# Optimizing the Spectral Characterisation of a CMYK Primer with Embedded CMY Printer Modelling
## 2 Methodology
### CYNSN-Based Forward Modelling  
In general, the plain Yule-Nielsen modified spectral Neugebauer (YNSN) model could be stated as follows,  
\[R(\lambda)=\left(\sum_{i}a_{eff,i}R_{i}(\lambda)^{\frac{1}{n}}\right)^{n}, \tag{5}\]  
where \(R(\lambda)\) is the predicted spectral reflectance at wavelength \(\lambda\), \(R_{i}(\lambda)\) denotes the spectral reflectance of the \(i\)th Neugebauer primary, \(n\) is an exponent proposed by [PERSON] and [PERSON] with the aim of accounting for optical dot gain in the halftone printing process [36; 37] and and \(a_{eff}\) represents the effective dot area of each Neugebauer primary, which is defined by the Demichel equation as shown in Equation (2). Note that when calculating \(a_{eff}\), the original device control values (i.e., the dot areas), \(c\), \(m\) and \(y\) are replaced with the effective dot area \(c_{eff}\), \(m_{eff}\) and \(y_{eff}\) respectively so as to compensate for the mechanical dot gain caused by the ink spreading effect [7]. Taking the cyan colorant as an example, the effective dot area \(c_{eff}\) is computed by a least-square method, as shown in Equation (6):  
\[c_{eff}=\Big{(}R(\lambda)-R_{p}(\lambda)\Big{)}/\Big{(}R_{s}(\lambda)-R_{p}( \lambda)\Big{)}, \tag{6}\]  
in which \(R(\lambda)\), \(R_{p}(\lambda)\) and \(R_{s}(\lambda)\) denote the spectral reflectance of the colour sample, the substrate (with a control value of 0), and the solid (with a control value of 1), respectively. As there are only two control values (0 and 1 in CMYK space) for the YNSN model, for a four-ink printing system only \(2\times 2\times 2\times 2=16\) Neugebauer primaries are adopted in Equation (5) to predict the printed spectrum, which indeed limits the model's performance. Therefore, to further increase the spectral prediction accuracy, the cellular Yule-Nielsen modified spectral Neugebauer (CYNSN) was derived [38]. According to such a model, the device colour space is uniformly divided into several subdomains (or cells) so that more Neugebauer primaries are generated. For instance, for a five-grid point CYNSN model, there are five sampling nodes (0, 0.25, 0.5, 0.75, and 1) in each ink channel, which results in \(5\times 5\times 5\times 5=625\) colour samples. The calculating protocol of the CYNSN model is quite similar to that of YNSN model, with the only exception that the original control values should firstly be scaled into a certain subdomain. For instance, for the cyan ink, the scaled control value \(c_{cell}\) should be computed by Equation (7), where \(c\) represent the original control value while \(c_{u}\) and \(c_{l}\) denote the upper and lower limit of the subdomain, respectively. \[c_{cell}=(c-c_{l})/(c_{u}-c_{l}), \tag{7}\]  
The above-mentioned implementations constitute the main workflow for setting up the CYNSN model. However, as such a model is based on several theoretical assumptions (e.g., it is assumed that the ink spreading and light scattering among different ink mixtures is always consistent), which, in fact, are not perfectly true, it still suffers from several limitations when spectrally characterising a printer. Therefore, to further optimise the performance of the CYNSN model, much effort has been made [19; 20; 26]. In 2010, [PERSON] et al. proposed the ink spreading CYNSN model (shortened as IS-CYNSN [26]) which can be considered as an effective modification of the plain CYNSN model. The performance of such a model is actually better, as it not only considers the ink spreading effect for an ink directly printed on the substrate, but also accounts for the condition of such an ink being in superposition with other solid inks. Meanwhile, it should be noticed that to build this IS-CYNSN model, more samples are needed for updating the effective dot area curves. In a recent work, [PERSON] et al. revisited the optimization of the CYNSN model [20].