Development of Computer Software to Calculate Fractal Dimension Using the Box-counting Method and Its Analytical Precision

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  • SAITOH Kazuya
    Department of Geoscience and Technology, Faculty of Engineering, Tohoku University
  • TSUCHIYA Noriyoshi
    Department of Geoscience and Technology, Faculty of Engineering, Tohoku University
  • NAKATSUKA Katsuto
    Department of Geoscience and Technology, Faculty of Engineering, Tohoku University

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Other Title
  • ボックスカウンティング法によるフラクタル次元算出システムの開発と解析精度
  • ボックスカウンティングホウ ニヨル フラクタル ジゲン サンシュツ システム

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Description

The box-counting method, BC method, is widely used to determine the fractal dimension of self-similar structure on a plane. The basic implementation of this method is, 1) covering the object with cells, 2) counting the number of cells occupied with the object, and 3) executing the former steps about the various size of cell. The fractal dimension is easily obtained from the slope of the approximation line on the log-log graph which shows relationship between cell size and the number of cell filled with the object. The minimum and maximum cell sizes were defined as lower and upper cut-off levels in order to restrict data points for evaluating the fractal dimension. The slope of approximation line which is equivalent to the fractal dimension is calculated using the data points in the range between lower and upper cut-off levels. Computer software FIVA-Fractal Image Visual Analyzer - was developed to evaluate fractal properties and to calculate fractal dimension on the basis of BC method.<BR>Lower and upper cut-off levels could be selected arbitrarily by FIVA and the fractal dimension was obtained from the restricted range. Optimum cut-off level was obtained by analyzing self-evident fractal objects. The lower cut-off level was larger than the resolution of object figure. The upper cut-off level was required 1/30 to the whole figure size of objects. The precision of fractal dimension calculated by FIVA was in the range from -5 to -1% on analyzing fractal objects of which resolution was smaller than 1/360. However, in the case of which those resolution was about 1/100, the precision was in the range from -11 to +5%.<BR>The fractal dimension of space filling of natural fracture pattern was obtained from data points on the log-log plot between the lower and upper cut-off levels. The resolution of natural fracture pattern used in this study was about 1/100 to the whole pattern size, therefore, the lower cut-off level was 1/90, the upper cut-off level was 1/30. By using the results of analyzing self - evident fractal objects, the fractal dimension of space filling of a fracture pattern was expected in the range from 1.1 to 1.3.

Journal

  • Geoinformatics

    Geoinformatics 8 (1), 23-30, 1997

    Japan Society of Geoinformatics

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