Numerical Algorithms

, Volume 48, Issue 1–3, pp 135–160 | Cite as

Polyharmonic multiresolution analysis: an overview and some new results

  • Christophe Rabut
  • Milvia Rossini
Original Paper


This paper first presents a condensed state of art on multiresolution analysis using polyharmonic splines: definition and main properties of polyharmonic splines, construction of B-splines and wavelets, decomposition and reconstruction filters; properties of the so-obtained operators, convergence result and applications are given. Second this paper presents some new results on this topic: scattered data wavelet, new polyharmonic scaling functions and associated filters. Fourier transform is of extensive use to derive the tools of the various multiresolution analysis.


Polyharmonic splines Wavelets MRA filters 

Mathematics Subject Classifications (2000)

65D07 65T60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bacchelli, B., Bozzini, M., Rabut, C., Varas, M.L.: Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets. Appl. Comput. Harmon. Anal. 18, 282–299 (2005)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bacchelli, B., Bozzini, M., Rabut, C.: A fast wavelet algorithm for multidimensional signal using polyharmonic splines. In: Cohen, A., Merrien, J.L., Schumaker, L.L. (eds.) Curves and Surfaces Fitting, pp. 21–30. NashboroPress, Brentwood, TN (2003)Google Scholar
  3. 3.
    Bacchelli, B., Bozzini, M., Rossini, M.: On the errors of a multidimensional MRA based on non separable scaling functions. Int. J. Wavelets, Multiresolution Inf. Process. 4(3), 475–488 (2006)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bozzini, M., Rabut, C., Rossini, M.: A multiresolution analysis with a new family of polyharmonic B-splines. In: Cohen, A., Merrien, J.L., Schumaker, L. (eds.) Curve and Surface Fitting: Avignon 2006. Nashboro Press (2007)Google Scholar
  5. 5.
    Bozzini, M., Rossini, M.: Approximating surfaces with discontinuities. Math. Comput. Model. 31(6/7), 193–216 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Buhmann, M.: Radial Basis Functions. Cambridge University Press (2003)Google Scholar
  7. 7.
    Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. R.A.I.R.O. Anal. Numer. 10(12), 345–369 (1976)MathSciNetGoogle Scholar
  8. 8.
    Kon, M.A., Raphael, L.A.: Characterizing convergence rates for muliresolution. In: Zeevi, Y.Y., Coifman, R.R. (eds.) Approximations, Signal and Image Representation in Compined Spaces, pp. 415–437 (1998)Google Scholar
  9. 9.
    Kon, M.A., Raphael, L.A.: A Characterization of wavelet convergence in Sobolev spaces. Appl. Anal. 78(3–4) 271–324 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Madych, W.R.: Polyharmonic Splines, Multiscale Analysis and Entire Functions. Multivariate Approximation and Interpolation, Duisburg, 1989. Intern. Er. Numer. Math., vol. 94, pp. 205–216. Birkhauser, Basel (1990)Google Scholar
  11. 11.
    Madych, W.R., Nelson, S.A.: Polyharmonic cardinal splines. J. Approx. Theory 60, 141–156 (1990)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Madych, W.R.: Some Elementary Properties of Multiresolution Analyses of L 2(R n). Wavelets, Wavelet Anal. Appl., vol. 2, pp. 259–294. Academic Press, Boston, MA (1992)Google Scholar
  13. 13.
    Micchelli, C., Rabut, C., Utreras, F.: Using the refinement equation for the construction of pre-wavelets, III: elliptic splines. Numer. Algorithms 1, 331–352 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Patel, R.S., Van De Ville, D., Bowman, F.D.: Determining significant connectivity by 4D spatiotemporal wavelet packet resampling of functional neuroimagin data. NeuroImage 31, 1142–1155 (2006)CrossRefGoogle Scholar
  15. 15.
    Rabut, C.: B-splines Polyarmoniques Cardinales: Interpolation, Quasi-interpolation, Filtrage, Thèse d’Etat. Université de Toulouse (1990)Google Scholar
  16. 16.
    Rabut, C.: Elementary polyharmonic cardinal B-splines. Numer. Algorithms 2, 39–62 (1992)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rabut, C.: High level m-harmonic cardinal B-splines. Numer. Algorithms 2, 63–84 (1992)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Renka, R.J., Brown, R.: Algorithm 792: accuracy tests of ACM algoritm for interpolation of scattered data in the plane. ACM TOMS 25, 78–94 (1999)MATHCrossRefGoogle Scholar
  19. 19.
    Rossini, M.: On the construction of polyharmonic B-splines. J. Comput. Appl. Math. (available online, 22 Oct 2007)Google Scholar
  20. 20.
    Rossini, M., Detecting discontinuities in two dimensional signal sampled on a grid. JNAIAM (in press)Google Scholar
  21. 21.
    Blu, T., Unser,M.: Self-similarity: part II–optimal estimation of fractal processes. IEEE Trans. Signal Process. 55(4), 1364–1378 (2007), AprilCrossRefMathSciNetGoogle Scholar
  22. 22.
    Van De Ville, D., Blu, T., Unser, M.: Isotropic polyharmonic B-Splines: scaling functions and wavelets. IEEE Trans. Image Process. 14(11), 1798–1813 (2005)CrossRefGoogle Scholar
  23. 23.
    Vo-Khac, K.: Distributions, Analyse de Fourier, Opérateurs aux Dérivées Partielles. tome 2, Vuibert, Paris (1972)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.INSA, IMTUniversité de ToulouseToulouse Cedex 4France
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanItaly

Personalised recommendations