Measuring the fractal dimensions of ideal and actual objects: implications for application in geology and geophysics

The box-counting algorithm is the most commonly used method for evaluating the fractal dimension D of natural images. However, its application may easily lead to erroneous results. In a previous paper (Gonzato 1998) we demonstrated that a crucial bias is introduced by insufficient sampling and/or by uncritical application of the regression technique. This bias turns out to be common in many practical applications. Here it is shown that an equally important additional bias is introduced by the orientation, placement and length of the digitized object relative to that of the initial box. Some additional problems are introduced by objects containing unconnected parts, since the discontinuities may or may not be indicative of a fractal pattern. Last, but certainly not least in magnitude, the thickness of the digitized profile, which is implicitly controlled by the scanner resolution versus the image line thickness, plays a fundamental role. All of these factors combined introduce systematic errors in determining D, the magnitudes of which are found to exceed 50 per cent in some cases, crucially affecting classification. To study these errors and minimize them, a program that accounts for image digitization, zooming and automatic box counting has been developed and tested on images of known dimension. The code automatically extracts the unconnected parts from a digitized shape given as input, zooms each part as optimally as possible, and performs the box-counting algorithm on a virtual screen. The size of the screen can be set to meet the sampling requirement needed to produce stable and reliable results. However, this code does not provide image vectorization, which must be performed prior to running this program. A number of image vectorizing codes are available that successfully reduce the thickness of the image parts to one pixel. Image vectorization applied prior to the application of our code reduces the sampling bias for objects with known fractal dimension to around 10–20 per cent. Since this bias is always positive, this effect can be readily compensated by a multiplying factor, and estimates of the fractal dimension accurate to about 10 per cent are effectively possible.