2. APPROACH and experiment

Notions of gradation trajectories and gradation surface remain the same. The gradation trajectory is a geometric place of the points in the CIELab space, which coordinates correspond to the CIE Labcoordinates of the individual patches of the initial colorants (C, M, Y, K). We assume that the tone part in the layout is denoted as a tparameter in the range from zero (paper) to one (full dye). Therefore, the trajectories might be analytically described by polynomials ofn th degree (1), since color changes continuously in accordance with continuous increment of t . The gradation surface of double colorants overlapping is a surface “stretched” on two gradation trajectories of the initial colorants. Extending the concept of gradation trajectories on the two-dimensional case, we define the gradation surface as a locus of points in the CIELab space where coordinates correspond to the CIE Labcoordinates of the individual patches of the binary halftone scale from zero (unprinted substrate) to two (full dye of 2 colorants). By analogy with the gradation trajectory, the gradation surface is also expressed by a polynomial of n th degree (2).
\(\left\{\par \begin{matrix}t\in\left[0;1\right],\\ a=a_{4}t^{4}+a_{3}t^{3}+a_{3}t^{3}+a_{1}t+a_{0},\\ \par \begin{matrix}b=b_{4}t^{4}+b_{3}t^{3}+b_{3}t^{3}+b_{1}t+b_{0},\\ L=\left(L_{0}-L_{\infty}\right)\times\exp\left({-L}_{3}t^{3}-L_{3}t^{3}-L_{1}t\right)+L_{\infty},\\ \end{matrix}\\ \end{matrix}\right.\ \) ()
\(\left\{\par \begin{matrix}\overline{m,\ n}\in\left[0;1\right],\\ a\left(m,n\right)=a_{0}+\sum_{i=1}^{4}{\sum_{j=0}^{i}a_{i-j,j}}\bullet m^{i-j}\bullet n^{j},\\ \par \begin{matrix}b\left(m,n\right)=b_{0}+\sum_{i=1}^{4}{\sum_{j=0}^{i}b_{i-j,j}}\bullet m^{i-j}\bullet n^{j},\\ L\left(m,n\right)=\left(L_{0}-L_{\infty}\right)\exp\left(-\sum_{i=1}^{3}{\sum_{j=0}^{i}L_{i-j,j}}\bullet m^{i-j}\bullet n^{j}\right)+L_{\infty},\\ \end{matrix}\\ \end{matrix}\right.\ \) ()
where \(L_{i},\ a_{i},b_{i}\) are some numerical coefficients,\(L_{0},\ a_{0},b_{0}\ \)are the CIE Lab coordinates of the unprinted substrate,\(\ L_{\infty}\) is the visual brightness of a hypothetical continuous ink layer of infinite thickness, t is the receipt of the tone in the layout, (m , n ) is the receipt of the binary overlap (in parts, percent or color “quanta”).
Color quantization in digital printing system with the color depth 8 bit per color channel is accounted as a virtual grid of 28=256 lines per 1 channel and 216=65536 possible halftones of a chosen pair of colorants. The halftone recipes achievable within a given printing system are in the virtual grid nodes. Each reproducible tone has its own recipe (m , n ) and a set of the CIE Lab coordinates.
Introducing the color quanta based on the above reasoning, we have to start from the fact that the tone fraction must take values from a particular discrete series. Thus, to account the color quantization in a real print system, the real tone fraction values should be transformed according to the formula
\(t_{\text{fact}}=\frac{uint8\ (t_{\text{des}}\times 255)}{255}\), ()
where \(t_{\text{des}}\) is the desired tone fraction,\(t_{\text{fact}}\) is the actually realizable tone fraction,uint8 is the Matlab function for rounding to an integer in the range 0–255 (8-bit integer). Thus, each patch recipe component (mn ) is rounded to the 8-bit integer that forms a grid of the color quanta replacing the initial colorant percentage.
We showed that the best trajectory of a binary surface is the geodesic on a gradation surface between (0, 0) and (1, 1) points. Hence, we will find the gradation trajectory of an ideal colorant assuming that, first, it exists and is unique, second, it is a geodesic of two adjacent colorants. The criterion for determining such a trajectory is the minimum color difference dE 2000 between each successive pair of the color quanta.
For the experiment, we have selected the electrophotographic Konica-Minolta Bizhub Pro C6000L as a printing system with the print resolution 1200×1200 dpi and the RIP EFI Fiery IC-306. Paper substrate: Moorim Neo Star Matt coated, 140 g/m2. The test gradation scales were synthesized using the ChartGenerator in theMeasureTool in the ProfileMaker package for the X-Rite i1 iSis automatic spectrophotometer. The test chart contained 2115 gradation patches that correspond to the 15-quanta tone increment for each colorant forming a binary surface. The experiment algorithm is shown in Fig. 2.
Gradation scales with an approximately 5% increment of the tone in the layout from 0 to 200% were implemented. Since the tone fraction must take values from the discrete series [0; 0.05; 0.1 … 0.95; 1], the fraction values should be transformed according to the discretization formula (3). The numerator of formula (3) is the expression of color tone in each color quanta. The expression of the color tone in quanta is initially formed in terms of the bit depth of the printing system and is preferable to use. Whichever way of specifying a color is chosen, formula (3) allows establishing an unambiguous relationship between them.