Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnic National University
Lviv Polytechnik National University

The article goes about the methods of training a neural network to recognize fractal structures with the rotation of iteration elements by means of an improved randomized system of iteration functions. Parameters of fractal structures are used to calculate complex parameters of physical phenomena. They are an effective tool in scientific works and used to calculate quantitative indicators in technical tasks. The calculation of these parameters is a very difficult mathematical problem. This is caused by the fact that it is very difficult to describe the mathematical model of the fractal image, it is difficult to determine the parameters of the iterative functions. The neural network learning will allow you to quickly determine the parameters of the first iterations of the fractal based on the finished fractal image and basing on them to determine the parameters of the iterative functions. The improved system of randomized iterative functions (SRIF) will allow to describe the mathematical process and to develop the software for generating fractal structures with the possibility of rotating elements of iterations. In its turn, this will make it possible to form an array of data for training a neural network. The trained neural network will be able to determine the parameters of the figures of the first iterations by means of which it will be possible to build a system of iterative functions. It will help to reproduce a fractal structure qualitatively. This approach can be used for three-dimensional fractal structures. After setting the parameters of the first iterations of the fractal, it will be possible to determine the geometric structure which is the basis of the fractal structure. In the future, this approach may be included in the system for recognizing objects under fractal structures, for example, under masking nets.

[1]     Al-shameri, W. F. H. Deterministic algorithm for constructing fractal attractors of iterated function systems. Eur. J. Sci. Res. 2015, 134, рр. 121–131.

[2]     Mandelbrot, B. B. The Fractal Geometry of Nature; W. H. Freeman & Company: New York, NY, USA, 1999.

[3]     O. Yunak, O. Shpur, B. Strykhaliuk, M. Klymash. Algorithm forming randomized system of iterative functions by based cantor structure. Information and communication technologies, electronic engineering, 2021, No. 1 (2), рp. 71–80.

[4]     M. C. Gutzwiller, Benoît B. Mandelbrot, C. J. G. Evertsz, et al. Fractals and Chaos: The Mandelbrot Set and Beyond. Springer New York, 2010. ISBN: 1441918973.

[5]     B. B. Mandelbrot. The Fractal Geometry of Nature; ‎ Echo Point Books & Media, LLC, 2021. 490 p. ISBN-10:‎ 1648370403.

[6]     Z. Z. Falconer, Kenneth Falconer. Techniques in Fractal Geometry. Wiley & Sons, Incorporated, John. 1997. 274 p. ISBN: 0471957240.

[7]     Юнак О. М., Пелещак Б. М., Охремчук Н. Л., Метлевич Я. Р. Перетворення зображення фрактальної структури типу “Фрактальний пил” (множина Кантора) в рандомізовану систему ітераційних фунцій, XII Міжнар. наук.-практ. конференція “Последните постижения на Европейската наука - 2016”, Том 13, София “Бял ГРАД-БГ” ООД, 2016. 90 с.

[8]     Mandelbrot, B. B. Fractals: Form, Chance and Dimension,  Echo Point Books & Media; Reprint ed. edition 2020. 656 p.

[9]     Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications 3rd Edition, 2014. 400 c.  ISBN-10: 111994239X.

[10]  The Mandelbrot Set and Beyond New York: Springer, 2004. 308 p. ISBN: 0-387-20158-0.

[11]  Peter R. Massopust. Fractal Functions, Fractal Surfaces, and Wavelets. Elsevier Science & Technology. Elsevier Science & Technology, 1995.  383 p. ISBN: 0124788408.