Accurate Estimation of Fractal Dimension of Binary Images by Box-Counting Method with Automatic Scale Selection

Fractal dimension can be used to describe the self-similarity, the complexity, or the irregularity of images. One of the most popular ways to estimate the fractal dimension of images is the box-counting method (BCM). Its naive estimates, however, tend to be inaccurate. This paper gives careful consideration to the sources of inaccurate estimates and modifies Buczkowski's method for accurate estimation of the fractal dimension of binary images. Buczkowski's method is the modified version of the BCM and can provide the most accurate estimates of all the methods based on the BCM so far. The proposed method automatically eliminates the scales which cause the degradation of estimation accuracy after the box-counting stage for whole scales available from a binary image. And then the method fits a regression line to the selected points on a log-log plot in order to obtain the estimate of the fractal dimension. Preliminary consideration indicates that the proposed method is less time-consuming than Buczkowski's method. Some experiments with deterministic fractals, random fractals, and Euclidean objects are also conducted to compare the proposed method to conventional ones in terms of estimation accuracy. The results show that the proposed method enables us to obtain more accurate estimates of the fractal dimension of binary images than the BCM and Buczkowski's method.