Belg. Math. Soc. 10, 201–215 (2003)

    MATH  Google Scholar 

  9. Brackx, F., De Schepper, N., Sommen, F.: Clifford-Jacobi polynomials and the associated continuous wavelet transform in euclidean space. In: Qian, T., Vai, M.I., Yuesheng, X. (eds.) Applied and Numerical Harmonic Analysis, pp. 185–198. Birkhäuser, Basel (2006)

    Google Scholar 

  10. Cerejeiras, P., Ferreira, M., Kähler, U.: Monogenic wavelets over the unit ball. J. Anal. Appl. 24(4), 841–852 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Cerejeiras, P., Ferreira, M., Kähler, U., Teschke, G.: Inversion of the noisy Radon transform on SO(3) by Gabor frames and sparse recovery principles. Appl. Comput. Harmon. Anal. 31(3), 325–345 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Christensen, O.: Frames and Bases: An Introductory Course. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel (2008)

    Book  MATH  Google Scholar 

  13. Cnops, J.: The wavelet transform in clifford analysis. Comput. Meth. Funct. Theory 1(2), 353–374 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cnops, J., Kisil, V.: Monogenic functions and representations of nilpotent Lie groups in quantum mechanics. Math. Meth. Appl. Sci. 22(4), 353–373 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Delanghe, R.: Clifford analysis: history and perspective. Comput. Math. Funct. Theory 1(1), 107–153 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Delanghe, R., Brackx, F.: Hypercomplex function theory and Hilbert modules with reproducing Kernel. Proc. Lond. Math. Soc. III Ser. 37, 545–576 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ebert, S.: Wavelets on Lie groups and homogeneous spaces. Ph.D. thesis, Department of Mathematics and Informatics, TU Bergakademie Freiberg (2011)

    Google Scholar 

  18. Ebert, S., Wirth, J.: Diffusive wavelets on groups and homogeneous spaces. Proc. R. Soc. Edinb. 141A, 497–520 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Felsberg, M., Sommer, G.: The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)

    Article  MathSciNet  Google Scholar 

  20. Ferreira, M.: Continuous wavelet transforms on the unit sphere. Ph.D. thesis, Universidade de Aveiro, Departmento de Matemàtica (2008)

    Google Scholar 

  21. Ferreira, M.: Spherical continuous wavelet transform arising from sections of the Lorentz group. Appl. Comput. Harmon. Anal. 26(2), 212–229 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Claredon Press, Oxford (1997)

    MATH  Google Scholar 

  23. Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  24. Held, S.: Monogenic wavelet frames for image analysis. Ph.D. thesis, Fakultät für Mathematik, TU München (2012)

    Google Scholar 

  25. Kisil, V.: Analysis in R (1, 1) or the principal function theory. Complex Var. Theory Appl. 40(2), 93–118 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kisil, V.: Wavelets in Banach spaces. Acta Appl. Math. 59(1), 79–109 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Marr, D.: Vision. Freeman, San Francisco (1982)

    Google Scholar 

  28. McIntosh, A.: Clifford algebras, Fourier theory, singular integrals and harmonic functions on Lipschitz domains. In: Ryan, J. (ed.) Clifford Algebras in Analysis and Related Topics, pp. 33–87. CRC Press, Boca Rotan (1996)

    Google Scholar 

  29. Mitrea, M.: Clifford Wavelets, Singular Integrals, and Hardy Spaces. Lecture Notes in Mathematics, vol. 1575. Springer, Berlin (1994)

    MATH  Google Scholar 

  30. Ungar, A.A.: Analytic Hyperbolic Geometry: Mathematical Foundation and Applications. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  31. Unser, M., Sage, D., Van De Ville, D.: Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform. IEEE Trans. Image Process. 18(11), 2402–3811 (2009)

    Article  MathSciNet  Google Scholar 

  32. Unser, M., Chenouard, N.: A unifying parametric framework for 2D steerable wavelet transforms. SIAM J. Imaging Sci. 6(1), 102–135 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vahlen, Th.: Über Bewegungen und complexe Zahlen. Math. Ann. 55, 585–593 (1902)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, Technische Universität Bergakademie Freiberg, Freiberg, Germany

    Swanhild Bernstein

Authors
  1. Swanhild Bernstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swanhild Bernstein .

Editor information

Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Basel

About this entry

Cite this entry

Bernstein, S. (2015). Wavelets in Clifford Analysis. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_17

Download citation

Publish with us

Policies and ethics

Search

Navigation