On Optimal Thickness of the Curve at Calculating the Fractal Dimension Using the Box-Counting Method
Ilya Pershin, Dmitrii Nikolaevich Tumakov · Journal of Computational and Theoretical Nanoscience · 2019
Fractal curves are found in various fields of science and technology, and the calculation of the dimensions of the curves is an actual direction. The fractal dimension (FD) is a metric used to characterize the filling of a plane with a curve. The choice of the thickness of the curve line at calculating its FD using the box-counting method (BCM) is not obvious. This is the subject of the present paper. First, we consider a straight horizontal line with different thicknesses, and calculate FD for different thicknesses, using BCM. Using the exact dimensional value for a straight line, the initial conclusion is made about the thicknesses that give a correct FD. In the next step, the dimensions of straight lines rotated by a certain angle in the range from to 90 degrees are calculated. At any angle of inclination of the line, the FD values should also remain equal to unity. The range of FD values depending on the rotation angle is analyzed. The result is an “optimal” line thickness equal to 3 px. At this value, the FD values are the closest to one for any angle of inclination. As a result it is received that at dimensional calculation by means of BCM, it is enough to choose the initial image of 200 on 200 pixels, and boxes to choose the sizes from 2 to 50 pixels with step of 2 pixels. The stability of dimension calculations on the grid offset has been verified. The algorithm’s error is estimated on the example of the calculation of FD for the Koch fractal.