A comfortable estimator for the fractal dimension of nearly any arbitrary structure is the box dimension.
In section 3.1 the length of a coastline was estimated
by where
is the number of steps needed for a roundtrip
along the coast and
is the opening of a pair of compasses. Alternatively
the coast can be covered with a grid of square cells with cell size
(see fig. 10).
Figure 10: The box dimension
The number of squares needed to cover the coastline is roughly equal
to the number of steps when using a pair of compasses with opening
.
This holds i.e. for small
.
Fig. 11 shows a double logarithmic plot of the number
of cells versus the boxsize
.
Figure 11: versus
(logarithmic)
The straight line corresponds to the relation:
is the box dimension of the coastline and can be determined
from the slope of the regression line as
. This matches
nicely with the compass dimension of
in
sect. 3.1.
Thus the coastlength can be expressed as:
For a calculation of the box dimension it doesn't matter if the number of boxes is counted for the entire structure. This will only result in another value of the constant in equ. (13). Thus it is possible to determine the box dimension of arbitrary binary structures. Fig. 12 shows a binary image of a fern leaf generated with iterated function systems [1].
Figure 12: Number of boxes needed to cover a structure
at different box sizes
(in pixels)
An analytical determination of the fractal dimension is not possible since the scaling factor within the structure is not constant and the object is called self affine.
For binary images it is appropriate to choose the grid length as numbers of pixels. Fig. 13 shows the double logarithmic plot of cells containing parts of the fern versus the gridsize in pixels.
Figure 13: Logarithmic -
-plot for determination of the
fern's box dimension
The box dimension which is an estimator of the Hausdorff Besicovitch dimension
(see sect. 2.4) is .
For structures in space the box counting procedure can be extended to three dimensions if cubes are used instead of squares.