Grey balance adjusting is one of the key moments in digital image processing. In the work, an experimental attempt was made to adapt the ideas about the Maxwell color triangle (additive color synthesis) to the patterns of autotypic color synthesis. To adjust the grey balance, it is suggested not to use colorants of autotypic synthesis (CMYK), but their double overlays (RGB). As double overlays, it is proposed to use geodesic lines on the gradation surfaces of the corresponding double overlays. All geometric objects (points, lines, and surfaces) are considered in the CIE Lab space. The metric of the space is determined by the magnitude of the color di erence CIE dE. Experimental veri cation of the approach and discussion of the results were carried out.
Grey balance is the basic and substantial control parameter in image processing, which operates by balancing the correlation of substrate features and print control parameters. The grey balance together with di erent color prediction models (CPMs) helps users to adjust colors in print and improve the color management. Another bene t of well-grey-balanced printing system is shortage of ink usage that is particularly important in the case of current multi-colors printers. Today, there are many CPMs. Such models take a set of inks as inputs and predict the resulting color in print, as speci ed by re ectance or tristimulus values.
Empirical surface color prediction models take into account superposition of ink halftones and do not deal with the light propagation and fading within the print. The models demonstrate the relationship between re ected light and surface coverages by colorants. Physically inspired models engage a more detailed analysis of light-print interaction based on the mathematical prediction of how the light paths go within a halftone print and what the resulting fade is. Ink spreading models characterize the e ective surface of an ink dot after it has been printed at a given nominal surface coverage compared to the e ective surface coverage that forms the physical dot gain. The models accounting the ink spreading in all ink superposition conditions rely on the ink spreading curves. The curves map the nominal surface coverages to e ective surface coverages for the surface coverages of single ink halftones and ones superposed with one and two solid inks. The further development of the color prediction models deal with spread-based light propagation and transportation probability. Spectral re ection prediction models study the impact of di erent factors in uencing the range of printable colors (the inks, substrate, illumination conditions, and halftones) and create the printer characterization pro les for the purpose of color management [1]. These models together with the ink-spreading models take into account physical dot gain and are able to predict the re ectance spectra as a function of ink surface coverage for 2{4 inks (binary and ternary color systems). The models uses multiple tone reproduction (ink spreading) curves (TRC) to characterize the physical dot gain of the ink halftones on the substrate and in all solid ink superposition conditions [2{4]. Di erent color prediction models have been successfully applied to color reproduction management in various contexts [5{9]. The major drawback of the mentioned models is the fact that all of them are computationally capacious, as n colorants require a solution of system of 2n equations; therefore, they cannot be implicated into real work ow.
The empirical approaches are more promising; however, they usually require a signi cant amount of print tests to do. Most practitioners prefer using relatively simple methods for setting up printing systems by analyzing gradation scales and applying the gradation-based techniques that are known as an indispensable attribute of color printing systems settings [10, pp. 88{89]. At the same time, we express doubts about the rational use of these characteristics in the digital printing technology. The main problem is in the fact that using the gradation curves in conventional 2D embodiment signi cantly reduces the quantity and quality of information extracted from them. In work [11], the 3D gradation trajectories are introduced as a further development of the gradation curves approach. Implication of the mathematical apparatus of di erential geometry for gradation trajectories analysis in 3D CIE Lab space allows one to reveal their intrinsic features of curvature and torsion. These features are applied to de ne the ink limits in ink-jet printing systems and to create the empirical approach based on trajectories' curvature and torsion behavior analysis.
Such an approach might be expanded onto the grey balance management. It is known that the digital printing system performance might be greatly improved and the requirements on number of color measurements per calibration initialization might be greatly reduced when using the grey balance control systems [12]. Moreover, grey balance has been widely used in the sheet-fed o set [13] and web-o set printing [14] themselves, as well as in application of a digital printing press for simulating the o set printing [15]. Normally, the grey balance may be accounted in the Grey Component Replacement (GCR) procedure in a Color Management System (CMS) [16]. Sometimes, CMY-K balance is a ected by spectral response properties of inks [17]. Despite the fact that calibrated printing devices possess smooth and gradual changed tone curves and correct grey balance, they have their own characteristics in output color gamut with various saturation performances in every hue sections [18] that must be taken into account when adjusting the color. Another signi cant point in the grey balance is in uence of chromaticity deviation of printing substrate that might be vital [19].
Methods of grey balance implication can vary in color management. The G7 speci cation by Idealliance consortium o ers a simple methodology for the grey balance even for ink-jet [20]. Recent works have drawn more complicated math apparatus to establish a connection between CIE Lab and CMY-K colorants in order to manage the gray balance, e.g. polynomial regression [21] and arti cial neural networks [22].
These approaches are based on certain model representations. In addition, the results obtained depend on the print type for which the assessment was carried out. Tradition of the gradation curves application in the characterization of printing devices led to widespread use of indirect (transformed) data in the calculations. The CIE Lab coordinates are not fully utilized, but only for determining the color di erence. Till now, there has not been proposed an invariant model that would describe the gray balance for any printing method with su cient accuracy.
This work is devoted to application of the gradation trajectories method for grey balance adjusting. We suggest using the Maxwell's Color Triangle as a basement of the model. We o er an empirical approach for a characterization, which implies a small number of prints and rapid calculations. 2
The Maxwell's Color Triangle relies on the Grassmann's law, which is an empirical result about human color perception that chromatic sensation can be described in terms of an e ective stimulus consisting of linear combinations of di erent light colors. It works for additive color synthesis only. The major feature of the triangle is the following: by combining equal parts of basic colors, the neutral grey being obtained. Based on various print standards, it can be argued that this rule does not work for the process colors (CMY). In the case of printing colorants (CMY), their paired double overlays (binaries) correspond to additive primary colors (RGB). Since RGB and CMYK spaces are both device-dependent models, there has been no simple or general conversion formula that converts between them, as concerning the gray balance, at least. We might suggest the way to develop such a conversion based on the ideas of gradation trajectories and surfaces.
To start with, we have to outline a concept of gradation surfaces. The gradation trajectory of double overlays is a surface constructed on the basis of gradation trajectories of two colorants. Let us consider the formation of color with the participation of a certain type of substrate and 2 (two) colorants. Take as an example the pair of Cyan n and Magenta m inks. Each colorant is able to take a halftone value from 0 (pure substrate) to 1 (full dye). Thus, In the case of continuous halftones, we obtain a square region of allowable recipes on the plane (n; m). In real print, a frequent grid occurs instead of a solid square. Each reproducible tone has its own recipe (n; m) and a set of Lab-coordinates. A smooth change in tone causes a smooth change of Lab-coordinates. We can represent it in the form, which describe smooth surfaces in Lab-space, a = a (n; m) ; b = b (n; m) ; L = L (n; m) :
Equation (
aCyan = a0 + P3 i
aMagenta = a0 + P3i=1 aCyan;i n ; (
i=1 aMagenta;i mi: :
We designate a binary gradation surface as \stretched" on the gradation
trajectories of generatrix pairs of colorants so that it contain them inside. In
system (
a = a (n; 0) ; b = b (n; 0) ; L = L (n; 0) : (
When gradation surface is stretched on the gradation trajectories, then,
taking into account (
3 i
aBlue = a0 + X X ai j;j ni j mj : (
i=1 j=0
The required degree of the polynomial (third) was approved experimentally.
The graph of the analytic surface (see Fig.1) is deviated from the experimental
data by a distance smaller than the measurement error of the
spectrophotometer. A polynomial of the third degree was quite accurate, so the fourth-degree
polynomial was super uous.
(
Further, we have to introduce a concept of geodesic lines. Characterization of a printing machine is made by a uniform distribution of points along the gradation curve. Therefore, it would be logical to assume that this principle should also be adhered to in the case of a double halftone surface. Since the gure of double superimposition in color space is not a curve, but a surface, the points should be located evenly within the surface i.e., at an equal distance from each other to be a grid. It is precisely this goal that an element of di erential geometry, such as geodesic lines, perfectly t. A geodesic line is an analog of a straight line on a plane for a surface, i.e., straight line, which, by the shortest path, connects two points on the surface. The basic property of a geodesic line is: on any su ciently small piece of surface through two points the only one arc of the geodesic line can be drawn, just as on a plane through two points the only one straight line passed.
Below, we describe an algorithm for nding out a geodesic in general form.
In our case, we consider a regular piece of the surface B (blue) de ned by vector
equations (
F (n; m) = dddnandddama 2+ dddnbndddmbb +2 ddLn dddmLL ; 2
G (n; m) =
dm
+ dm
+ dm
A regular piece of the surface B with the rst fundamental form is a
twodimensional Riemannian space referred to the coordinates (n, m). If we consider
a surface as a Riemannian space, then vectors, tensors, scalar products, and
covariant di erentiation can be de ned on it [24]. The three-index Christo el
symbols have the following form for the surface B :
(
We can carry out similar calculations for the Red and Green colors that are corresponding binaries of Magenta-Yellow (M-Y) and Yellow-Cyan (Y-C), respectively.
For convenience of the following writing, we replace the notation. Instead of
the variables m, n (
The following order of mutual dependence of variables is accepted: the
variable placed earlier in abbreviation CMY will be considered independent.
Therefore, the variable C is always independent, and Y is always dependent. M
(magenta) is considered independent in the calculation of the Red geodesic, and acts
as dependent when calculating the Blue one. Dependences (
After obtaining analytical dependencies (
C =
C [9:2] + C[9:3] 2 ; M =
M [9:1] + M [9:3] 2 ; Y =
Y [9:1] + Y [9:2]
2
:
(
For the experiment, we use the 4-color (CMYK) wide-format solvent ink-jet printer Mimaki CJV30-160BS. Print mode: 720 720 dpi, small dot. Substrate: coated paper FancyEmboss 110 g/m2 as an absorbent substrate. The measurement tools: spectrophotometer x-Rite iOne iSis + x-Rite Pro leMaker package.
A halftone scale contains 14800 elds. The scale represents the multidimensional grid of test values. The total number of patches is 11 (0, 0.1, 0.2. . . 1) values per channel to the power of the number of colorants (four). They are synthesized using the Argyll CMS and TestChartGenerator in Pro leMaker for the automatic iOne iSis spectrofotometer. Such a frequent grid allow one to distinguish all possible surfaces within the device's color space. We utilized surfaces of binaries only, i.e., it formed 121 patch per each surface. Thus, 330 patches were actually used. Matrix variables, which contained n, m, L, a, and b data of each color patch as columns, are imported in MatLab, where they are carried out in further mathematical processing.
The proposed method of gradation surfaces is based on the interpolation of
experimental data by polynomials (
Results of geodesics calculations by (
The maximum possible value of the neutral color recipe (without the involvement of K-black) is determined by the component that has the highest brightness of the full tone (red). The errors in terms of deviations from the linear trends we associate with the rounding error, because it is impossible to put the formulation into the source code with a precision of better than a few percent. The developed and printed test chart for grey balance evaluation is shown in Fig. 3b.
Figure 4 shows the lines of red, green, and blue tones in the CIE Lab space and a line of synthesized neutral colors.
Analysis of the nature of deviation of chromaticity from the neutral tone indicates the accumulation of a systematic error. This is probably due to inadequate accuracy of the condition of equal lightness when the tone of red, green, and blue increases. Nevertheless, as it can be seen from Fig.5, the line of neutral colors is close to the vertical.
For convenience, we show the projections of points of the neutral tone on the chromaticity plane, which are grouped near zero, with the exception of the latter. A visual comparison of synthesized neutral elds with blacks matched in brightness (K) has shown an interesting phenomenon. It is impossible to determine, which eld is composite, and what is black, in spite of some di erence in the shades.
A new three-dimensional interpretation of the grey balance and application of gradation curves in 3D CIE Lab-space for the grey balance evaluation are proposed for up-to-date digital image processing. Gradation trajectories in terms of gradation surfaces, as well as the method of their analytical speci cation, are described. Gradational trajectories are conceived as continuous and bounded on the gradation range curves where the gradation surfaces are stretched on.
Gradation trajectories introduced by the described manner are the global features of ink-jet image processing that are depended only on type of a substrate and properties of the ink. They are not a ected by rasterizing method, number of passes, and measuring technique.
Grey balance evaluation with the help of the gradation trajectories of inks binaries (R, G, B) instead of inks themselves (C, M, Y) might be utilized as a powerful and fast-acting tool for printing systems characterization.
Further development of the approach implies introduction of 3D gradation surfaces as a method describing interconnection between two or even three colorants, especially in the case of regular and light inks pairs and not only for ink-jet, but, also, for all kinds of print.