Box-counting fractal dimension

The box-counting fractal dimension is calculated from the log-log plot of the number of occupied boxes, versus side length, /,. The slope of this line is used to determine the fractal dimension. Both very small and very large box sizes should be exempt from the calculation to reduce surface artifacts (Awad and Marangoni 2005). This method is most sensitive to the degree of fill as well as particle size and it can be expected that if there are void volumes it will have a characteristic low fractal dimension. 

TABLE 3. Particle-Counting Fractal Dimension (Df), Box-Counting Fractal Dimension (Db), and Mean Microstructural Element Area (MEA) for Anhydrous Milkfat Crystallized at Various Rates of Cooling and Storage Times at 5°C. 

Donnelly, D.R, Wilkins, M.R and Boddy, L. (1995). An integrated image analysis approach for determining biomass, radial extent and box-count fractal dimension of macroscopic mycelial systems. Binary, 7,19. 

The particle-counting fractal dimension, Dj, was not sensitive to crystal shape, size, AF or the distribution orderliness. It was found that Df was affected by the radial distribution pattern of the fat crystals as shown in Figure 17.28. The simulation results were found to be consistent with experiments (Litwinenko et al. 2002 Tang and Marangoni 2006). Devalues close to 2 indicated more homogenously distributed fat crystals. It is important to note, the values of Dfm y exceed the dimensionality of the embedding space. This is not the case for the box-counting dimension or the Fourier transform fractal dimension. 

It is the local dimension that describes the irregularity of the self-affine fractal. The local dimension can be determined by such methods as the box-counting method1,61,62,65 and the dividerwalking method.61,66 The box dimension dEB is defined by the... 

Figure 11. Schematic diagram shows the incremental decreases in box size used for the particle counting method for the determination of the microscopic fractal dimension Df.

Figure 24. Linear correlations between yield force and (A) microstructural element area, (B) fractal dimension by particle-counting. (C) fractal dimension by box-counting, and (D) fractal dimension by rheology. Data shown represent all points collected at all cooling rates and storage times.

Box counting method is commonly used to obtain mass fractal dimension from an aggregate s projected area. 

Sarkar, N. and Chaudhuri, B.B. An efficient differential box-counting approach to compute fractal dimension image, IEEE Trans. Systems, Man Cybernetics, 24,115, 1994. 

Figure 1.1 A fractal structure overlaid by a grid of squares. In three dimensions the squares are replaced by boxes. Surface, mass and pore fractal dimensions can be estimated by counting the number of squares occupied by the object. Two-dimensional images of three-dimensional objects do not contain the full information about the object, so fractal dimensions of three-dimensional objects need to be estimated in three dimensions [11]. Reproduced with permission of John WUey Sons, Ltd.

Figure 2.22 Influence of physical pixel size (image pixel size translated into actual physical length on the soil profile) on a number of mass fractal dimensions of the preferential pathway in Figure 2.21. Open symbols correspond to images thresholded with the intermeans algorithm, whereas full symbols correspond to the minimum-error algorithm. Circles, squares, and diamonds are associated with the box-counting, information and correlation dimensions respectively. (Modified from [81].)...

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