## 2 Materials and Methods  
### The MDCIM  
In order to apply the MDCIM [10], color measurements on physical samples must be taken [19]. These require a spectrophotometer, resulting in measurements of the reflectance spectral distribution, from which the tristimulus values \(X\), \(Y\), and \(Z\) can be derived. To calculate the digital \(R\), \(G\), and \(B\) values for the red, green, and blue color channels in an image that best represents the color of a physical sample, the tristimulus values are converted to normalized luminance values (each scaled between 0 and 1) for each of the three channels. The standard device-independent color space, sRGB, is based on the Gain-Offset-Gamma model (GOG model) by [PERSON] et al. [20; 21]. This model linearly relates tristimulus values \(X\), \(Y\), and \(Z\) to luminance values \(Y_{R}\), \(Y_{G}\), and \(Y_{B}\) of the red, green, and blue channels of a display:\[ \begin{pmatrix}\chi\\ Y\\ Z\end{pmatrix} =M \begin{pmatrix}Y_{R}\\ Y_{G}\\ Y_{B}\end{pmatrix}  \tag{1}\]  
The GOG model assumes a primarily exponential relationship between these normalized luminances and the digital values of each channel. In the MDCIM, an initial linear section is also considered for low digital values:  
\[Y_{i}= \begin{cases}\beta\cdot d_{i}&\text{if }d_{i}<d_{0}\\ (k_{1,i}\cdot d_{i}~{}+~{}k_{2,i})^{\gamma_{R}}&\text{if }d_{i}\geq d_{0}\end{cases}  \tag{2}\]  
These three functions are called Tone Rendering Curves (TRCs) or electro-optical transfer functions. For displays with \(3\times 8\)-bit color representation, the digital values \(d_{R}\), \(d_{G}\), and \(d_{B}\) are equal to the \(R\), \(G\), and \(B\) values that range from 0 to 255, divided by \(2^{8}-1=255\). The parameter \(\gamma_{i}\) represents the exponent of each curve, while \(k_{1,i}\) and \(k_{2,i}\) refer to the gain and offset of the curve, respectively. Additionally, \(\beta_{i}\) is the slope of the linear segment, which applies to lower luminance values at the threshold \(d_{0,i}\). In sRGB space, it is assumed that the matrix \(M\) in Equation (1) is given by the following expression [22]:  
\[M_{sRGB}~{}=~{} \begin{pmatrix}0.4124&0.3576&0.1805\\ 0.2126&0.7152&0.0722\\ 0.0193&0.1192&0.9505\end{pmatrix}  \tag{3}\]  
For sRGB, it is also assumed that for all three color channels, the parameters gamma \(\gamma=2.4\), gain \(k_{1}=1/1.055\), offset \(k_{2}=0.055/1.055\), linear coefficient \(\beta=1\) / 12.92 and threshold \(d_{0}=0.0392\). The threshold parameter \(d_{0}\) and linear coefficient \(\beta\) can be obtained from purely mathematical and pragmatic arguments [10; 12]. Furthermore, the elements of matrix \(M\) as appearing in Equation (1) are related to the values of the CIE 1931 chromaticity coordinates \((x,y)\) and maximum luminance \(Y_{\text{max}}\) of the 3 channels [23; 24]:  
\[M~{}=~{} \begin{pmatrix}\frac{x_{R}}{y_{R}}Y_{R,max}&\frac{x_{G}}{y_{G}}Y_{G, max}&\frac{x_{R}}{y_{R}}Y_{B,max}\\ Y_{R,max}&Y_{G,max}&Y_{B,max}\\ \frac{x_{R}}{y_{R}}Y_{R,max}&\frac{x_{G}}{y_{G}}Y_{G,max}&\frac{x_{B}}{y_{B}}Y _{B,max}\end{pmatrix}  \tag{4}\]  
Therefore, in Equation (3) the values of the elements for the matrix \(M\) for the sRGB color space correspond to particular values for display specifications. These are shown in Table 1 and agree with the values for high-definition television as defined already in 1991 by the Comite Consultatif International pour la Radio (now replaced by the International Telecommunication Union) in recommendation BT.709 [25]. It should be acknowledged that in the display specifications, as tacitly assumed when using sRGB space, the total display luminance (\(Y=Y_{R,max}+Y_{G,max}+Y_{B,max}\)) when displaying a white image is scaled to 100 according to the values shown in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline
**sRGB** & **Red** & **Green** & **Blue** \\ \hline \(x\) & 0.64 & 0.30 & 0.15 \\ \hline \(y\) & 0.33 & 0.60 & 0.06 \\ \hline \(Y_{max}\) & 21.26 & 71.52 & 7.22 \\ \hline \end{tabular}
\end{table}