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 most preeminent 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, Wang 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 ¬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. With the help of such a modification, we could make the performance of a 5-grid point CYNSN model match that of a 6-grid-point CYNSN model, while reducing the modelling samples by more than one quarter. In the backward modelling process, a sequential gamut judging method was raised, which significantly accelerated the optimal cell searching process. Finally, the superiority of the entire workflow was demonstrated by comparing our methods with other typical methods [
19].
Basing on our previous workflow, in this paper we further optimised the spectral characterisation of a CMYK printer with the aid of an embedded CMY-printer modelling. Our key insight in this contribution is to improve the printer characterisation in light-tone regions which is limited by the cellular-sampling manner of CYNSN models together with the influence of the black ink. This paper is organised as follows.
Section 2 briefly reviews several key procedures of spectral printer characterisation and then introduces our proposed method.
Section 3 describes the experimental details while
Section 4 discusses the feasibility of our proposed methods based on the experimental results. Finally,
Section 5 summarizes the conclusions of the paper.
3. Experiments
In the experiment, a Canon IPF 5100 printer equipped with 12 ink sets was employed. Since only the CMYK inks 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 × 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 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 i1iSis 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 Urban et al. was adopted. As for each substrate, sixteen 21-level uniformly sampled Neugebauer primary ramps were printed and the maximum total ink amount
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 × 5 × 5 × 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 × 5 × 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. That is, we randomly generated 50 samples both in CMY and CMYK colour spaces and printed those 100 colours through an ICC workflow [
43] provided by the Onyx software. As that ICC workflow had its own mechanism for colour transformation, we were blind to the exact control values sent to the printer but could guarantee that the colours it generated indeed located inside the gamut. Additionally, as the majority of the colour samples in CMY space located in light and middle tones, such a setting made the sampling more uniform in CIELAB colour space.
4. Results and Discussion
Table 1 reveals the maximum total ink amount for the 16 Neugebauer Primary Ramps, as defined in Equation (1), where canvas, glossy and water colour, respectively, refer to the Black Diamond 220 g canvas paper, the Canon 170 g glossy photo paper and the Pinnacle 245 g water colour paper.
From this table, it is clear to see that the glossy paper could hold the most ink while the water colour paper held the least ink. Since the spectral reflectances of these three substrates do not vary significantly, the gamut of those substrates is mainly dependent on the maximum ink amounts. Therefore, the glossy paper exhibits the largest gamut while the water colour paper exhibits the smallest gamut.
Figure 2 shows the spectral and colorimetric accuracy of the three forward models when predicting the 100 testing samples as described in
Section 3. Note that for CMYK and CMY models, different groups of samples were adopted. Since the results regarding to these three substrates are quite consistent, only the case for the canvas papers is presented. The root mean square error (RMSE) [
39] and the CIEDE2000 colour difference [
31] under the D50/2 condition were adopted to denote the spectral and colorimetric accuracy respectively. As stated above, the light tone, middle tone and dark tone were defined by dividing the whole gamut with two thresholds, which refer to the lightness of (c = 0.3, m = 0.3, y = 0.3, k = 0) and (c = 0.7, m = 0.7, y = 0.7, k = 0). In this case, such two lightness values are 59 and 39 in CIELAB colour space (D50/2) respectively.
From
Figure 2, it is clear that our formerly proposed BPn-CYNSN model obviously outperformed the plain CYNSN model for the CMYK printer, which is consistent with our previous work [
19]. Besides, it can be concluded that those models always encountered relatively large errors for light-tone regions while exhibiting the best performance for dark tones, which indeed highlights our concern upon the accuracy for light tones.
For the light-tone colours, the embedded CMY-CSN model exhibited the best performance despite the fact that this model is of the lowest level among the three (i.e., without Yule–Nielsen
n value modification). As far as we are concerned, such a finding could be ascribed to the nonuniform sampling of those CYNSN models.
Figure 3 depicts the colour distribution of the training and testing samples for the canvas paper, where the ordinate denotes the percentage of different colour tones. As described above, 625 samples were printed for the CMYK printer, whereas 125 samples were printed for the embedded CMY printer when building the cellular structure of these models. In addition, there were 100 testing samples for the CMYK and CMY models respectively. From
Figure 3, it can be seen that the training and testing samples for the CMYK models mostly located in the dark tone regions due to the strong light absorption of the black ink. On the contrary, few light-tone colours were generated. As those models computed the printed reflectance by a weighted sum of those primary reflectances, such sampling overemphasised the dark tones while resulting in relatively large errors for the light tones. Similarly, as for the proposed embedded CMY printer modelling, as many more light-tone colours were generated, the prediction performance for this region should become significantly better.
In addition to this, the colour coordinates of the 625 samples for CMYK models as well as the 125 samples (a subset of the 625 samples) for CMY models were plotted in CIELAB colour space, as shown in
Figure 4. By that figure, the above statements are further validated. Meanwhile, another significant finding is that the colour gamut of the CMY printer in light and middle tones approximately matches that of the CMYK printer while for dark tones it is quite obvious that the gamut of CMY printer is much smaller, especially for the extreme dark tones.
Note that although the embedded CMY printer modelling emphasised the sampling in middle tones and de-emphasised the sampling in dark tones, the corresponding accuracy did not vary considerably when compared to the CMYK models. In our opinion, several possible reasons mutually contribute to this phenomenon, including the modelling protocols of these three models, the magnitude of the spectral reflectance of different tones, the nonlinearity between the spectral accuracy and colorimetric accuracy as well as the measurement error in the extremely dark regions [
19,
27]. Moreover, it should be mentioned that the CMY and CMYK models actually employed different sets of testing samples with different colour gamut for dark tones (very similar to the condition depicted in
Figure 4), therefore the performance comparison for dark region in this condition is actually not very meaningful; regardless, the aim of this study is to optimise the performance of CYNSN-based forward and backward models with black ink, so it is quite obvious that the colour gamut of a CMY printer is too small for the dark tone.
As mentioned above, 100 in-gamut colour samples were reproduced in order to test the performance of different separation methods.
Table 2 depicts the mean separation accuracy of the CMYK-BPn-CYNSN-based backward model and the CMY-CSN-based backward model for different colour tones on different substrates. As the superiority of CMYK-BPn-CYNSN-based spectral separation over that of CMYK-CYNSN model had been comprehensively proved by our previous work [
19], in this study the CMYK-CYNSN model was not inverted.
It is obvious in
Table 2 that the results for these three substrates are quite consistent. For light tones, the separation accuracy of the CMY-based model always outperformed that of the CMYK based model while for middle tone colours, the improvement was not so significant. Meanwhile, as for the performance of the CMYK based separation, the accuracy of the dark tones was always higher than that of the middle tones while the accuracy for the light tones was always the worst. Such findings actually correlate well with the results of the forward modelling, as shown in
Figure 2. Similarly, they could also be attributed to the nonuniform sampling of the CYNSN models as well as the strong light absorption of the black ink.
In addition, from
Table 2 it was found that the separation accuracy of CMY-based separation model exhibited much larger errors in dark tones. This could be explained by the fact that some of the colours in this region were so dark that they located outside the gamut of the embedded CMY printer, as indicated in
Figure 4. To further illustrate this statement, three examples are provided in
Figure 5, where the spectral colours from different colour tones were respectively reproduced by CMY-based separation and CMYK based separation. In the dark-tone case, the overprint of the CMY printer could not achieve a colour with extreme darkness. Fortunately, since the accuracy of CMYK based separation for the dark tones is already excellent and the proposed CMY modelling is only adopted in light and middle tones, there is no need to be concerned about this problem.
Besides, the ink limitation settings as mentioned in
Table 1 had strong influence on the accuracy of those models. From
Table 1 and
Table 2, it is obvious that the glossy paper, which holds the most black ink, always suffered from the largest errors. Such a finding, as mentioned above, should be attributed to the measuring error of the colour measurement device for extremely dark tones [
19,
27,
28]. Meanwhile, it was found that for the glossy paper, the errors of the CMY model in dark tones were relatively smaller than those of the other substrates. A possible explanation is that the moderate ink limitation of the CMY inks for this substrate to some extent enabled three such inks to synthesize the colour of the black ink.
Figure 6 further demonstrates why the proposed method adopted the embedded CMY model for the characterisation in light and middle tones while maintaining the CMYK model for the dark tones. In this figure, taking the canvas paper as an example, the relationship between the backward accuracy and the final separation accuracy for different colour tones are illustrated. It is quite clear that the testing colour samples are more uniform among different colour tones when compared to the case of
Figure 3, because we have modified the sampling for test colour samples as described in
Section 3.
As stated before, current separation methods always inverted the CYNSN models based on the backward accuracy, which merely refers to the smallest error between the target and the predicted colour according to the forward model. However, if the forward model is problematic, the corresponding separation may not target the optimal control values. In that condition, a failure in final colour reproduction will occur.
From
Figure 6, it can be concluded that other than the case of the CMY-CSN-based colour separation for dark tone colours, there is always some obvious difference between the values of the backward accuracy and the final accuracy. (For instance, the samples in the dashed ellipses in
Figure 6, which exhibit sound backward accuracy but relative lower final accuracy.) Such a finding indeed highlights the importance of the accuracy of the forward model. In other words, when the final colour separation accuracy is within a small error range (e.g., mean RMSE less than 0.02, mean DE2000 less than 2), the influence of the errors of the forward model should not be neglected when the highest precision is required.
To be specific, it is clear that for the light tones the embedded CMY-based separation tends to lead to better performance. According to the least-square-fitting lines, for the same backward accuracy the CMY-based separation may correspond to a higher final accuracy (lower RMSE or DE2000). Such results soundly validate the rationality of the proposed method.
For the middle tones, it can be concluded that the two models performed almost equally. In fact, from the viewpoint of mean values, the performance of the CMY-based separation only provided a very small improvement. Therefore, for the middle tones it is not quite essential to replace the CMYK separation with CMY separation. As mentioned above, our optimal device values-choosing strategy (i.e., to choose by the backward modelling accuracy) is only intended for avoiding the possible colour shift around the threshold regions. In fact, since the final errors for this condition are quite small, it is highly possible that human observers could not discriminate so small colour shift if there is any.
As depicted in
Figure 6, the results of the two separations for the dark tones are completely different. Since the dark colours actually located outside the CMY gamut, there is no doubt that the CMYK separation should be adopted in this condition. Meanwhile, the slope and the Y-intercept of the CMYK least-square-fitting line are quite small, which soundly validates the performance of the CMYK based separation for this condition.
Lastly, the ultimate comparison of our formerly proposed CMYK-BPn-CYNSN-based separation [
19] and the proposed CMY optimization approach is depicted in
Figure 7. It is quite clear that for the three experimental substrates, the proposed method in this study outperformed its counterpart, with an error reduction of 10–30%. It should be mentioned that such comparison was based on the 100 testing samples, where the sampling for the light and middle tones had been emphasised. It is easy to infer that the proposed method will become more useful when there are more light-tone colours in the original image. On the contrary, if the majority of the target colours located in dark-tone regions, the improvement may turn out to be insignificant.