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Advances in Computational Mathematics

, Volume 5, Issue 1, pp 51–94 | Cite as

Spherical wavelet transform and its discretization

  • Willi Freeden
  • Ulrich Windheuser
Article

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 (C0) (like Abel-Poisson or Gauß-Weierstraß operators) lead in a canonical way to (pyramidal) algorithms.

Keywords

Spherical continuous wavelet transform scale discretizations wavelet packets Daubechies wavelets (R-)multiresolution analysis 

AMS subject classification

41A58 42C15 44A35 45E99 

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References

  1. [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. [2]
    A.P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113–190.Google Scholar
  3. [3]
    A.P. Calderon and A. Zygmund, On a problem of Mihlin, Trans. Amer. Math. Soc. 78 (1955) 205–224.Google Scholar
  4. [4]
    C.K. Chui,An Introduction to Wavelets (Academic Press, 1992).Google Scholar
  5. [5]
    S. Dahlke and P. Maass, A continuous wavelet transform on tangent bundles of spheres, Preprint (1994).Google Scholar
  6. [6]
    I. Daubechies,Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, vol. 61 (SIAM, Philadelphia, 1992).Google Scholar
  7. [7]
    I. Daubechies, The wavelet transform, time-frequency analysis, and signal analysis, IEEE Trans. Inform. Theory 36 (1990) 961–1005.CrossRefGoogle Scholar
  8. [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.CrossRefGoogle Scholar
  9. [9]
    W. Freeden, Über eine Klasse von Integralformeln der Mathematischen Geodäsie, Veröff. Geod. Inst. RWTH Aachen, Heft 27 (1979).Google Scholar
  10. [10]
    W. Freeden and M. Schreiner, Nonorthogonal expansions on the sphere, Math. Meth. Appl. Sci. 18 (1995) 83–120.Google Scholar
  11. [11]
    T. Gronwall, On the degree of convergence of Laplace series, Trans. Amer. Math. Soc. 15 (1914) 1–30.MathSciNetGoogle Scholar
  12. [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.CrossRefGoogle Scholar
  13. [13]
    C.E. Heil and D.F. Walnut, Continuous and discrete wavelet transform, SIAM Rev. 31 (1989) 628–666.CrossRefGoogle Scholar
  14. [14]
    D. Marr,Vision (Freeman, San Francisco, CA, 1982).Google Scholar
  15. [15]
    C. Müller,Spherical Harmonics, Lecture Notes in Mathematics, vol. 17 (Springer, 1966).Google Scholar
  16. [16]
    B. Torresani, Phase space decomposition: Local Fourier analysis on spheres, Preprint (1993).Google Scholar
  17. [17]
    U. Windheuser, Sphärische Wavelets: Theorie und Anwendung in der Physikalischen Geodäsie, Doctoral Thesis, University of Kaiserslautern, Laboratory of Technomathematics, Geomathematics Group (1995).Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1996

Authors and Affiliations

  • Willi Freeden
    • 1
  • Ulrich Windheuser
    • 1
  1. 1.Laboratory of Technomathematics, Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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