Advertisement

Random Fractals: characterization and measurement

  • Richard F. Voss

Abstract

Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer. A brief mathematical characterization of random fractals is presented with emphasis on variations of Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statistical self-similarity and self-affinity will be reviewed. The various methods and difficulties of estimating the fractal dimension and lacunarity from experimental images or point sets are summarized.

Keywords

Brownian Motion Fractal Dimension Fractional Brownian Motion Intuitive Notion Random Fractal 
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.
    Mandelbrot, B.B. The Fractal Geometry of Nature, (Freeman, New York) 1982 and references therein. See also, Fractals: Form, Chance, and Dimension, W. H. Freeman and Co., San Francisco (1977).Google Scholar
  2. 2.
    Voss, R.F, “Random Fractal Forgeries:”, Proc. NATO A.S.I on Fundamental Algorithms in Computer Graphics, Ilkley, Yorkshire, UK (1985).Google Scholar
  3. 3.
    Mandelbrot, B.B. and Wallis, J. W. “Fractional Brownian motions, fractional noises, and applications”, SIAM review, 10 (1968) 422–437.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Mandelbrot, B.B. “Fractals in Physics: Squig Clusters, Diffusions, Fractal Measures, and the Unicity of Fractal Dimensionality”, J. Stat. Phys. 34 (1984) 895–930.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mandelbrot, B.B., Gefen, Y., Aharony, A., and Peyeiere, J. “Fractals, their transfer matrices and their eigen-dimensional sequences”, J. Phys. A 18 (1985) 335–354.MathSciNetCrossRefGoogle Scholar
  6. 6.
    A good discussion is found in Reif, F. Statistical and Thermal Physics,McGraw-Hill Book Co., New York, (1965), Chapter 15, “Irreversible Processes and Fluctuations.”Google Scholar
  7. 7.
    For example see: Freeman, J. J. Principles of Noise, John Wiley & Sons, Inc., New York, (1958), Chapter 1, “Fourier Series and Integrals.” or Robinson, F.N.H. Noise and Fluctuations, Clarendon Press, Oxford, (1974).Google Scholar
  8. 8.
    Mandelbrot, B.B. “Intermittent turbulence in self-similar cascades: divergence of higher moments and dimension of the carrier”, J. Fluid Mech. 62 (1974) 331–358.zbMATHCrossRefGoogle Scholar
  9. 9.
    Hentschel, H.G.E. and Procaccia, I. “The infinite number of generalized dimensions of fractals and strange attractors”, Physica 8D (1983) 435–444.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Richard F. Voss
    • 1
    • 2
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA
  2. 2.Division of Applied ScienceHarvard UniversityCambridgeUSA

Personalised recommendations