# Optimizing the Spectral Characterisation of a CMYK Primer with Embedded CMY Printer Modelling
## 3 Experiments  
In the experiment, a Canon IPF 5100 printer equipped with 12 ink sets was employed. Since only the CMYK links were used, the other ink channels were switched off by the Onyx Production House 10 software, which also enabled us to control the CMYK channels independently. During the test, the printing resolution was set to \(2400\times 1200\) dpi and a stochastic screening method was adopted for the halftone process. To test the robustness of the proposed approach, three different kinds of substrate  
Figure 1: Flow chart of the proposed optimization approach. were used, including Black Diamond 220 g canvas paper, Canon 170 g glossy photo paper and Pinnacle 245 g water colour paper. An X-Rite i1 iSis automated spectrophotometer was employed to obtain the spectral reflectances (from 380 nm to 730 nm in 10 nm intervals) of the printed colours. In the ink limitation stage, the above mentioned multilinear interpolation method [34] proposed by [PERSON] et al. was adopted. As for each substrate, sixteen 21-level uniformly sampled Neugebauer primary ramps were printed and the maximum total ink amount \(\Phi_{total-max}\) for each ramp, as defined in Equation (1), was determined by visual judgment. Note that we only implemented such ink limitation before sending the device control values to the printer. As for forward and backward modelling, the corresponding original control values ranging from 0 to 1 were used, which is more straightforward and convenient. To build the forward models, 1111 colour samples were printed and measured, which included four uniformly sampled single-ink ramps for obtaining the effective dot gain curves described in Equation (6) (11 levels each, 44 samples in total), the sampling nodes corresponding to a five-grid point CYNSN model (\(5\times 5\times 5\times 5=625\) colour samples) as well as 442 randomly generated colour samples in CMYK space in order to train the model. Three models were built based on these samples for each substrate. For the plain CMYK-CYNSN model, the optimal \(n\) value was fitted by minimizing the average prediction error for the 442 training samples, while for the CMYK-BPn-CYNSN model, the 442 training samples were used to fit 442 optimal \(n\) values corresponding to each ink combination and thus for training a neural network [19]. The embedded CMY-CSN model was set up based on the existing colour samples whose control values for the black channel were zero (i.e., 3 single-ink ramps for the effective dot gain curves and the sampling nodes corresponding to a five-grid point cellular model with \(5\times 5\times 5=125\) colour samples). As noted above, the Yule-Nielsen \(n\) value for this model was uniformly set to 1 since no extra samples were generated to fit the optimal \(n\) value for this model. By this setting, we intended to demonstrate that the proposed approach could perform better while maintaining the modelling efficiency. To test the performance of the forward models, for each substrate 100 colour samples were randomly generated in CMY and CMYK spaces. After printing, the forward modelling accuracy could be obtained by calculating the colour and spectral differences between the predicted reflectances and the printed reflectances. For backward modelling, 100 in-gamut colours were printed and considered as the targets. The same approach proposed in our previous study [19] was employed to implement the spectral separation, which used a sequential gamut judging method to target the optimal cell and a sequential quadratic programming algorithm [42] to invert the forward model in the subdomain. Afterwards, the final reproduction accuracy was obtained by computing the colour and spectral difference between the printed and target reflectances. Note that no out-gamut colours were adopted, as for such colours the true characterisation accuracy would be masked by the errors introduced by gamut mapping. In addition, to avoid the above-mentioned oversampling problem in dark tones, a moderate modification was implemented.