1. Introduction

In the color management, the color reproduction adjustment is closely related to the fine tuning of the basic colorants gradations. This approach works in conjunction with various color prediction models embedded in the color management systems and aims to adjust colors, save amount of inks used, and to predict the resulting color in print. Initial colorants in the frame of the standard subtractive color model are represented by three basic colors (CMY) and one additional Black (K). The color formation in such a model is a separate complicated task, although all color prediction models have to rely on those four colorants and their overlapping.
On the other hand, a few is known about the initial colorants in real printing systems. Usually, each manufacturer uses its own recipe of the CMYK inks formation based on the fossil pigments. Hence, each color reproduction system, in fact, uses a unique initial colorants set, especially, considering the non-ideality of the printing system itself. Anyway, each system reproduces quite accurate colors after the procedure of calibration and color tuning with the help of color prediction models used in the color management software is carried out.
There are many different color prediction models applied in color management. The empirical surface models take into account superposition of ink halftones and do not deal with the light propagation and fading within the print. The physically inspired models engage a more detailed analysis of light-print interaction based on the prediction of how the light paths go within a halftone print and what the resulting fade is. The ink spreading models characterize the effective surface of an ink dot after it has been printed at a given nominal surface coverage compared to the effective surface coverage that forms the physical dot gain. The spectral reflection color prediction models deal with spread-based light propagation-transportation probability. These models study the impact of different factors influencing the range of printable colors (the inks, substrate, illumination conditions, and halftones) and create the printer characterization profiles for the purpose of color management [1]. These models together with the ink-spreading models take into account the physical dot gain and are able to predict the reflectance spectra as a function of ink surface coverage for 2–4 inks (binary and ternary color systems). The models use the multiple tone reproduction (ink spreading) curves to characterize the physical dot gain of the ink halftones on the substrate and under all solid ink-superposition conditions [2–4].
The color prediction models have been successfully applied to the color reproduction management [5–9], however, all the proposed approaches require a significant number of measurements, computations and checks. However, since most of them are based on empirical relationships, the accuracy of predicting some shades of the reflection spectrum remains low.
If one looks at the problem of color reproduction from the other side, the use of the “ideal” initial colorants might be an excellent solution providing the basic color stability. An ideal colorant is such a combination of the initial colorants, which ensures minimal fluctuations of color tone in the range from a white substrate to a full dye. In this case, the ideal colorant will be a specific mixture of two initial colorants.
In previous works, the gradation trajectories (GT) as a three-dimensional interpretation of tone reproduction curves (TRC) in CIE Lab space were introduced [10–12] (see also Fig.1 (a, b)). Thus, any color is uniquely described by a three-component vector in the 3D Lab space. The metric of the space is the color difference dE (or ΔE ), which is defined at least by the root squared sum of color components differences. Presence of any metric in the sense of the method of measuring distances turns the color space into a metric one.