A comfortable estimator for the fractal dimension of a binary image is the
box dimension [6]. The image can be covered with a grid of
square cells with cell size . For binary images the cell size is
expressed as numbers of pixels. Fig. 4 shows the
Sierpinsky gasket stored as a
-matrix overlayed with a grid
of squares. The number of grids containing a part of the structure is
.
Figure 4: Sierpinsky gasket overlayed with a grid of squares
The number of squares needed to cover the structure is given by a
power law:
is the box dimension.
The image processor
IDOLON
uses a modified algorithm which determines the area of cells containing parts
of the structure. Using equ. (1) the total area
covered by the squares of size
is:
Fig. 5 shows this area for different box sizes.
Figure 5: Richardson-Mandelbrot plot for the area of a
binary structure at different box sizes
(in pixels)
can be determined by the slope
of the regression line in
fig. 5:
This matches nicely with the Hausdorff-Besicovitch and self-similarity
dimension respectively, which is
(see [6]).
For binary structures in space the box counting procedure can be extended to three dimensions if cubes are used instead of squares. Thus the box counting procedure can be also applied to three dimensional patterns.