Minimal Coverage of Investigated Object when Seeking for its Fractal Dimension
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.
KeywordsFractal Dimension Hausdorff Dimension Special Subset Minimal Coverage True Dimension
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- M. Barnsley, Fractals everywhere, Boston, Academical Press, Inc. 1988.Google Scholar
- 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 http://www.pa.msu.edu/~ /bauer/cancer/cancer.pdf.Google Scholar
- 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
- K. Falconer, Fractal geometry, Mathematical Foundations and Applications, John Wiley \& Sons, New York 1990.Google Scholar
- B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York 1977, 1983.Google Scholar
- V. Nordmeier, Fractals in Physics – From Low-Cost Experiments to Fractal Geometry, University of Essen, Germany, http://appserv01.uni-duisburg.de/hands-on/files/autoren/nordm/nordm.htm.Google Scholar
- 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
- 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
- A. Szustalewicz, Choosing best subsets for calculation of fractal dimension by the box-counting method, manuscript (in preparation).Google Scholar
- 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 http://materialscience.uoregon.edu/taylor/art/Taylor/CCS2002.pdf.Google Scholar