Measuring the fractal dimensions of ideal and actual objects: implicationsfor 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 eas ily lead to erroneous results. In a previous paper (Gonzato et al. 1998) we demonstrated that a crucial bias is introduced by insufficient sampling an d/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, placeme nt and length of the digitized object relative to that of the initial box. Some additional problems are introduced by objects containing unconnected p arts, since the discontinuities may or may not be indicative of a fractal p attern. Last, but certainly not least in magnitude, the thickness of the di gitized profile, which is implicitly controlled by the scanner resolution v ersus the image line thickness, plays a fundamental role. All of these fact ors combined introduce systematic errors in determining D, the magnitudes o f which are found to exceed 50 per cent in some cases, crucially affecting classification. To study these errors and minimize them, a program that acc ounts for image digitization, zooming and automatic box counting has been d eveloped and tested on images of known dimension. The code automatically ex tracts the unconnected parts from a digitized shape given as input, zooms e ach 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 c ode does not provide image vectorization, which must be performed prior to running this program. A number of image vectorizing codes are available tha t successfully reduce the thickness of the image parts to one pixel. Image vectorization applied prior to the application of our code reduces the samp ling bias for objects with known fractal dimension to around 10-20 per cent . Since this bias is always positive, this effect can be readily compensate d by a multiplying factor, and estimates of the fractal dimension accurate to about 10 per cent are effectively possible.