Fractal dimension being a measurement in the way in which the fractal occupies space, to determine it we have to draw up the relationship between this way of occupation of space and its variation of scale. If a linear object of size L is measured with a self-similar object of size l, then number of self similar objects within the original object N(l) is related to L/l as
n=(L T <2лі)
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where D is the fractal dimension. From where
For the self-similar fractals, L/l represents the magnification factor and l/L the reduction factor. Nevertheless, when one tries to determine fractal dimension of
natural objects, one is often confronted with the fact that the direct application of Eq. (2.12) is ineffective. In fact, the majority of the natural fractal objects existing in our real world are not self-similar but rather self-affine. The magnification factor and the reduction factor are thus difficult to obtain since there is not an exact selfsimilarity. Other methods are then necessary to estimate the fractal dimensions of these objects.
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In practice, to measure a fractal dimension, several methods exist, some of which are general, whereas others are applicable only to special classes of fractals. This section, focuses on the more commonly used methods namely, Box-counting dimension and Minkowski-Bouligand dimension which are based on the great works of Minkowski and Bouligand (Minkowski 1901; Bouligand 1928) and from which derive several other algorithms.
If one plots ln(N(є)) versus 1п(1/є), the slope of the straight line gives the estimate of the fractal dimension Db in the box-counting method.
Figure 2.2 gives an example illustrating this method. The object E (a curve) is covered by a grid of squares of side є1 = 1/20, and for this value of є total number of squares contained in the grid is 202 = 400 and the number of squares intersecting the curve E is 84 (Fig. 2.2a). In Fig. 2.2b, which is obtained using different values of є, the slope of the straight line fitted by a linear regression constitutes the fractal dimension of the curve E.
Minkowski-Bouligand dimension: This method is based on Minkowski’s idea of dilating the object which one wants to calculate the fractal dimension with disks of radius є and centered at all points of E. The union of these disks thus creates a
Minkowski cover.
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Let Б(є) be the surface of the object dilated or covered and Dm the Minkowski – Bouligand dimension. Bouligand defined the dimension Dm as follows
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or, rearranged
The fractal dimension can then be estimated by the slope of the log-log plot: ln(S(є) /є) = f (ln(1/є)) fitted by the least squares method. Figure 2.3a shows the Minkowski covering E(є) composed of the union of disks of radius є.
3.3 Discussion of the Two Methods
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According to the analysis of Dubuc et al. (Dubuc et al. 1989), the Box-counting dimension and the Minkowski-Bouligand dimension are mathematically equivalent
in limit thus, Db = Dm. However, they are completely different in practice because of the way that limits are taken, and the manner in which they approach zero.
Experimental results published in the literature (Dubuc et al. 1989; Maragos and Sun 1993; Zeng et al. 2001) showed that these two methods suffer from inaccuracy and uncertainty. Indeed, according to Zeng et al. (2001) the precision of these estimators are mainly related to the following aspects:
– Real Value of the Fractal Dimension D: With big values of D, the estimation error is always very high. This can be explained by the effect of resolution (Huang et al. 1994). When the value of D increases, its estimates can not reflect the roughness of the object and higher resolution is then needed.
– Resolution: In the case of the temporal curves, the resolution consists of observation size of the curve (minute, hour, day…). According to Tricot et al. (1988) estimated fractal dimension decreases with the step of observation. This is due to the fact that a curve tends to become a horizontal line segment and appears more regular.
– Effect of Theoretical Approximations: Imprecision of the Box-counting and the Minkowski-Bouligand methods is also related to constraints occurring in theoretical approximations of these estimators. For example, the Box-counting dimension causes jumps on the log-log plots (Dubuc et al. 1989) which generate dispersion of the points of the log-log plots with respect to the straight line obtained by linear regression. Moreover, the value of N (є) must be integer in this method. The inaccuracy of the method of Minkowski-Bouligand is due to the fact that the Minkowski covering is too thick.
– Choice of the Interval [є0, єтax]: The precision of the estimators is influenced much by the choice of the interval [є0, єтах] through which the line of the log-log plots is adjusted. є0 is the minimum value that can be assigned to the step. When є0 is too large, the curve is covered per few elements (limp or balls). Conversely, when the value єтах is too small, the number of elements which cover the curve is too large and each element covers few points or pixels. Some researchers tried to choose this “optimal” interval in order to minimize the error in estimation (Dubuc et al. 1989; Huang et al. 1994). For example, Liebovitch and Toth (1989) proposed a method for determining this interval, Maragos and Sun (1993) used an empirical rule to determine єтах for temporal signals. In practice, these optimal intervals improve considerably the precision of the fractal dimension estimate for special cases but not in all cases.
