(44-1) 15 * << * >> * Russian * English * Content * All Issues

Mathematics and practice of color space invariants by the example of determining the gray balance for a digital printing system

D.A. Tarasov 1, O.B. Milder 1

Ural Federal University, Ekaterinburg, Russia

 PDF, 1699 kB

DOI: 10.18287/2412-6179-CO-580

Pages: 117-126.

Full text of article: Russian language.

In modern printing, a large number of tasks are associated with the mutual transformation of color spaces. In particular, the most common pair of hardware-dependent color spaces is RGB and CMYK, the mutual transformation of colors in which is ambiguous, which creates significant problems in color reproduction. To solve this problem, we propose using color space invariants — gradation trajectories and gradation surfaces, which are analogs of gradation curves for initial colorants and their binary overlays, constructed in the absolute color space of the CIE Lab. Invariants are introduced on the basis of the mathematical apparatus of the differential geometry of spatial curves and surfaces. Practical application of color space invariants involves certain difficulties associated with their complex analytical description; moreover, for most practical problems, the high accuracy of the model is redundant. For the practical application of invariants, we propose a simpler approach using natural color sampling in digital printing systems. As an example, the procedure for determining the gray balance for an electrophotographic printing press is given.

grey balance, gradation trajectories, gradation surfaces, digital printing.

Tarasov DA, Milder OB. Mathematics and practice of color space invariants by the example of determining the grey balance for a digital printing system. Computer Optics 2020; 44(1): 117-126. DOI: 10.18287/2412-6179-CO-580.


  1. Bandyopadhyay S, Mandal S. Effect of gray component replacement on color reproduction. Image Processing, Image Quality, Image Capture, Systems Conference 2000: 188-191.
  2. Lo M, Chiang R. Characterization models for multi-colored CMYKRGB printing process. Proc TAGA 1998: 242-254.
  3. Spiridonov I, Shopova M, Boeva R. Study of the effect of gray component replacement level on reflectance spectra and color reproduction accuracy. Proc SPIE 2013; 8770: 87700W.
  4. Zeng H. Neutral gray adjustment in printer ICC profiles. Proc SPIE 2003; 5008: 341-348.
  5. Woolfe G, Spaulding K, Giorgianni E. Optimal color spaces for balancing digital color images. Final Program and Proceedings of the 11th Color and Imaging Conference 2003: 66-70.
  6. Breede MH. Testing the validity of measuring gray balance with the GATF Color Circle, grayness and hue error values. Proc TAGA 2006: 292-314.
  7. Hsieh Y-C, Wu Y-J. The relationship between the chroma values of neutral gray and important print attributes. Proc TAGA 2005: 57-58.
  8. Rong X. 7 method for indigo press calibration and proofing. 24th International Conference on Digital Printing Technologies and Digital Fabrication 2008: 603-606.
  9. Habekost M, Blum V. G7 for every print device? Proc TAGA 2015: 204-221.
  10. Garg NP, Singla AK, Hersch RD. Calibrating the Yule-Nielsen modified spectral Neugebauer model with ink spreading curves derived from digitized RGB calibration patch images. J Imag Sci Tech 2008; 52(4): 040908.
  11. Arney JS, Engeldrum PG, Zeng H. An expanded Murray-Davis model of tone reproduction in halftone imaging. J Imag Sci Tech 1995; 39: 502-508.
  12. Livens S. Optimisation of printer calibration in the case of multi density inks. Conf CGIV 2002: 633-638.
  13. Chagas L, Blayo A, Giraud P. Color profile: methodology and influence on the performance of ink-jet color reproduction. International Conference on Digital Printing Technologies 2004: 655-659.
  14. Wu Y-J. Reducing ink-jet ink consumption with RIP software for POP display media. Digital Fabrication and Digital Printing: NIP30 Technical Program and Proceedings 2014: 108-111.
  15. Titova MYu, Milder OB, Tarasov DA. Determination of saturation bands set using LAB color system. In Book: Information: transfer, processing, perception. Ekaterinburg: UrFU Publisher; 2016: 120-125.
  16. Milder OB, Tarasov DA, Titova MYu. Inkjet printers characterization using 3D gradation curves. CEUR Workshop Proceedings 2017; 1814: 74-83.
  17. Milder O, Tarasov D. Ink-jet printer’s characterization by 3D gradation trajectories on an equidistant color difference basis. In Book: Paul M, Hitoshi C, Huang Q, eds. Image and Video Technology. Springer International Publishing AG; 2018: 40-52.
  18. Tarasov DA, Milder OB. Gradation trajectories as an analog of gradation curves in the metric CIE Lab space: discrete approach. Computer Optics 2019; 43(1): 132-136.
  19. Sharma G, Wu W, Dalal EN. The CIEDE2000 color-difference formula: Implementation notes, supplementary test data, and mathematical observations. Color Research and Application 2005; 30(1): 21-30.
  20. Milder O, Tarasov D. Gradation surfaces as a method for multi-color ink-jet printers color specifications management. Image and Video Technology (PSIVT 2017) 2018: 53-61.
  21. Korn GA, Korn TM. Mathematical handbook for scientists and engineers: Definitions, theorems, and formulas for reference and review. Courier Corporation; 2000.
  22. Sergeev AP, Tarasov DA, Arapov SY, Arapova SP. Qualimetric researches of educational resources: Standardizing of light conditions in the light booth // Procedia – Social and Behavioral Sciences 2015; 174: 1285-1291.


© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: ko@smr.ru ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20