From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 21

grid position is often subsequently moved and the minimal number of occupied grid
squares is retained (see, e.g., Jiang and Liu 2012 ). We are currently testing a method
allowing free positioning of the boxes for which the number necessary to cover the
texture is optimized by means of a genetic algorithm.
A similar method is dilation analysis, based on the algorithm introduced by
Minkowski and Bouligand. In dilation analysis, each occupied point is surrounded
by a square of size ", the surface area of which is considered to be completely
occupied. The size of these squares is then gradually enlarged, and we measure the
total surface area covered A (") at each stage. As the squares are enlarged, any details
smaller than " are overlooked and we gradually obtain an approximation of the
original shape. This is reminiscent of a gradual change in the degree of cartographic
detail in drawing. Because more and more squares overlap, the total area occupied
A (dil) (") for a particular value e is less than what it would be if the same number
of occupied points that make up the original shape were surrounded individually.
By dividing this total area by the area A (dil) (") D " 2 of a test square, we get the
number of elements N (") necessary to cover the whole and we obtain a relation
corresponding to relation ( 2.3 ). The corresponding fractal dimension D (dil) is known
as the Minkowski dimension or dilation dimension.

2.4.3 Mass-Distance Relations

A rather different method is radial analysis. It provides information about the spatial
organization around a chosen counting point. A circle is drawn around this point,
and the radius " is gradually increased. At each step, the total number of occupied
points N (") inside the circle is counted. Here, the fractal law takes the form

N."/ D a " D

(2.6)

As pointed out for more complex structures, different fractal behaviors may be
mixed, the radial analysis provides specific information about local fractal behavior,
and so the scaling exponent is known as the local fractal dimension.
It is possible to realize such analysis for each occupied, i.e., built-up, site within
a given zone and to compute the mean number of occupied points observed for each
distance value " (Vicsek 1989 ). An equivalent fractal law is associated with the
method. This method first proposed by Grassberger and Procaccia ( 1983 ) is known
as correlation analysis. The information obtained provides mean information about
the zone analyzed, as in grid analysis. However, the information is more detailed
than for grid analysis, since the exact position of occupied sites is explored and not
just the fact of lying within a box of a given size. Hence, other dimension values
can be expected for more complex patterns. In fractal terms, this is referred to as a
second-order dimension and refers thus to multifractal logic.