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Fractals and Dimensions

Roland Kraft
Munich University of Technology - Weihenstephan
Department of Agricultural and Horticultural Sciences
Mathematics, Statistics and Data Processing Institute
D-85350 Freising / Germany
Phone: +049-(0)8161-713727, Fax: +049-(0)8161-714409



Dimension analysis is a tool to quantify structural information of artificial and natural objects. For a meaningful interpretation it is necessary to take note of different concepts of dimensions.

The Euclidean dimension is introduced as the dimension of an object's embedding space, whereas the topological dimension of a structure is equated with its cover dimension. The Hausdorff-Besicovitch dimension and the related self-similarity dimension are used to define a fractal dimension. With these definitions a fractal is a set of points whose Hausdorff-Besicovitch dimension exceeds the topological dimension.

Fractal dimensions of many artificial and natural objects cannot be determined analytically. Estimators of the fractal dimension are compass and box dimension, which can easily be calculated if the structure is stored in matrix form. Thus these measures are appropriate for image processors like IDOLON to obtain structural information from digital images [5].

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R. Kraft