Minimal Coverage of Investigated Object when Seeking for its Fractal Dimension

  • Adam Szustalewicz


The measuring of the complexity of river shapes is very important in erosion problems. A perfectly useful way for it is calculating the fractal dimension of curves representing rivers on maps. The shapes of rivers vary in form from smooth to very complicated and algorithm for calculating their fractal dimension should be universal and give good results for both. The paper considers the essence of fulfilling the very difficult and numerically very expensive assumption of the fractal theory about the minimal coverage of the measured objects. The actual version of the algorithm taking care to fulfill the assumption is compared, using 59 objects of different kinds and known fractal dimensions, with four easier and cheaper algorithms of covering. For some objects the obtained results are similar, but generally the elaborated algorithm is the best.


Fractal Dimension Hausdorff Dimension Special Subset Minimal Coverage True Dimension 
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.


  1. [1]
    M. Barnsley, Fractals everywhere, Boston, Academical Press, Inc. 1988.Google Scholar
  2. [2]
    W. Bauer, C. D. Mackenzie, Cancer Detection via Determination of Fractal Cell Dimension, presented at the Workshop on Computational and Theoretical Biology, Michigan State University, April 24, 1999, see also /bauer/cancer/cancer.pdf.Google Scholar
  3. [3]
    L. E. Da Costa, J.A. Landry, Synthesis of Fractal Models for Plants and Trees: First Results, Fractal 2004, Complexity and Fractals in Nature, Vancouver, Canada, April 4-7, 2004.Google Scholar
  4. [4]
    K. Falconer, Fractal geometry, Mathematical Foundations and Applications, John Wiley \& Sons, New York 1990.Google Scholar
  5. [5]
    G. Gonzato, F. Mulargia, M. Ciccotti, Measuring the fractal dimensions of ideal and actual objects: implications for application in geology and geophysics, Geophys. J. Int. 142, 2000, 108-116.CrossRefGoogle Scholar
  6. [6]
    B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York 1977, 1983.Google Scholar
  7. [7]
    V. Nordmeier, Fractals in Physics – From Low-Cost Experiments to Fractal Geometry, University of Essen, Germany, Scholar
  8. [8]
    H.-O. Peitgen, H Jurgens, D. Saupe, Fractals for the Classroom Part 1, 2, Springer-Verlag, New York 1992 (polish edition by PWN, Warszawa 2002).Google Scholar
  9. [9]
    A. Szustalewicz, Numerical problems with evaluating the fractal dimension of real data, in Enhanced Methods in Computer Security, Biometric and Artificial Intelligence Systems, ed. J. Pejas, A. Piegat, by Kluwer Academic Publishers, Springer, New York 2005, 273-283.CrossRefGoogle Scholar
  10. [10]
    A. Szustalewicz, A.Vassilopoulos, Calculating the fractal dimension of river basins, comparison of several methods, in Biometrics, Computer Security Systems and Artificial Intelligence Applications, ed. Khalid Saeed, Jerzy Pejas, Romuald Mosdorf, by Springer, 2006, pp. 299-309.Google Scholar
  11. [11]
    A. Szustalewicz, Choosing best subsets for calculation of fractal dimension by the box-counting method, manuscript (in preparation).Google Scholar
  12. [12]
    R. P. Taylor, B. Spehar, C. W. Clifford, B. R. Newell, The Visual Complexity of Pollock’s Dripped Fractals in Proceedings of the International Conference of Complex Systems, 2002, see also Scholar
  13. [13]
    K. J. Vinoy, K. A. Jose, V. K. Varadan, V. V. Varadan, Hilbert Curve Fractal Antenna: A Small Resonant Antenna for VHF/UHF Applications, Microwave & Opt. Technol. Lett. 29 No. 4, 215-219 (2001).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Adam Szustalewicz
    • 1
  1. 1.Institute of Computer Science Wroclaw Universityul. Joliot-Curie 15Wrocńaw

Personalised recommendations