Advances in the implementation of the box-counting method of fractal dimension estimation

The box-counting analysis is an appropriate method of fractal dimension est imation for images with or without self-similarity. However, this technique , including processing of the image and definition of the range of box size s, requires a proper implementation to be effective in practice. The object ives of this study were thus (1) to determine how to prepare an image for b ox-counting analysis; (2) to define reasonable preferences for using the Fr actal Dimension Calculator software; and (3) to develop a routine procedure for defining the most appropriate range of box sizes for any one-piece ima ge. Four fractal images were chosen for this study: the Koch curve, Koch co astline, Koch boxes, and Cross-tree. Our results show that the skeletons pr ovide better material for the box-counting method since only lines and/or c urves are responsible for the fractal dimension value. In the procedure of box counting for fractal dimension estimation, the image must be surrounded by a four-square frame with the least possible area and the condition of l inear relationship must be satisfied in a log-log plot. Fractal dimension i s to be estimated over the minimum number of boxes covering the image for e ach box size, after superimposing a reasonable number of grid offsets. In m any cases, 25% of the shorter image side may provide an appropriate value f or largest box size. However, for noisy or dispersed patterns, a smaller bo x size than this is needed. In the log-log plot with 12 box sizes, some poi nts corresponding to smaller box sizes deviate from the straight line from a certain point on. The box size corresponding to this breakpoint will prov ide an appropriate smallest box size. The exercise of determining the most appropriate range of box sizes must be performed repeatedly for every indiv idual image. (C) 1999 Elsevier Science Inc. All rights reserved.