# Optimizing the Spectral Characterisation of a CMYK Primer with Embedded CMY Printer Modelling
## 2 Methodology
### Ink Limitation  
For different ink combinations, there are always limitations for the maximum total ink amount. When such thresholds are exceeded, the printed artefacts such as ink bleeding and bronzing occur,which significantly deteriorate the print quality. To avoid this, printer manufacturers and colour management software suppliers always provide an embedded ink limitation module in their products, by which users can print some ink limitation ramps and set the ink limits themselves. As for researchers, they tend to develop their own ink limitation algorithms so that the inks can be controlled more thoroughly [11; 33; 34]. Among current algorithms, the multilinear interpolation method [34] proposed by [PERSON] et al. is typical and it is highly related to CYNSN models. According to this approach, the ink limitation process \(T_{\text{ink-limitation}}\) could be depicted as Equation (1), in which \(\Omega_{dee}\)ice and \(\Omega_{\text{ink-limitation}}\) respectively, denote the original device-value space and the printable region for a printer equipped with \(n\) inks, \(\Phi\) represents the device control values after ink limitation, \(\Phi_{\text{total-max}}\) denotes the maximum total ink amount, \(a(i)\) is the dot area of the \(i\)th Neugebauer primary defined by the Demichel equation [35], \(g(i)\) defines the control values of the Neugebauer primaries.  
\[T_{\text{ink-limitation}}=\left\{\begin{array}{l}\Omega_{dee \text{i}\text{e}\text{i}\text{e}\text{-}\Omega_{\text{ink-limitation}}\\ \Phi=\sum\limits_{i=0}^{2^{n}-1}a(i)\cdot\frac{\min(\|\Phi_{\text{total-max}} \|g(i)\|_{\text{k}}\|)}{\|g(i)\|}\end{array}\right., \tag{1}\]  
For instance, for a CM printer whose device control values are respectively denoted as \(c\) and \(m\), the Demichel equation could be written as follows, where \(a(0)-a(3)\) represent the four Neugebauer primaries, respectively.  
\[\begin{array}{l}a(0)=(1-c)(1-m);\ a(1)=c(1-m);\\ a(2)=(1-c)m;\ a(3)=cm;\end{array}\right., \tag{2}\]  
In addition, in this case \(g(i)\) in Equation (1) could be defined as  
\[\begin{array}{l}g(0)=(0,0);\ g(1)=(1,0);\\ g(2)=(0,1);\ g(3)=(1,1);\end{array} \tag{3}\]  
For instance, assumed that the maximum total ink amount \(\Phi_{\text{total-max}}\) for the CM printer is (0.6, 0.6), then according to Equation (1) the output for an input value (c, m) = (0.8, 0.7) could be computed as  
\[\begin{array}{l}a(0)\!=0.06;\ \frac{\min(\|\Phi_{\text{total-max}}\|_{1},\|g(0)\|_{11})}{\|g(0)\|_{11}}g(0)\!=0\times(0,0);\\ a(1)\!=0.24;\ \frac{\min(\|\Phi_{\text{total-max}}\|_{1},\|g(1)\|_{11})}{\|g(1)\|_{11} }g(1)\!=1\times(0,1);\\ a(2)\!=0.14;\ \frac{\min(\|\Phi_{\text{total-max}}\|_{1},\|g(2)\|_{11})}{\|g(2)\|_{11} }g(2)\!=1\times(1,0);\\ a(3)\!=0.56;\frac{\min(\|\Phi_{\text{total-max}}\|_{1},\|g(3)\|_{11})}{\|g(3) \|_{11}}g(3)=\frac{12}{2}\times(1,1)=(0.6,0.6)\\ (c,m)_{\text{output}}=0.06\times(0,0)+0.24\times(0,1)+0.14\times(1,0)+0.56 \times(0.6,0.6)=(0.476,0.576)\end{array} \tag{4}\]  
Therefore, in this condition an input value of (c, m) = (0.8, 0.7) will be restricted to (c, m) = (0.476, 0.576) according to the ink limitation algorithm as well as the maximum total ink amount.