Mass fractal dimension image analysis

Figure 3.9 Box-counting analysis of a projection of a computer-generated diffusion-limited cluster aggregate of 10 000 particles with mass fractal dimension of 1.88. The images have box sizes L of (top left to bottom right) 1, 2,4,8,16, 32 and 64 pixels and require 205 245,59519, 17 062, 4895,1436, 462 and 135 squares respectively to cover the image.

One of the easiest ways to measure fractal dimension with this technique is to capture images of slices through the structure and measure the fractal scaling of the image as discussed above. The dimension measured in this way is not the projected area dimension Dp, discussed earlier, because the image is a slice not a projection. It turns out that the dimension measured in this way is numerically equal to the mass fractal dimension minus one, by virtue of the codimension rule [58]. The measurement of fractal dimensions by this technique is not subject to the restriction of geometric transparency, as is the case with the analysis of projected images, and so fractal dimensions well over two can be measured. 

---