Modeling of Banknote Paper Surface Based on the Theory of Fractals
DOI:
https://doi.org/10.20535/1810-0546.2016.2.66801Keywords:
Paper with water marks, Surface structure of paper, Theory of fractals, Fractal dimension, Amplitude coefficient, Mathematical model of the surface, ProfilometryAbstract
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.
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