Multifractal Analysis on the Sphere

  • Emilie Koenig
  • Pierre Chainais
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5099)

Abstract

A new generation of instruments in astrophysics or vision now provide spherical data. These spherical data may present a self-similarity property while no spherical analysis tool is yet available to characterize this property. In this paper we present a first numerical study of the extension of multifractal analysis onto the sphere using spherical wavelet transforms. We use a model of multifractal spherical textures as a reference to test this approach. The results of the spherical analysis appear qualitatively satisfactory but not as accurate as those of the usual 2D multifractal analysis.

Keywords

Multifractal Analysis Omnidirectional Image Spherical Data Cosmic Microwave Background Data Spherical Wavelet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Frisch, U.: Turbulence, The legacy of A. Kolmogorov. Cambridge University Press, Cambridge (1995)Google Scholar
  2. 2.
    Abry, P., Flandrin, P., Taqqu, M., Veitch, D.: Wavelets for the analysis, estimation and synthesis of scaling data. In: Self-Similar Network Traffic and Performance Evaluation, Wiley-Interscience, Chichester (2000)Google Scholar
  3. 3.
    Arneodo, A., Bacry, E., Graves, P.V., Muzy, J.F.: Characterizing long-range correlations in dna sequences from wavelet analysis. Physical Review Letters 74(16), 3293–3296 (1995)CrossRefGoogle Scholar
  4. 4.
    Turiel, A., Perez-Vincente, C.J., Grazzini, J.: Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study. Journal of Computational Physics 216(1), 362–390 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Marinucci, D., Pietrobon, D., Balbi, A., Baldi, P., Cabella, P., Kerkyacharian, G., Natoli, P., Picard, D., Vottorio, N.: Spherical needlets for cosmic microwave background data analysis. Mon. Not. R. Astron. Soc. 383(2), 539–545 (2007)Google Scholar
  6. 6.
    Tosic, I., Bogdanova, I., Frossard, P., Vanderghynst, P.: Multiresolution motion estimation for omnidirectional images. In: EUSIPCO (2005)Google Scholar
  7. 7.
    Chainais, P.: Infinitely divisible cascades to model the statistics of natural images. IEEE Trans. on Pattern Anal. Mach. Intell. 29(1) (2007)Google Scholar
  8. 8.
    Mandelbrot, B.: The fractal geometry of Nature. W.H. Freeman and Co, New York (1982)MATHGoogle Scholar
  9. 9.
    Lashermes, B., Abry, P., Chainais, P.: New insights on the estimation of scaling exponents. Int. J. of Wavelets, Multiresolution and Information Processing 2, 497–523 (2004)MATHCrossRefGoogle Scholar
  10. 10.
    Jaffard, S.: Multifractal formalism for functions. S.I.A.M. 28(4), 944–998 (1997)MATHMathSciNetGoogle Scholar
  11. 11.
    Schröder, P., Sweldens, W.: Spherical wavelets: Efficiently representing functions on a sphere. In: Computer Graphics Proceedings SIGGRAPH 1995, vol. 29, pp. 161–172 (1995)Google Scholar
  12. 12.
    Gao, Y., Nain, D., LeFaucheur, X., Tannenbaum, A.: Spherical wavelet itk filter. In: Ayache, N., Ourselin, S., Maeder, A. (eds.) MICCAI 2007, Part I. LNCS, vol. 4791, Springer, Heidelberg (2007)Google Scholar
  13. 13.
    Starck, J.L., Moudden, Y., Abrial, P., Nguyen, M.: Wavelets, ridgelets and curvelets on the sphere. Astron. astrophys. 446, 1191–1204 (2006)CrossRefGoogle Scholar
  14. 14.
    Gorski, K., Hivon, E., Banday, A., Wandelt, B., Hansen, F., Reinecke, M., Bartelmann, M.: Healpix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J. 622(2), 759–771 (2005)CrossRefGoogle Scholar
  15. 15.
    Antoine, J.P., Demanet, D., Jacques, L., Vandergheynst, P.: Wavelets on the sphere: implementation and approximations. Appl. Comp. Harmon. Anal. 13, 177–200 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jacques, L.: Ondelettes, repères et couronne solaire. PhD thesis, Université catholique de Louvain (2004)Google Scholar
  17. 17.
    Driscoll, J., Healy, J.: Computing fourier transforms and convolutions on the 2-sphere. Adv. in Appl. Math. 15, 202–250 (1994)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhou, K., Bao, H., Shi, J.: 3d surface filtering using spherical harmonics. Computer-Aided Design 36(4), 363–375 (2004)CrossRefGoogle Scholar
  19. 19.
    Kostelec, P., Rockmore, D.: S2Kit: A Lite Version of SpharmonicKit, Department of Mathematics, Dartmouth College, Hanover (2004)Google Scholar
  20. 20.
    Gonzalez-Nuevo, J., Argueso, F., Lopez-Caniego, M., Toffolatti, L., Sanz, J., Vielva, P., Herranz, D.: The mexican wavelet family: Application to point source detection in cmb maps. Mon. Not. R. Astron. Soc. 369(4), 1603–1610 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Emilie Koenig
    • 1
  • Pierre Chainais
    • 1
  1. 1.LIMOS UMR 6158University of Clermont-Ferrand IIAubire CedexFrance

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