Documentation

1. Measure of fractal dimension

1.1 Principles

Fractalyse implements different methods (box counting, radius mass, correlation...) to measure fractal dimension which corresponds to different dimensions (Hausdorff, Minkowski, correlation...). The process of the measure is split in two part :

the counting method
the estimation module.

1.1.1 The counting method

Counting method goes step by step following an iteration principle. At each iteration step, the method involved counting the number of black pixels contained in a counting window. From one step to the next, the size of the counting window is enlarged. By doing that, we artificially change the level of analysis of the image. So, for each method we have two elements varying according to the counting step (iteration step) (i):

the number of counted elements (which is roughly the number of black pixels present in the window) (N)
the size of either the counting window or the reference element (ε).

Then, we obtain a series of points that can be represented on a Cartesian graph. The Y-axis corresponds to the number of counted elements (N) and the X-axis corresponds to the size of the counting window or to the size of the reference element ε, with ε increasing from step to step.

	-> [Counting method] ->		-> [Estimation module] ->	D=1.7
Image		Curve		Fractal dimension

1.1.2 The estimation module

Mathematically, the series of points is a curve (named the empirical curve). The next stage is to fit this empirical curve with another one, the estimated curve. If the empirical curve follows a fractal law, the estimated curve has the form of a power law (parabolic or hyperbolic), and D represents the fractal dimension.

N = ε^D or N = ε^-D