# Optimizing the Spectral Characterisation of a CMYK Primer with Embedded CMY Printer Modelling
## 2 Methodology
### Embedded CMY Printer Modelling  
The separation approaches described above are all based on the backward modelling accuracy (quantified by the difference between target colour and model predicted colour). However, it must be mentioned that such accuracy is actually not the final colour reproduction accuracy (quantified by the difference between target colour and actually printed colour), but only refers to the smallest error between the target and the predicted colour according to a forward model, as shown in Equation (8). Therefore, if the accuracy of the forward model is problematic, it is quite possible that the optimal control values cannot be accurately targeted and thus the final colour reproduction will not be so precise.  
For instance, when setting up the CYNSN model (or its variants) for a CMYK printer, a certain number of colour samples are generated in CMYK colour space by uniform sampling (the cellular nodes) and random sampling (the training samples). However, such sampling in CMYK space actually could not lead to uniformly distributed colours in a device independent colour space, such as CIELAB [41], due to the strong light absorption of the black ink. Meanwhile, the overprint of cyan, magenta and yellow may also result in a dark grey colour. Thus, it is obvious that the sampling of a CYNSN model indeed overemphasises the colour sampling in dark tones while relatively neglecting the sampling in light tones. Therefore, it is quite possible that the accuracy of the CYNSN model for the light tones is not as robust as that for the dark tones.  
Basing on this consideration, the proposed optimization was developed, as illustrated in Figure 1. The core of our proposed method is an embedded CMY printer modelling, which is simple but effective. That is, a cellular spectral Neugebauer (CSN) model for the embedded CMY colour space was built up, using the existing colour samples whose control values for the black channel were zero.  
As we believed, such a CMY-CSN model could also provide better forward modelling performance in the light tones. As no extra training samples were generated with the aim of maintaining modelling efficiency, the Yule-Nielsen \(n\) value for this model was uniformly set to 1. Afterwards, two lightness thresholds were empirically defined by which the printer gamut was divided into light tones, middle tones and dark tones. Such thresholds corresponded to the lightness values in CIELAB space (D50/2) of the colour samples with the control values of 0.3 and 0.7 in the cyan, magenta and yellow channel. For instance, for a CMYK printer and a given substrate, if the lightness of the target colour is higher than the lightness of (c = 0.3, m = 0.3, y = 0.3, k = 0), then it locates in the light-tone region; if the lightness of the target colour is lower than the lightness of (c = 0.7, m = 0.7, y = 0.7, k = 0), then it is a dark-tone colour; otherwise, that colour is considered as a middle-tone colour. In the work of [PERSON] et al., similar division approach was adopted [24].  
In this method, the colour separation protocol is modified as follows. When a target spectral colour is to be reproduced, we first decide which tone it belongs to according to its CIELAB lightness value. If it is a light-tone colour, then the CMY-CSN model is inverted to calculate the device control values; if it is a dark tone colour, then the BPn-CYNSN model for the CMYK printer was inverted. Meanwhile, as for a mid-tone colour, both the CMY model and CMYK model are inverted, and the device values with a better backward modelling accuracy will be finally selected. Such an implementation for the middle tone is intended to avoid the possible colour shift around the threshold regions.