Skip to main content
Springer Nature Link
Log in

Spherical wavelet transform and its discretization

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to wavelet packets and scale discrete wavelets. The essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (C 0) (like Abel-Poisson or Gauß-Weierstraß operators) lead in a canonical way to (pyramidal) algorithms.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Berens, P.L. Butzer and S. Pawelke, Limitierungsverfahren mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten, Publ. Res. Inst. Math. Sci. Ser. A4 (1968) 201–268.

    Google Scholar 

  2. A.P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113–190.

    Google Scholar 

  3. A.P. Calderon and A. Zygmund, On a problem of Mihlin, Trans. Amer. Math. Soc. 78 (1955) 205–224.

    Google Scholar 

  4. C.K. Chui,An Introduction to Wavelets (Academic Press, 1992).

  5. S. Dahlke and P. Maass, A continuous wavelet transform on tangent bundles of spheres, Preprint (1994).

  6. I. Daubechies,Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, vol. 61 (SIAM, Philadelphia, 1992).

    Google Scholar 

  7. I. Daubechies, The wavelet transform, time-frequency analysis, and signal analysis, IEEE Trans. Inform. Theory 36 (1990) 961–1005.

    Article  Google Scholar 

  8. M. Duval-Destin, M.A. Muschietti and B. Torresani, Continuous wavelet decompositions, multiresolution, and contrast analysis, SIAM J. Math. Anal. 24 (1993) 739–755.

    Article  Google Scholar 

  9. W. Freeden, Über eine Klasse von Integralformeln der Mathematischen Geodäsie, Veröff. Geod. Inst. RWTH Aachen, Heft 27 (1979).

  10. W. Freeden and M. Schreiner, Nonorthogonal expansions on the sphere, Math. Meth. Appl. Sci. 18 (1995) 83–120.

    Google Scholar 

  11. T. Gronwall, On the degree of convergence of Laplace series, Trans. Amer. Math. Soc. 15 (1914) 1–30.

    MathSciNet  Google Scholar 

  12. A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984) 723–736.

    Article  Google Scholar 

  13. C.E. Heil and D.F. Walnut, Continuous and discrete wavelet transform, SIAM Rev. 31 (1989) 628–666.

    Article  Google Scholar 

  14. D. Marr,Vision (Freeman, San Francisco, CA, 1982).

    Google Scholar 

  15. C. Müller,Spherical Harmonics, Lecture Notes in Mathematics, vol. 17 (Springer, 1966).

  16. B. Torresani, Phase space decomposition: Local Fourier analysis on spheres, Preprint (1993).

  17. U. Windheuser, Sphärische Wavelets: Theorie und Anwendung in der Physikalischen Geodäsie, Doctoral Thesis, University of Kaiserslautern, Laboratory of Technomathematics, Geomathematics Group (1995).

Download references

Author information

Authors and Affiliations

  1. Laboratory of Technomathematics, Geomathematics Group, University of Kaiserslautern, D-67663, Kaiserslautern, Germany

    Willi Freeden & Ulrich Windheuser

Authors
  1. Willi Freeden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ulrich Windheuser
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by C.K. Chui

Supported by the “Graduiertenkolleg Technomathematik, Kaiserslautern”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freeden, W., Windheuser, U. Spherical wavelet transform and its discretization. Adv Comput Math 5, 51–94 (1996). https://doi.org/10.1007/BF02124735

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02124735

Keywords

AMS subject classification

Search

Navigation