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A note on fractal sets and the measurement of fractal dimension

Author

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  • Sandau, Konrad

Abstract

The fractal dimension of a set in the Euclidean n-space may depend on the applied concept of fractal dimension. Several concepts are considered here, and in a first part, properties are given for sets such that they have the same fractal dimension for all concepts. In particular, self-similar sets hold these properties. The second part deals with the measurement of fractal dimension. An often-used method to empirically compute the fractal dimension of a set E is the box-counting method where the slope of a regression line gives the estimate of the fractal dimension. A new interpretation, which concentrates on the visible complexity of the set, uses counted boxes to define generators for adjacent self-similar sets. Their maximal fractal dimension is assigned to the set as a measurement of fractal dimension or of visible complexity. The results from the first part guarantee that all measurements are independent of the considered concepts. The construction suggests a new method, which is called extended box-counting method, to estimate fractal dimension or to measure complexity of an image in a given range of magnification. The method works without linear regression and has the advantage to nearly preserve the union stability (maximum property).

Suggested Citation

  • Sandau, Konrad, 1996. "A note on fractal sets and the measurement of fractal dimension," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 233(1), pages 1-18.
    • Handle: RePEc:eee:phsmap:v:233:y:1996:i:1:p:1-18
    DOI: 10.1016/S0378-4371(96)00248-8
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