Documentation

1. Measure of fractal dimension

1.3 The estimation module

The next stage of the process is to fit the empirical curve (in blue) from counting method, with another one, the estimated curve (in red). If the empirical curve follows a fractal law, the estimated curve has the form of a power law (parabolic or hyperbolic) N = ε^D or N = ε^-D. Because an image is not a pure fractal (it is not a continuous function but a discrete and finite one), it is only possible to approximate the fractal law. It explains why Fractalyse propose some option to approximate the empirical curve :

non linear regression
linear logarithmic regression
non linear differentiate regression (only with radial and correlation methods)

1.3.1 Nonlinear regression

In this method, we can approximate the empirical curve with four different equations : the fractal law N(ε) = ε^D and three others wich permit to measure also the fractal law deviation

N(ε) = a.ε^D
N(ε) = ε^D+c
N(ε) = a.ε^D+c

D is the fractal dimension.
c corresponds to the point of origin on the Y-axis.
a is called the “pre-factor of shape”. It gives a synthetic indication of the local deviations from the estimated fractal law.
Default is last equation.

To ensure that Fractalyse finds the best approximation, the software can performs the estimation with two different methods. The first one is based on the use of partial derivate whereas the second one is based on evolutionary strategies (that is, genetic algorithms). In most cases, the two methods give exactly the same results.
Default method is partial derivate, but you can use both by checking "ES estimation".
Last option is the error function. Default is least square method : f_err = (f_emp - f_estim)², but you can choose normalized least square : f_err = (f_emp-f_estim)² / f_emp² by checking "Normalized estim." (testing only).

1.3.2 Linear logarithmic regression

In this method, the curve is transformed by logarithmic transformation. Then the power law become linear equation :

log(N(ε)) = log(ε^D) => log(N(ε)) = D.log(ε)

In the software, we used log(N(ε)) = D.log(ε)+c

1.3.3 Nonlinear differenciate regression

This method has been added to compare results with non linear regression.

1.3.4 Quality of estimation

The quality of the estimation is quantified using a correlation ratio. If the fit between the two curves (empirical and estimated ones) is bad, two conclusions are possible: either the pattern under study is not of a fractal nature or it is of a multi-fractal nature. In the second case, the empirical curve has to be divided into several portions, each of them corresponding to a different estimated curve (i.e. according to the considered portion of curve, the non linear regression gives different values for the three parameters a and D and c).

Correlation ratio : 1 − ∑ ( y i − y i ˆ ) 2 ∑ ( y i − y ˉ ) 2 1 - sum (y_{i}-hat y_{i})^{2} over sum (y_{i}-bar y)^2

1.3.5 Other curves

Curve of the scaling behaviour : α(ε) = d log(N(ε)) / d log(ε)
Error curve : difference between empirical and estimated curve f_diff = f_emp - f_estim
Objective curve

<- 1.2 The counting methods