## 4 Mathematical Models of Dyeing as a Basis for Programming  
Experimental measurements are the basis for the development of regression equations of the connection between the visual and near-infrared spectra. The color difference includes measured values from the colorimetry range respecting the parameters: L*a*b* and the value of \(\Delta\)E (that is less than 3) [15]. The iterative process of printing and measuring the value of L*a*b* is performed on the same material on which the final sample is printed. Twin dye recipes for ink-jet printing are proposed. The reproduction of clothing uses its own printing nks.  
Our IRD method is the fusion of two images [16] in the CMYK printing system. The first image (X\({}_{0}\), K = 0%) is the one intended for the visible spectrum, while the second image (X\({}_{40}\), K = 40%) is detectable only with an NIR-Z camera. Values C\({}_{0}\), M\({}_{0}\), and Y\({}_{0}\) are the initial values of coverage for every individual pixel in the X\({}_{0}\) image (V). These values decrease depending on the quantity of K in image Z. The programming has a starting point (X\({}_{0}\)) and the calculated point at 40% coverage of the carbon black toner in image Z. The remaining values between X\({}_{0}\) and X\({}_{40}\) are proportionally interpolated. The measurements of twin dyes (column X\({}_{40}\)) are interpreted (regression analysis) as analytical solutions for calculating other color tones in the visible image V.  
The programming of dye mixtures is planned in two ways. The first plan is tabulated, taking the values of the mixtures. Only the offered colors, checked for the minimum value of \(\Delta\)E, are used in this algorithm. A program is previously developed that tests the invisibility of the touch of the twin dyes with 20%, 40% and 0% content of the black component.  
The second plan is an algorithm that uses regression Equation (1). This algorithm allows the use of all color shades. The regression model (1) is derived from the values obtained experimentally, which implies a limited number of calculated twin dyes.  
A model with three independent variables is presented here. Each dye formulation depends on all three values of the initial (X\({}_{0}\)) components. The parameters in the regression Equation (1) of the twin dyes recipe are valid for ink-jet printing on a plotter and with inks for HP Latex 330 printing on white canvas. The twin dyes' regression formulations (1) refer to 40% black ink coverage. In this paper, a relation used for programming CMYKIR separation is proposed.  
\[\begin{array}{l}\text{C}_{40}=-0.0441\times\text{Y}_{0}-0.1988\times\text{ M}_{0}+1.1142\times\text{C}_{0}-19.03\\ \text{M}_{40}=-0.2313\times\text{Y}_{0}+1.0540\times\text{M}_{0}-0.3078\times \text{C}_{0}-7.44\\ \text{Y}_{40}=1.1414\times\text{Y}_{0}-0.1349\times\text{M}_{0}-0.1169\times \text{C}_{0}-32.05\end{array} \tag{1}\]  
Relations determine the quantity of dyes C\({}_{40}\), M\({}_{40}\) and Y\({}_{40}\), which manifest in the NIR and V spectrum, for the same color that manifests only in the visible spectrum C\({}_{0}\), M\({}_{0}\), and Y\({}_{0}\). The interdependence measurement X\({}_{40}\), X\({}_{20}\) and X\({}_{0}\), (Table 1) are continuously supplemented for different materials and printing techniques. Our research focuses on experimental printing on canvas and making a sample at the \"prototype\" level.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Measured Data: C, M, Y, for X\({}_{0}\), X\({}_{20}\), X\({}_{40}\)**} & \multicolumn{2}{c}{**L*a*b*} & \multicolumn{2}{c}{**Color**} \\  & **K = 0\%** & **K = 20\%** & **K = 40\%** & & \\ \hline  & 89, 65, 45 & 78, 46, 25 & 65, 23, 0 & 31, \(-\)4, \(-\)22 & 1 \\  & 73, 40, 42 & 64, 23, 24 & 53, 3, 2 & 49, \(-\)14, \(-\)10 & 2 \\  & 48, 69, 51 & 35, 55, 33 & 18, 39, 11 & 44, 26, 5 & 3 \\  & 40,