# Optimizing the Spectral Characterisation of a CMYK Primer with Embedded CMY Printer Modelling
## 1 Introduction  
Printer characterisation aims to reveal the relationship between the input and the output of the halftone printing process, and thus generates the optimal ink combination for a target colour [1; 2; 3; 4; 5; 6]. The first half of printer characterisation relates to a forward modelling in which a colour prediction model [7; 8; 9; 10] is set up to predict the printed colours from the device control values (i.e., the dot areas). The second half of printer characterisation, which is termed as a backward process, aims at to invert the forward model by certain optimization algorithms and decomposes the target colour into control values of individual inks such as cyan, magenta, yellow and black (CMYK). Therefore, in several studies such a model is also named as a colour separation model [11; 12; 13; 14]. In addition, since the technical progress in digital imaging has activated the concept of spectral colour reproduction for the reduction of metamerism [15; 16; 17; 18], the studies of printer characterisation have also been extended to the spectral domain [19; 20; 21]. Many mathematical models have been adopted for forward modelling, which includes a least-squares based polynomial function [22], the Kubelka-Munk model [23], the artificial neural network (ANN) [24], the spectral Neugebauer model [25], etc. Among these models, the cellular Yule-Nielsen spectral Neugebauer (CYNSN) model, together with its variants, is perhaps the mostpreeminent model due to its intuition and precision [19; 20; 26; 27; 28]. Such models calculate the printed reflectance spectrum with a weighted sum of participating Neugebauer-primary spectra and take several influential effects into consideration, such as mechanical and optical dot gain [7]. Moreover, its cellular structure provides many more sub-models and primaries and thus remarkably improves the prediction accuracy. Mathematically speaking, backward modelling is the inverted implementation of forward models and thus the forward accuracy directly impacts the backward accuracy. That is, one cannot set up a sound separation model unless this prediction model is precise. Until now, several optimization algorithms have been proposed for backward modelling which unexceptionally provide acceptable colour separation performance [11; 13; 14]. Meanwhile, as for the backward modelling for CYNSN models, specific algorithms are needed to target the optimal subdomain (namely, the optimal cell), as the colour separation process eventually takes place in a defined cellular region [19; 21; 27]. At present, the high accuracy of the CYNSN-based spectral printer characterisation could be easily achieved by increasing the sampling nodes in the cellular structure. For instance, based on several 7-grid-points CYNSN models, [PERSON] et al. set up a multi-ink printer characterisation workflow and achieved excellent forward and backward modelling accuracy [27; 28]. However, in that contribution more than 10,000 colour samples were printed and measured, which is definitely impractical for daily applications. Therefore, several researchers have focused on reducing the amount of modelling samples while maintaining an acceptable precision by using some nonuniformly sampling approaches [29; 30]. Unfortunately, it is not easy to apply those nonuniform sampling methods into CYNSN models due to their cellular structures, which actually raise challenges for related studies. According to current studies, to achieve the final colour reproduction accuracy of \(\
eg 1\) CIE DE2000 unit [31], at least 1000 training samples are needed [19; 27; 28]. In our recent work, an effective and efficient modelling workflow was proposed which systematically optimised several crucial steps in spectral printer characterisation [19; 32; 33]. For forward modelling, a backward propagation artificial neural network (BPANN) modified CYNSN model was proposed in order to simulate the nonlinearity between different ink mixtures and the optical dot gain.