Measurements of fractal dimension by box-counting: a critical analysis of data scatter

The multifractal concept was introduced in the 1980s by Mandelbrot. This theory arose from the analysis of complex and/or discontinuous objects. In this study, we analyzed the data scatter obtained by a modified box-counting method. Considering the curved shape of the data scatter, it is noticeable that there is more than one slope corresponding to different fractal behavior of an object. In this work, to discriminate different fractal dimensions from data scatter obtained by box counting, we suggest a rigorous selection of data points. The results show that large epsilon's usually characterize the embedding surface of the whole object and that small epsilon's approximate the dimension of the substructure for discontinuous objects. They also show that a dimension can be associated with a density distribution of singularities. (C) 1998 Elsevier Science B.V. All rights reserved.