Advertisement

Communications in Mathematical Physics

, Volume 258, Issue 3, pp 513–539 | Cite as

Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents

  • Stéphane JaffardEmail author
  • Clothilde Mélot
Article

Abstract

Let Open image in new window be a domain of Open image in new window . In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of Open image in new window . Classifications based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local L p regularity of the characteristic function Open image in new window or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of Open image in new window . We develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains.

Keywords

Neural Network Statistical Physic Complex System Characteristic Function Nonlinear Dynamics 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arneodo, A., Audit, B., Decoster, N., Muzy, J.-F., Vaillant, C.: Wavelet-based multifractal formalism: applications to DNA sequences, satellite images of the cloud structure and stock market data. In: The Science of Disasters; Bunde, A., Kropp, J., Schellnhuber, H. J. (eds.), Berlin-Heidelberg-New York: Springer, 2002; pp. 27–102Google Scholar
  2. 2.
    Arneodo, A., Bacry, E., Jaffard, S., Muzy, J-F.: Oscillating singularities on Cantor sets: A grandcanonical multifractal formalism. J. Stat. Phys. 87, 179–209 (1997)Google Scholar
  3. 3.
    Arneodo, A., Bacry, E., Muzy, J-F.: The thermodynamics of fractals revisited with wavelets. Physica A 213, 232–275 (1995)Google Scholar
  4. 4.
    Aubry, J.-M., Jaffard, S.: Random wavelet series. Commun. Math. Phys. 227, 483–514 (2002)CrossRefGoogle Scholar
  5. 5.
    Calderòn, A. P., Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Studia Math. 20, 171–227 (1961)Google Scholar
  6. 6.
    Catrakis, H. J., Dimotakis, P. E.: Mixing in turbulent jets: scalar measures and isosurface geometry. J. Fluid Mech. 316, 369–406 (1996)Google Scholar
  7. 7.
    Dalang, R., Walsh, J.: Geography of the level sets of the Brownian sheet. Prob. Th. Rel. Fields, 96(2), 153–176 (1993)Google Scholar
  8. 8.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. Pure and App. Math. 41, 909–996 (1988)Google Scholar
  9. 9.
    Dubuc, B., Zucker, S. W., Tricot, C., Quiniou, J.F., Wehbi, D.: Evaluating the fractal dimension of surfaces. Proc. R. Soc. Lond. A 425, 113–127 (1989)Google Scholar
  10. 10.
    Falconer, K.: Fractal geometry. New York: John Wiley and Sons, 1990Google Scholar
  11. 11.
    Gousseau, Y.: Distribution de formes dans les images naturelles. Thèse de l’Université Paris- Dauphine, 2000Google Scholar
  12. 12.
    Gouyet, J.-F., Russo, M., Sapoval, B.: Fractal surfaces and interfaces. In: Bunde, A., (ed.), Fractals and disordered systems. Berlin-Heidelberg-New York: Springer Verlag, 1996Google Scholar
  13. 13.
    Guiheneuf, B., Jaffard, S., Lévy-Véhel, J.: Two results concerning Chirps and 2-microlocal exponents prescription. App. Comput. Harm. Anal. 5(4), 487–492 (1998)CrossRefGoogle Scholar
  14. 14.
    Jaffard, S.: Pointwise smoothness, two-microlocalization and wavelet coefficients. Publ. Mat. 35, 155–168 (1991)Google Scholar
  15. 15.
    Jaffard, S.: Multifractal formalism for functions. Part 1: Results valid for all functions and Part 2: Selfsimilar functions. SIAM J. Math. Anal. 28, 944–998 (1997)CrossRefGoogle Scholar
  16. 16.
    Jaffard, S., Meyer, Y.: Wavelet methods for pointwise regularity and local oscillation of functions. Mem. Am. Math. Soc. 123, 587 (1996)Google Scholar
  17. 17.
    Jaffard, S., Meyer, Y., Ryan, R.: Wavelets: Tools for Science and Technology. Philadelphia, PA: S.I.A.M., 2001Google Scholar
  18. 18.
    Mallat, S.: A Wavelet tour of signal processing. London-New York: Academic Press, 1998Google Scholar
  19. 19.
    Mandelbrot, B.: On the geometry of homogeneous turbulence with stress on the fractal dimension of the isosurfaces of scalars. J. Fluid Dyn. 72, 401–416 (1975)Google Scholar
  20. 20.
    Melot, C.: Sur les singularités oscillantes et le formalisme multifractal. Thèse de l’Université Paris XII, 2002Google Scholar
  21. 21.
    Meneveau, C., Sreenivasan, K.: Interface dimension in intermittent turbulence. Phys. Rev A 41(4), 2246–2248 (1990)CrossRefGoogle Scholar
  22. 22.
    Meyer, Y.: Principe d’incertitude, bases Hilbertiennes et algèbres d’opérateurs. Séminaire Bourbaki, n. 662 (Fev. 1986)Google Scholar
  23. 23.
    Meyer, Y.: Ondelettes et opérateurs. Paris: Hermann, 1990Google Scholar
  24. 24.
    Meyer, Y.: Wavelets, Vibrations and Scalings. CRM Ser. AMS, Vol. 9, Montréal: Presses de l’Université de Montréal, 1998Google Scholar
  25. 25.
    Meyer, Y., Xu, H.: Wavelet analysis and chirps. Appl. Comput. Harmon. Anal. 4(4), 366–379 (1997)CrossRefGoogle Scholar
  26. 26.
    Mimouni, S.: Analyse fractale d’interfaces pour les instabilités de Raleigh-Taylor. Thèse de l’Ecole Polytechnique, 1995Google Scholar
  27. 27.
    Mimouni, S., Laval, G., Scheurer, B.: Fractal interface. EUROTHERM Seminar 39, Nantes, 1994Google Scholar
  28. 28.
    Mimouni, S., Laval, G., Scheurer, B., Jaffard, S.: Morphology of the mixing layer in the Raleigh-Taylor instability. In: Small scale structures in three-dimensional hydrodynamics and magnetohydrodynamic turbulence; Springer, Lect. Notes in Phys. 462, Berlin-Heidelberg-New York: Springer, 1995; pp. 179–192Google Scholar
  29. 29.
    Moffat, H. K.: Simple topological aspects of turbulence velocity dynamics. In: Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena in Fluids, ed. Tatsumi Amsterdam: Elsevier/North Holland, 1984, p. 223ffGoogle Scholar
  30. 30.
    Parisi, G., Frisch, U.: On the singularity structure of fully developed turbulence, Appendix to Fully developed turbulence and intermittency, by U. Frisch, Proc. Int. Summer School Phys. Enrico Fermi, Amsterdam: North Holland, 1985, pp. 84–88Google Scholar
  31. 31.
    Redondo, J. M.: Fractal models of density interfaces. In: IMA Conf. Ser. 13, Frage, M., Hunt, J.C.R., Vassilicos, J. C. (eds.) Oxford: Clarendon Press/Elsevier, 1993; pp. 353–370Google Scholar
  32. 32.
    Sapoval, B.: Universalités et fractales, Paris: Flammarion, 1997Google Scholar
  33. 33.
    Saether, G., Bendiksen, K., Muller, J., Frøland, E.: The fractal dimension of oil-water interfaces in channel flows. In: IMA Conf. Ser. 13, Frage, M., Hunt, J.C.R., Vassilicos, J. C. (eds.) Oxford: Clarendon Press/Elsevier, 1993, pp. 371–378Google Scholar
  34. 34.
    Sethian, J. A.: Level Set Methods: Evolving interfaces in geometry, fluid mechanics, computer vision and material sciences. Cambridge: Cambridge University Press, 1996Google Scholar
  35. 35.
    Vassilicos, J.C.: The multispiral model of turbulence and intermitency. Moffat, H.K. et al. ed. Topological aspects of the dynamics of fluids and plasmas, Kluwer acad. pub., 1992, pp. 427–442Google Scholar
  36. 36.
    Vassilicos, J.C., Hunt, J.C.R.: Fractal dimensions and spectra of interfaces with application to turbulence. Proc. Roy. Soc. Series A, 435(1895), 505–534 (1991)Google Scholar
  37. 37.
    Xiao, Y.: Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Prob. Th. Rel. Fields, 109(1), 129–157 (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité Paris XIICréteil CedexFrance
  2. 2.LATPCMI, Université de Provence, 39 rue F. Joliot-CurieFrance

Personalised recommendations