109.

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image. Our experience suggests that L_max=15 is sufficient to obtain a good estimate of the fractal dimension (Tso, 1997).

Gonzato (1998) notes that, in order to generate a reliable estimate of the fractal dimension D, a large range of box sizes should be used, and for each step size a large number of boxes should be occupied. He provides a C program to carry out the box calculations and the estimation of D.

Sarkar and Chaudhuri (1994) propose an approach called differential box counting to estimate the fractal dimension. Consider an M×M image that has been divided into overlapping or abutted windows of size L×L, where M/2≥L≥3, and L is an integer. For each window, there is a column of accumulated boxes, each box with a size of L×L×L'. If G denotes the grey level range of the image (e.g. 256), L' is calculated by:

(5.10)

Note that the symbol indicates the ‘integer part’ of the argument. For instance, if L=3, image side length M=13, and grey level G=33, then L' is computed as:

(5.11)

Assign the numbers 1, 2,…, n in turn to each box in the column from bottom to top. Let the minimum and maximum grey levels of the pixels in the (i, j)th window fall in boxes numbered u and b, respectively. The number of boxes spanned on the (i, j)th window is:

(5.12)

For instance, in Figure 5.6a, the window size is set to L=3, and each window thus contains nine pixels. If grey level G is 16 then L'=4. We can then build up a column of boxes (see Figure 5.6a). Suppose that a column of boxes is centred on an image co-ordinate (3, 6), and the corresponding pixel values are as shown on the left-hand side of Figure 5.6b. After intersecting the pixel values with the column of boxes, we get u=4, b=2, and finally calculate n_L(3, 6)=4−2+1=3 in terms of Equation (5.12) (see Figure 5.6b). The same calculation is performed for all windows, and the total number of boxes needed to cover the whole image with box size L×L×L' is:

(5.13)

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