Box-counting 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. 

Even without having recourse to this graphical method, the similarity dimension is clearly very straightforward to compute. Unfortunately, it is meaningful only for a small class of strictly self-similar sets. It cannot be used to evaluate the dimension of self-affine, statistically self-similar or statistically self-affine sets. For these, other easily measurable dimensions are necessary, like the box-counting dimension. 

Based on the definition of Equation (2.19), it is convenient to define a new dimension, called the box-counting dimension, denoted here by Dbc, as... 

A number of alternative dimensions have been proposed to overcome the difficulties associated with the traditional box-counting dimension Dbc- They include the lower and upper modified box-counting dimensions [5] and the packing or Tricot dimension [5, 33, 34]. Unfortunately, these dimensions reintroduce all the difficulties of calculation associated with Du, and in some cases are even more awkward to use ... 

One key advantage of the box-counting dimension Dbc over the similarity dimension Ds is that Due can be used to evaluate the dimension of self-affine sets. In these sets, however. Due is not uniquely defined instead, it assumes two different values a local or small-scale value and a global or large-scale value [e.g. [10 (p. 187), 31 (p. 55), 35 (p. 8)]. In the case of the fractional Brownian motion (Section 2.2.5), the local Due value is equal to the Hausdorff dimension and is given hy2- H, where H is the Hurst exponent, whereas the global value of Dbc = 1 [e.g. 10 (p. 189)]. 

It is also possible to immerse the object in a lattice space and count the number of lattice units intersecting the object (see Fig. 2.7) such a method is directly applicable to image analysis, in two dimensions. The fractal dimensions obtained by these methods are referred to as box-counting dimensions. 

As with the Box-Counting dimension, the intrinsic dimension can be estimated from the correlation dimension by examining the plot of log Cm (r) versus log r and estimating the linear part of the curve. 

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