BI-PHASE BOX COUNTING: AN IMPROVED METHOD FOR FRACTAL ANALYSIS OF BINARY IMAGES

Abstract

Many natural systems are irregular and/or fragmented, and have been interpreted to be fractal. An important parameter needed for modeling such systems is the fractal dimension, D. This parameter is often estimated from binary images using the box-counting method. However, it is not always apparent which fractal model is the most appropriate. This has led some researchers to report different D values for different phases of an analyzed image, which is mathematically untenable. This paper introduces a new method for discriminating between mass fractal, pore fractal, and Euclidean scaling in images that display apparent two-phase fractal behavior when analyzed using the traditional method. The new method, coined "bi-phase box counting", involves box-counting the selected phase and its complement, fitting both datasets conjointly to fractal and/or Euclidean scaling relations, and examining the errors from the resulting regression analyses. Use of the proposed technique was demonstrated on binary images of deterministic and stochastic fractals with known D values. Traditional box counting was unable to differentiate between the fractal and Euclidean phases in these images. In contrast, bi-phase box counting unmistakably identified the fractal phase and correctly estimated its D value. The new method was also applied to three binary images of soil thin sections. The results indicated that two of the soils were pore-fractals, while the other was a mass fractal. This outcome contrasted with the traditional box counting method which suggested that all three soils were mass fractals. Reclassification has important implications for modeling soil structure since different fractal models have different scaling relations. Overall, bi-phase box counting represents an improvement over the traditional method. It can identify the fractal phase and it provides statistical justification for this choice.