6ee29754-7e22-409f-99be-de1ba645dbd320210216094045859wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS1109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40512720202720201910.37394/23206.2020.19http://wseas.org/wseas/cms.action?id=23185OlegMilderUral Federal University, Mira 32, Ekaterinburg 620002, RUSSIADmitryTarasovUral Federal University, Mira 32, Ekaterinburg 620002, RUSSIAColor management in printing processes has been traditionally based on an analysis of the behavior of tone reproduction curves (TRC) calculated for the initial color channels. The tone curves, as well as, the color channels, are considered separately. This approach does not take into account the mutual influence of colorants when they overlap. We propose replacing two-dimensional tone reproduction curves with three-dimensional gradation trajectories in the CIE Lab metric space. When two colors overlap, one considers the space between two gradation trajectories that forms a gradation surface. These objects are described using the apparatus of differential geometry of spatial curves and surfaces, respectively, and are also invariants of color spaces. In this paper, we offer their analytical description.228202022820201116https://www.wseas.org/multimedia/journals/mathematics/2020/a045106-070.pdf10.37394/23206.2020.19.2http://www.wseas.org/multimedia/journals/mathematics/2020/a045106-070.pdfBala, R. (2003) Device characterization. Digital Color Imaging Handbook. G. Sharma (ed.). CRC Press, Boca Raton, FL, 269–379. Balasubramanian, R. (1999) Optimization of the spectral Neugebauer model for printer characterization. J Elec. Imag. 8, 156–166. 10.1117/12.588907Hersch, R.D., Crété, F. (2005) Improving the Yule–Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions. Proc. SPIE 5667, 434–445. 10.1002/(sici)1520-6378(200002)25:1<4::aid-col3>3.0.co;2-xWyble, D.R., Berns, R.S. (2000) A critical review of spectral models applied to binary color printing. Col.Res.&App. 25, 4–19. 10.2352/j.imagingsci.technol.(2008)52:4(040908)Garg, N.P., Singla, A.K., Hersch R.D. (2008) Calibrating the Yule–Nielsen Modified Spectral Neugebauer Model with Ink Spreading Curves Derived from Digitized RGB Calibration Patch Images. J.Imag.Sci.Tech. 52(4), 040908–040908-5.Arney, J.S., Engeldrum, P.G., Zeng, H. (1995) An expanded Murray-Davis model of tone reproduction in halftone imaging. J.Imag.Sci.Tech. 39, 502–508.Livens, S. (2002) Optimisation of Printer Calibration in the Case of Multi Density Inks. Conference on Color in Graphics, Imaging, and Vision, CGIV 2002. Final Program and Proceedings, 633–638.Chagas, L., Blayo, A., Giraud P. (2004) Color Profile: methodology and influence on the performance of ink-jet color reproduction. IS&T's NIP20. 2004 International Conference on Digital Printing Technologies, 655–659. Wu, Y-J. (2014) Reducing Ink-jet Ink Consumption with RIP software for POP Display Media. Digital Fabrication and Digital Printing. NIP30 Technical Program and Proceedings, 108–111. Titova, M., Milder, O., Tarasov, D. (2016) Gradation trajectories: acquisition and analysis. Proceedings of the Conference «Information: transmission, processing, perception», UrFU, 2016, 120–125 (in Russian). Milder, O.B., Tarasov, D.A., Titova M.Yu (2017) Inkjet Printers Characterization Using 3D Gradation Curves. CEUR Workshop Proceedings. Vol.1814. 74–83. 10.1007/978-3-319-75786-5_4Milder, O., Tarasov, D. (2018) Ink-jet printer’s characterization by 3D gradation trajectories on an equidistant color difference basis. Lecture Notes in Computer Science. Vol. 10749, 2018, 40–52.Jeffreys H. & Jeffreys B. S. (1988) Weierstrass's Theorem on Approximation by Polynomials" and "Extension of Weierstrass's Approximation Theory. §14.08-14.081 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, 446–448.Pogorelov A.V. (1959) Differential geometry. Noordhoff, 171p. 10.1145/3369973.3369981Mohcine Boudhane, Ojars Balcers, Underwater Image Enhancement Method using Color Channel Regularization and Histogram Distribution for Underwater Vehicles AUVs and ROVs, International journal of circuits, Systems and Signal processing, pp.570-578, Volume 13, 2019.10.1051/itmconf/20181602004Idoko John Bush, Rahib Abiyev, Mohammad Khaleel Ma’aitah, Intelligent Machine Learning Algorithms for Colour Segmentation, WSEAS Transactions on Signal Processing, Volume 13, 2017, pp. 232-240.10.1007/978-3-319-75786-5_5Milder, O., Tarasov, D. (2018) Gradation surfaces as a method for multi-color ink-jet printers color specifications management. Lecture Notes in Computer Science (LNCS). Vol. 10749, 2018, 53–61.Korn, G.A., Korn, T.M. (2000) Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Courier Corporation, 1130p.