Multifractals and the Local Connected Fractal Dimension

Classification of Early Chinese Landscape Paintings
  • Richard F. Voss
  • James C. Y. Wyatt

Abstract

Local and global applications of multifractals to the analysis of digitized image intensities, I(x,y), are discussed. The magnitude of the local slope, |ΔI(x,y)|, is shown to be a more useful measure than I(x, y). A global fractal dimension, D, can be estimated from the spectral density, \(S(\vec k)\), the angle-averaged pair-correlation, C(r), and mass-radius M(R). The concept of local fractal dimension can be used to construct a color-coded dimensional image. Applications to the classification of early Chinese landscape paintings, however, suggest that the local connected fractal dimension provides the best agreement with the human eye for highlighting and discriminating between images.

Keywords

Europe Coherence Autocorrelation Hunt Geophysics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Fiel87]
    Field, D.J., Relations between the statistics of natural images and the response properties of cortical cells, Jour. Opt. Soc. Amer., Vol. A4, pp. 2379–2394, 1987.CrossRefGoogle Scholar
  2. [Mand68]
    Mandelbrot, B.B., and van Ness, J.W., Fractional Brownian motion, fractional noises and applications, SIAM Review, Vol. 10, pp. 422–437, 1968.MathSciNetCrossRefMATHGoogle Scholar
  3. [Mand74]
    Mandelbrot, B.B., Intermittent turbulence in self similar cascades; Divergence of high moments and dimension of the carrier, Jour. Fluid Mechs., Vol. 62, pp. 331–358, 1974.CrossRefMATHGoogle Scholar
  4. [Mand75]
    Mandelbrot, B.B., Les objects fractals: forme, hasard et dimension, Paris: Flammarion, 1975.Google Scholar
  5. [Mand77]
    Mandelbrot, B.B., Fractals: Form, Chance, and Dimension, San Francisco, CA: W.H. Freeman, 1977.MATHGoogle Scholar
  6. [Mand82]
    Mandelbrot, B.B., The Fractal Geometry of Nature, San Francisco: W.H. Freeman, 1982.MATHGoogle Scholar
  7. [Mand85]
    Mandelbrot, B.B., Self-affine fractals and fractal dimension, Physica Scripta, Vol. 32, pp. 257–260, 1985.MathSciNetCrossRefMATHGoogle Scholar
  8. [Mand88]
    Mandelbrot, B.B., An introduction to multifractal distribution functions, in Fluctuations and Pattern Formation (Cargese 1988), Stanley, H.E., and Ostrowsky, N., Eds., Boston: Kluwer, Dordrecht, pp. 345–360, 1988.Google Scholar
  9. [Mand89]
    Mandelbrot, B.B., Multifractal measures for the geophysicist, Pure and Applied Geophysics, Vol. 131, pp. 5–42, 1989.CrossRefGoogle Scholar
  10. [Reif65]
    Reif, F., Irreversible processes and fluctuations, Chap. 15 in Fundamentals of Statistical and Thermal Physics, New York: Mc-Graw Hill, 1965.Google Scholar
  11. [Robi74]
    Robinson, F.N.H., Noise and Fluctuations, Oxford: Clarendon Press, 1974.Google Scholar
  12. [Rogo90]
    Rogowitz, B.E., and Voss, R.F., Shape perception and low dimension fractal boundary contours, in Proc. Conf. on Human Vision: Methods, Models and Applications, Rogowitz, B.E., and Allenbach, J., Eds., SPIE/SPSE Symposium on Electronic Imaging, Vol. 1249, Santa Clara, CA, 1990.Google Scholar
  13. [Stau85]
    Stauffer, D., Introduction to Percolation Theory, London: Taylor and Francis, 1985.CrossRefMATHGoogle Scholar
  14. [Vass91]
    Vassilicos, J.C., and Hunt, J.C.R., Fractal dimensions and spectra of interfaces with application to turbulence, Proc. Roy. Soc. London Ser. A, Vol. 435, pp. 505–534, 1991.MathSciNetCrossRefMATHGoogle Scholar
  15. [Voss82]
    Voss, R.F., Laibowitz, R.B., and Alessandrini, E.I., Fractal (scaling) clusters in thin gold films near the percolation transition, Phys. Rev. Lett., Vol. 49, pp. 1441–1444, 1982.CrossRefGoogle Scholar
  16. [Voss85]
    Voss, R.F., Random fractals: characterization and measurement, in Scaling Phenomena in Disordered Systems, Pynn, R., and Sjeltorp, A., Eds., New York: Plenum Press, 1985.Google Scholar
  17. [Voss88]
    Voss, R.F., Fractals in nature: from characterization to simulation, in The Science of Fractal Images, Peitgen, H.-O., and Saupe, D., Eds., New York: Springer-Verlag, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Richard F. Voss
  • James C. Y. Wyatt

There are no affiliations available

Personalised recommendations