Correcting for finite spatial scales of self–similarity when calculating fractal dimensions of real–world structures

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Abstract
Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self–similarity over any range of scales. Unlike ideal fractals, real–world structures exhibit self–similarity only over a finite range of scales. Here we present a new technique for quantitatively determining the scales over which real–world structures show statistical self–similarity. The new technique uses a combination of curve–fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self–similarity. Consequently, we estimate D only over the statistically identified region of self–similarity and introduce the finite scale– corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self–similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self–similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real–world branching structures.