# Modeling of Banknote Paper Surface Based on the Theory of Fractals

## Authors

• Тетяна Юріївна Киричок National Technical University of Ukraine “Kyiv Polytechnic Institute”, Institute of Printing and Publishing, Ukraine
• Тетяна Євгенівна Клименко National Technical University of Ukraine “Kyiv Polytechnic Institute”, Institute of Printing and Publishing, Ukraine
• Олександр Владиславович Рибак National Technical University of Ukraine “Kyiv Polytechnic Institute”, Institute of Physics and Technology, Ukraine

## Keywords:

Paper with water marks, Surface structure of paper, Theory of fractals, Fractal dimension, Amplitude coefficient, Mathematical model of the surface, Profilometry

## Abstract

Background. The article describes the experimental study of the structure of the paper surface.

Objective. The surface structure modeling of printed and unprinted banknote paper using fractal theory. Calculation of the main parameters describing the macro- and microstructure of the banknote paper surface within and beyond water marks.

Methods. A method for studying the structure of the surface paper based on the theory of fractals is used. This method is based on profilograms analysis obtained by contact and optical profilometry.

Results. The approximate method of measuring the fractal dimension of arbitrary shape was determined $S_{i}\cdot h_{i}^{D}\approx c,$ where c – some constant, and D – fractal dimension figure. Function of the surface paper sheet was received $f(h)\approx L\cdot c\cdot h^{-(1+D_{\mathrm{np}})}.$ Determined the fractal dimension $D=1+D_{\mathrm{np}}$ and amplitude coefficient of the surface of the paper $S=C\eta^{(2D-3)}\tau ^{(4-2D)}.$ Graphical model structure of the surface paper on the basis fractal dimension and amplitude coefficient was constructed.

Conclusions. A method studying the structure of the paper surface based on the theory of fractals was developed. This method is based on analysis of profilograms obtained by contact and optical profilometry. A mathematical model of the surface of unprinted and printed banknote paper was developed. Algorithm and software to determine the fractal dimension and the amplitude coefficient of the microstructure of the unprinted and printed paper was developed. The value of the fractal dimension and the amplitude coefficient of the surface of the banknote paper was obtained.

## Author Biographies

### Тетяна Юріївна Киричок, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Institute of Printing and Publishing

Tetiana Yu. Kyrychok, doctor of tehnical sciences, associate professor, assistant professor at the Technology Printing Production Department, director

### Тетяна Євгенівна Клименко, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Institute of Printing and Publishing

Tetiana Ye. Klymenko, PhD, assistant at the Technology Printing Production Department

### Олександр Владиславович Рибак, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Institute of Physics and Technology

Oleksandr Vlad. Rybak, assistant at the Information Security Department

## References

M. Perron, “Patenting of new banknote security features”, Billetaria. Int. Rev. Cash Management, iss. 9, pp. 22–23, 2010.

T.J. Kyrychok et al., “Badania odpornosci na zuzycie papieru banknotowego” = [“Testing of banknote paper wear and tear”], Przegland Papierniczy, no. 12, pp. 653–656, 2013.

L.S. Korochkyn, Materials and Methods of Protection of Special Papers and Documents from Counterfeiting. Minsk, Belarus: NTUP “Kryptotekh”, 2001, 264 p. (in Russian).

The Technology of Pulp and Paper Production, vol. 2, Manufacture of Paper and Paperboard, part 1, Technology of Production and Processing of Paper and Paperboard. St Petersburg, Russia: Polytekhnyka, 2005, p. 20 (in Russian).

V.Y. Bobrov et al., “Application of fractal theory to describe and simulate the surface of printed materials”, Vestn. MHUP, no. 6, pp. 76–81, 2010 (in Russian).

A.V. Holunov and L.H. Varepo, “Determination of the fractal dimension of the surface profile papers (cardboards)”, in Proc. IX Int. Conf. Visual Culture: Design, Advertising, Information Technology, Omsk, Russia, 2010, pp. 175–177 (in Russian).

J.M. Geusebroek and A.W.M.A Smeulders, “Physical explanation for natural image statistics”, in Int. Workshop on Texture Analysis and Synthesis, Amsterdam, Netherlands, 2002, pp. 47–52.

V.Y. Kurdyukov et al., “Development of fractal mathematical model of the surface roughness”, Vestnik Kuzbasskogo Gosudarstvennogo Tehnicheskogo Universiteta, no. 5, pp. 43–45, 2008 (in Russian).

B. Mandelbrot, Fractal Geometry of Nature. Moscow, Russia: IKI, 2002, 656 p. (in Russian).

V.I. Rolduhin, “Fractal structures in materials”, Materialoznavstvo, no. 5, pp. 19–26, 2005 (in Ukrainian).

T. Kyrychok et al., “Microgeometry of the paper surface with watermarks”, Tekhnolohiya i Tekhnika Drukarstva, no. 4, pp. 130–137, 2009 (in Ukrainian).

M.Y. Kulak, Fractal Mechanics of Materials. Minsk, Belarus: Vysshaja Shkola, 2002, 304 p (in Russian).

M.Y. Kulak, Methods of the Fractal Theory in the Technological Mechanics and Control Processes: Printing Materials and Processes. Minsk, Belarus: Belarusskaja Nauka, 2007, 419 p. (in Russian).

D.M. Medyak, “Fractal geometry of the surface structure of the paper with water marks”, Trudy BGTU, no. 9, pp. 51–56, 2011 (in Russian).

V.Yu. Kobenko, “Determination of the surface quality of the paper method of fractal analysis”, Omskij Nauchnyj Vestnik, no. 3, pp. 330–334, 2011 (in Russian).

M. Bigerelle, “A new method to calculate the fractal dimension of surfaces: application to human cell proliferation”, Int. J. Comp. Math. Applications, vol. 42, no. 1-2, pp. 241–252, 2001.

E. Feder, Fraktals. Moscow, Russia: Mir, 1991 (in Russian).

T. Kormen et al., Algorithms: Construction and Analysis, 2nd. Moscow, Russia: Vylyams, 2005 (in Russian).

R. Kallan, The Basic Concepts of Neural Networks. Moscow, Russia: Vylyams, 2001, 287 p. (in Russian).

Paper. Methods for Determination of Transparency and Opacity, GOST 8874–80, 1981 (in Russian).

Paper and Board Determination of Air Permeance (Medium Range), part 3, Bendtsen Method, ISO Standard 5636–3:1992.

Paper and Cardboard. Viznachennya Shorstkostі Method (Method іz Zastosuvannyam Bandwidth Povіtrya). Bendtsen Method, State Standard ISO 8791-2:1990, 1990 (in Russian).

Paper and Cardboard. Method for Determination of Surface Absorption of Water at One-Sided Wetting (Cobb Method), GOST 12605–97, 2001 (in Russian).

Paper and Cardboard. Method for Determination of Surface Absorption by Drip Method, GOST 12603–67, 1967 (in Russian).

Т. Kyrychok et al., “Badanie szorstkosci powierzchni papierow ze znakami wodnynmi” = [“The investigation of roughness for paper with watermarks”], Przegland Papierniczy, no. 7, pp. 414–417, 2012.

S.V. Maydanyuk et al., “Module for measuring shapes”, Visnyk ZhDTU, no. 2 (41), pp. 15–18, 2007 (in Ukrainian).

S.R. Yhnatovych et al., “Instrumental complex for micro- and nano- surface testing materials and coatings”, Tehnical Systems, no. 5 (49), pp. 72–78, 2009 (in Ukrainian).