2.1 Box-count estimator

Box-count estimator is one of the most popular and commonly used estimators, and it is motivated by the following scaling law:

where D _BC is the Box-counting dimension, N(ϵ) denotes the smallest number of boxes of width ϵ in the Euclidean space with dimension d, which can cover the point set:

i.e. graph of time series or spatial data X _t observed at finite time set T⊂R.

The main idea of this method is that initially the series graph is covered by a single bounding box, which is divided in four quadrants. This process is repeated subsequently with each of the quadrants until the resulting box width matches the resolution of the data recording the number of boxes required at each step. Thus, if N(ϵ) is the number of boxes required at scale ϵ, Box-count estimator equals the slope in an ordinary least squares regression fit of log N(ϵ) on log ϵ. This method can be used to quantify the fractal dimension of any set of points in the plane and, particularly, for single-valued interfaces or time series data.

Some authors identified problems with the original Box-count estimator, which includes all scales in the least squares regression fit of log N(ϵ) on log ϵ, and proposed modifications that allow obtaining reliable estimations of D for some kinds of data¹⁵.

2.2 Hall-Wood estimator

Peter Hall y Andrew Wood¹⁶ introduced a Box-count estimator modification for fractal dimension D _HW , which operates to the smallest observable scales. Let A(ϵ) be the total area of the boxes with scale ϵ that intersect with the linearly interpolated graph of X, then if N(ϵ) is the total number of such boxes, hence

which results in a reformulation of definition (1) being:

On scale ϵl=l/n , where n is the total number of data series and l=1,2,...,n, Hall-Wood estimator works similarly to Box-count estimator, but the main difference is that the scale application operates only over the “columns" and counts A(l/n).

This estimator has better accuracy as compared to the original method of Box-count ¹⁵, but its disadvantage is that it can only be applied to the case in which the set of points X is a single-valued profile (interface) of form (2), that is, the time series graph or spatial data observable on a finite set T⊂R of times or equally spaced points ¹⁰.

2.3 Madogram and Variogram estimators

In the case when the data is given as a time series data or spatial data observed on a finite set T⊂R of times or equally spaced points, the best results for the estimation of D are provided by the variational estimators, particularly Madogram and Variogram estimators¹⁵, and, potentially, their use can be extended to anisotropic and non-stationary processes.

If {Xt : t∈T, Xt∈Rd} is a Gaussian process with stationary increments whose variogram or structure function is:

where γ2(t) is the average value of the squared difference of the values in two points separated by a distance or space t and E(.) denotes the mathematical expectation, then it holds:

where α∈(0,2, β≥0, y C2>0. Here α is the fractal index, β is roughness exponent, |⋅| denotes the Euclidean norm, so, a sample’s path graph, has the fractal dimension:

The Variogram estimator is obtained using the classical method of moments estimator for γ2(t). In the publication of Bez and Bertrand¹⁷ it was suggested that the Madogram estimator, that is, variational estimator obtained from the Variogram of order p > 0 of the form:

For p = 1, is more efficient and the relation (5) between the fractal dimension D and the fractal index α is more robust as compared to the default case p = 2 due to the fact that

Generally, the lower is the value of roughness exponent β, the harder is the estimation of fractal index α, and is more pronounced the finite sample bias of estimators of the fractal index or fractal dimension¹⁵.

3.1 Fractionating estimator (FE)

Using as reference the Hall-Wood estimator with scale ϵl=l/n adjusted equidistantly to a data series of form (2), namely, single valued profiles, we developed a procedure, which allows estimating the fractal dimension from a self-avoiding curve embedded in R2 obtained from an interface image. This method, called Fractionating estimator (FE), consists in the following. At the first step the self-avoiding curve is covered by a bounding box and after that at the k- th step, where k = 2, 3, …, m, his box is divided into k ² equal boxes, with this goal in mind each side of the bounding box is divided into k equal segments (Fig. 3a). Then the intersection points between the self-avoiding curve and the obtained grid are determined and a broken line is constructed by joining consecutively the intersection points (Fig. 3b). Thereafter, the segments of the broken line are covered by individual bounding rectangles with consecutive vertices in break points (Fig. 3c) and the number of such rectangles Ñ (k) at step k is counted, and so on (Fig. 3d). The FE estimation of fractal dimension is obtained as the linear regression fitfrom the log-log plot, of Ñ (k) vs. k ^-1 , for k = 1, 2, …, m. Here the number of steps m is an unbounded natural number, which depends on the image resolution and computational capacity, the greater the number m the greater the accuracy of the estimate of D.

Figure 3. Operation of FE on a self-avoiding curve: a) grid (k = 6), b) adjusted broken line (k = 6), c) the box cover of the adjusted broken line (k = 6), d) the box cover of the adjusted broken line (k = 10). 

Note that meanwhile the number of step k increases the broken line of FE approaches the interface curve. Furthermore, FE splits the rough self-avoiding curve not equidistantly, and in this way, it takes into account as much information from the rough curve both in rows and columns, “breaking it down" into fractional straight segments, preserving the Hall-Wood scale idea but counting boxes as in the Box-count estimator.

3.2 Fractal dimension estimation of Koch curve, modified Koch curves and Weierstrass curve

To verify the efficiency of FE, proposed above, a routine in Visual Basic for Applications for Corel Draw X3 was developed. The developed program was used for the estimation of fractal dimension by FE of the Koch curve (Fig. 4a) with theoretical D = 1.1609, single-value transformation of Koch curve (Fig. 4b), modified Koch curve (Fig. 4c) with theoretical D = 1.1609, G² modified Koch curve (Fig. 4d) with theoretical D = 1.2924 and the Weierstrass curve (Fig. 4e) of the form:

for a = 0.5, b = 33 and theoretical D = 1.8017.

Figure 4. a) Koch curve generated with 4200 nodes, b) Single-valued transformation of Koch curve, c) modified Koch curve generated with 3200 nodes, d) G2 modified Koch curve generated with 6680 nodes, e) Weierstrass curve generated with 1571 nodes. 

In addition the Box-count estimator included in software Benoit 1.31 and the plug-in for Image J software of FracLac 1.48, and the estimators Box-count, Hall-Wood, Variogram and Madogram of fractaldim package of software R, where used for the estimations of D for the considered fractals curves.

The FE routine was performed directly to the generated curve without transforming it to a single-valued profile, and the original vector graphic image was exported to .bmp format for Benoit 1.31 and FracLac 1.48, where the D was also estimated directly from the image using the Box-count estimator.

As the software R works only with numerical data the Koch curve was transformed into a single-valued curve (Fig. 4b) in .png format. After that the binarized image was interpreted as a matrix of ones and zeros, each column of ones are accounted to get a series of single-valued profile, from which D was estimated by Box-count, Hall-Wood, Variogram and Madogram estimators.

The results of the estimates of the single-valued Koch curve (Fig. 4b), are presented in a synthesized way by Boxplot graph in Fig. 5, where the theoretical and expected dimension D = 1.2619 of the Koch curve is marked with dotted horizontal line.

Figure 5. Fractal dimension estimations from the transformed single-valued Koch curve. 

Numerical results of the average estimates for a single-valued profile generated from the Koch curve obtained using the fractaldim package are presented in Table I.

Table I. Average values of fractal dimension estimation for the single-valued Koch curve. 

From Table I it can be seen that the Hall-Wood estimator has the best performance, followed by Variogram, but none of the considered estimators gives a reliable estimate. Above mentioned results confirm the fact that the transformation of the original Koch curve into a single-valued profile leads to a significant errors in fractal dimension estimation.

Now, performing the estimation of D for the original Koch curve by FE, t was observed, as expected, that for self-avoiding interfaces the basic proportionality of Hall-Wood method (3) does not hold (Fig. 6).

Figure 6. Hall-Wood relation (3) of Koch curve. 

At the same time, from Fig. 7, one can observe a power law behavior with the coefficient of determination R² very close to 1 and the estimated value of D obtained by the proposed FE method is 1.2609 and approximates the theoretical D = 1.2619.

Figure 7. Fractal dimension estimation for Koch curve obtained by FE method. 

In Fig. 7 the points indicated as triangles correspond to outliers and were not taken into account for the linear regression fit, as the value of the coefficient of determination R² in that case reduces to R² = 0.9873. The fractal dimension of the original Koch curve was also estimated by the Box-count estimator using the software Benoit and FracLac. The same estimators were used for modified Koch curve (Fig. 4c), G² modified Koch curve (Fig. 4d) and Weierstrass curve (Fig. 4e).

The values of estimates of fractal dimensions of the Koch curve, modified Koch curve, G₂ modified Koch curve and Weierstrass curve obtained by FE, Benoit and FracLac methods are presented in Table II.

Table II. Fractal dimension estimates of theoreticall self-avoiding curves: Koch curve, modified Koch curve, G ₂ modified Koch curve and Weirstrass curve. 

As it can be seen from Tables I and II, the developed FE method gives significantly better estimates for both curves studied as compared to the commonly used methods of estimation.

Furthermore, as the scaling behavior of a profile with overhangs (Fig. 2), can lead from self-affinity to multi-affinity by the removal of overhangs in the representation of real interface by single-valued profile²⁴, the proposed method avoid this problem and there is no need of further study scaling and universality for a varied self-avoiding curves.