Box-counting

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... 

The capacity dimension tells us how the number of balls changes as the size of the balls is decreased. This method is usually called box counting since the method is implemented in computers with a variety of algorithms utilizing rectangular grids instead of balls . Dimensions df and dc are quite similar, and the differences between them are usually much smaller than the error of estimates [3]. 

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. 

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. 

Box counting methods. These are simple statistical approaches to describe the mixing of a small... 

Another method used to examine microstructure is the box counting or grid dimension analysis. This dimension is referred to as a grid dimension because for mathematical convenience, the boxes are usually part of a grid that is laid over the image (65). The box dimension is dehned as the exponent D in the relationship ... 

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. 

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.

This method has been extended to creating rectangular boxes instead of polygons. In this box counting method, >l can be determined as follows ... 

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

Tominaga, Y. (1998a). Data Structure Comparison Using Box Counting Analysis. J.Chem.Inf Comput.ScL, 38,867-875. 

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