# Random Forest-Based Prediction of the Optimal Solid Ink Density in Offset Lithography
## 2 Theory
### Methods for Testing the Quality of Printed Matter
#### 2.1.2 Colorimetry Detection  
The color perception of an object is determined by the stimulation from an external light source and the visual properties of the human eye. To calibrate the concept of color, the CIE (International Commission on Illumination), by comparing the results of two visual experiments--[PERSON] and Gilder--and after resin conversion, specified a set of triadic values required to match the corresponding spectral colors, the \"CIE 1931 Standard Color Observer Spectral Triple Stimulus Values for X, Y, and Z\" [16]. The standard equation for the triple stimulus value is:  
\[\left\{ \begin{array}{l}X=K\int_{\lambda}\varphi(\lambda)^{\overline{x}}( \lambda)d\lambda\\ Y=K\int_{\lambda}\varphi(\lambda)\overline{y}(\lambda)d\lambda\\ Z=K\int_{\lambda}\varphi(\lambda)\overline{z}(\lambda)d\lambda\end{array} \right. \tag{4}\]  
where each colorimetric value is multiplied by the color stimulus function by the CIE spectral triple stimulus value, respectively, and then this product is integrated by bounding it by the entire spectral range. The value of color \(\varphi\) (relative spectral power) is different for light sources or objects. For example, for an ink with irradiance factor \(\beta\), the tri-stimulus value is calculated by the formula:  
\[\left\{ \begin{array}{l}X=K\int_{\lambda}\rho(\lambda)S(\lambda)\overline{x} (\lambda)d\lambda\\ Y=K\int_{\lambda}\rho(\lambda)S(\lambda)\overline{y}(\lambda)d\lambda\\ Z=K\int_{\lambda}\rho(\lambda)S(\lambda)\overline{z}(\lambda)d\lambda\end{array} \right. \tag{5}\]Approximate the integral by summing:  
\[\left\{ \begin{array}{l}X=K\underset{\lambda}{\sum}\varphi(\lambda)^{\overline{x} }(\lambda)d\lambda\\ Y=K\underset{\lambda}{\sum}\varphi(\lambda)\overline{y}(\lambda)d\lambda\\ Z=K\underset{\lambda}{\sum}\varphi(\lambda)\overline{z}(\lambda)d\lambda \end{array} \right. \tag{6}\]  
\(K\) in the above formula is called the adjustment factor, the meaning of which is calculated when the Y value of the light source or object is increased to 100:  
\[\mathrm{K}=\frac{100}{\underset{\lambda}{\sum}S(\lambda)\overline{y}(\lambda)d\lambda} \tag{7}\]  
Although the CIE 1931 standard colorimetric system effectively addresses the issue of color representation, when calculating color differences, the calculated values for different color regions (spaces) can significantly differ from the human visual system [17]. Therefore, to achieve color control and ensure that changes in the values of different color regions align with the human eye's perception, a special color space--CIELAB uniform color space--was developed, based on which the color difference formula was established [18]. The CIE 1931 standard colorimetric system undergoes a nonlinear transformation, converting the X, Y, and Z Cartesian coordinates to cylindrical polar coordinates, with the colorimetric values associated with the human visual system mapped to L*a*b* [19], as follows:  
\[\mathrm{luminance}\ L^{*}=116\sqrt[3]{\frac{Y}{Y_{n}}}-16 \tag{8}\]  
\[\mathrm{saturation}\ (\mathrm{color}\ \mathrm{theory})\left\{ \begin{array}{l}a^{*}=500 0\left(\sqrt[3]{\frac{X}{X_{n}}}-\sqrt[3]{\frac{Y}{Y_{n}}}\right)\\ b^{*}=200\left(\sqrt[3]{\frac{Y}{Y_{n}}}-\sqrt[3]{\frac{Z}{Z_{n}}}\right) \end{array} \right. \tag{9}\]  
In comparison to the density detection method, the chromaticity detection method directly provides numerical results that align with the visual perception of the human eye and more accurately reflect the ink's color performance. However, its abstract color coordinates and other data are not suitable for guiding and controlling industrial printing quality. Moreover, to date, there is no comprehensive theoretical framework for colorimetric detection, which limits its industrial application. In industrial production, the use of different printing papers or inks can lead to significant differences in color performance, even with identical physical parameters. Chromaticity detection and control methods can enable printed materials on different substrates to achieve nearly identical color quality.