In the experiment, an IT8.7/5 (TC1617x) printer test chart was utilized. The IT8.7/5 (TC1617x) Printer Test Chart is a recently developed target utilized to characterize CMYK printers. The IT8.7/5 (TC1617x) Printer Test Chart contains 1,617 patches, the same number as IT8.7/4. However, 29 duplicate patches have been removed and replaced by 29 patches from columns 4 and 5, which were not present in IT8.7/4. As a result, the IT8.7/5 (TC1617x) Printer Test Chart offers 1,617 different colors.

The IT8.7/5 test chart was printed on the XL75 four-color offset press, manufactured by Heidelberg, Germany. The machine was purchased in 2008. Heidelberg Supermaster, Germany, was used as the printing plate maker. 200 LPI AM screening was used, with a dot pattern for the square and round mesh. The screening resolution was 2,540 dpi. The test charts were printed on 157 g/m² double-coated art paper from Asia Pulp & Paper Co., Ltd. (APP). The printing environment was maintained at a temperature of 23–25°C, with a humidity of 55% ± 5, a pH value of 4.8–5.3, an alcohol concentration of 10% by volume, and a dampening solution temperature set at 10–12°C.

The printing sequence used was black, cyan, magenta, and yellow. The printed target follows the ISO 15339-2-2015 CRPC6 standard, as shown in Table 1.

The primaries color target and CIEDE00 color difference between printed samples and ISO 15339-2:2015 CRPC6 standards. In the table, S represents the paper substrate, and C,M,Y,K,R,G,B represent the solid colors cyan, magenta, yellow, black, red, green, blue, respectively.

The printed paper was naturally dried for 1 hour prior to measurement to ensure that the colors were sufficiently dry. Each sheet was printed using the same target to ensure consistent results from print to print. The spectral reflectance measurements were carried out using an iSisXL2 spectrophotometer from xrite, United States, purchased in January 2023. The photometer of the measurement device has a 45/0 ring geometry and measures at 10 nm intervals in the range of 400–700 nm. Chromaticity values are generated according to D50 illumination and the CIE 1931 2° standard observer. Figure 1 displays the spectral reflectance of IT8.7/5 color patches from 400 nm to 700 nm.

The PCA method is a data dimensionality reduction algorithm, used as a preprocessing method for the prediction model. It explains most of the interpretation of the original sample through a few common key factors to reduce the computational complexity of the model. In the prediction model for printing image spectra, assuming that the original spectral data contain n-dimensional features, the PCA method maps the n-dimensional features described in the original data to the d-dimensional principal components. The d-dimensional principal components use fewer data indicators to describe most of the information contained in the original data, achieving the purpose of reducing the dimensionality of the data. Assuming that the number of samples is N, with n-dimensional features, the original data matrix X is described as in Equation 1. X = x 11 ⋯ x 1 n ⋮ ⋮ x N 1 ⋯ x N n = X 1 , X 2 , ⋯ , X n (1) where x i j denotes the j-th dimensional feature of the i-th sample (i ≤ N, j ≤ n), and Xn denotes the characteristic of the n-th dimension of the original data.

Using the PCA method, the characteristics of dimensions n can be mapped in dimension d to obtain d new variables, where d ≤ n; the new variables obtained are called the dth principal component Y d of the original data, and the extracted are used as the output of the LSSVM model, which is expressed as in Equation 2. Y 1 = a 11 X 1 + a 12 X 2 + ⋯ + a 1 n X n Y 2 = a 21 X 1 + a 22 X 2 + ⋯ + a 2 n X n ⋮ Y d = a d 1 X 1 + a d 2 X 2 + ⋯ + a d n X n (2) where a i j are the principal component coefficients and satisfy a i 1 2 + a i 2 2 + ⋯ + a i n 2 = 1 , i = 1 , 2 , ⋯ , d , each principal component is uncorrelated with each other, and the variance of the principal components satisfies V a r Y 1 > V a r Y 2 > ⋯ > V a r Y d .

Machine learning is a technique for making decisions by learning previous results and relying on patterns. Machine learning applications fall into classification or regression categories. DNN is a standard technique for developing regression models and is widely used for both discrete and continuous pattern recognition. DNN parameters analyze various patterns to ensure optimal performance. DNNs commonly comprise a fully connected feedforward architecture, which incorporates an input layer, a hidden layer, and an output layer [24–26]. The input layer receives instances from the data set. Instances from the input layer are transmitted through multiple neurons, and the information is processed through a series of weighted connections and biases. Several activation functions, including Sigmoid and Relu, are used to carry out activation in the hidden layer. The number of target classes determines the size of the output layer. The output layer in regression DNN is designed to predict the spectral reflectance or other continuous quantities of a printed image. The prediction process in regression DNN focuses on minimizing the difference between the network’s predicted values and the actual measurements.

The Whale Optimization Algorithm (WOA) was introduced in 2016, drawing inspiration from the distinctive “bubble-net” feeding strategy observed in humpback whales in their natural habitat [28, 29]. In the framework of WOA, each whale’s position represents a viable solution within the problem space. During the collective feeding process, individual whales exhibit three key behaviors: pinpointing the location of prey and encircling it, herding the prey using a bubble net, and participating in random exploration for prey. Assuming the position X of each whale in an n-dimensional solution space is X = x 1 , x 2 , ⋯ , x n .

Each whale randomly selects between encircling and herding the prey, with an equal probability for both behaviors (p = 50% for each). When encircling the prey (p < 0.5), the whale swims towards the individual with the optimal position to approachthe prey and its position update formula is shown in Equation 3. X i t + 1 = X b e s t t − A C · X b e s t t − X i t (3) where, X i t + 1 is the position of an individual whale at time t+1, X b e s t t is the position of the best individual in the population at time t, X i t is the position of an individual whale at time t, and A and C are coefficients.

When driving prey (p ≥ 0.5), in order to form a bubble net, the whale swims toward the prey in a spiral motion and its positional update equation is shown in Equation 4. X i t + 1 = X best t − X i t · e b l · ⁡ cos 2 π l + X best t (4) where b is a constant used to define the shape of the helix; L is a random number distributed in [-1,1].

In addition, the whale randomly looks around for prey, randomly selecting an individual in the group and approaching it, with its position updated by the equation: X i t + 1 = X rand t − A C · X rand t − X i t (5) where X rand t represents the position of a random individual at time t.

The discrepancy between Equations 3, 5 lies in the differing range of values for parameter A. Specifically, the decision of whether whales opt to encircle the prey or engage in random search depends on this magnitude. When a whale opts to encircle the prey, the whale collective contracts its search radius to target the prey, facilitating a localized search approach, albeit with slower convergence. Conversely, when whales embark on a random search for prey, the whale population expands its search radius to encompass a broader scope, allowing a global search approach, resulting in faster convergence. However, with a linear decrease in the control parameter a, the algorithm’s convergence process is not linear. This discrepancy underscores that a linearly decreasing control parameter a does not accurately depict the optimization process. Hence, this paper proposes a novel control parameter based on the variation of the cosine law. The modified expression for this control parameter is as follows Equation 6. a = a f i n a l + a i n i t i a l − a f i n a l 1 + cos t − 1 π / t max − 1 2 (6) where, a i n i t i a l and a f i n a l are the initial and final values of the control parameter a. In this paper, we obtain a i n i t i a l = 2 , a f i n a l = 0 , t is the current iteration number, and t max is the maximum iteration number.

The control parameter a exhibits a slow decrease during the initial iterations, thus prolonging its retention at a relatively high value for an extended duration. Consequently, this maintains a high value for parameter A over a longer period, thereby enhancing search efficiency. In contrast, in the later iterations, a decreases rapidly, facilitating its persistence at a relatively low value for an extended period. Consequently, this maintains A at a low value for a prolonged duration, thus improving the search accuracy. Thus, this method effectively balances the algorithm’s global and local search capabilities, ensuring search stability.

The spectral model of the printing press is characterized by inputting the four characteristic quantities of cyan (C), magenta (M), yellow (Y), and black (K) into the color reflectance prediction model ( x i = C , M , Y , K i ), and the corresponding measured spectral reflectance as the output of the model ( y i = Y 1 , Y 2 , ⋯ Y m i ). Therefore, the dataset containing N sets of samples used for the characterization of the printing press spectral model is x i , y i , i = 1 , 2 , ⋯ , N .

The proposed DNN model comprises an input layer, three to five hidden layers, and an output layer (Figure 2). The input layer includes neurons representing the sixteen input features, and the output layer includes seven neurons representing the seven principal components of the 31 spectral reflectance we want to predict.

The framework of the proposed model for color reflectance is shown in Figure 3. The framework consists mainly of three parts. The first part is the preprocessing of the input data. This includes expanding the input data considering the ink overlay situation into 16 Neugebauer primary colors as new inputs based on the Demichel equation [30, 31]. The spectral dimension of the output data is then reduced by extracting principal components through PCA technology. The second part mainly involves training and testing the DNN model optimized by IWOA. The third part includes data denormalization, data analysis, and model evaluation.

The proposed model contains the following steps:

Step 1: Pre-processing of data. Before establishing the model, normalize all experimental data to [0, 1] using the Equation 7.

The CIELAB average color differences (ΔE_ab) is a colorimetric assessment metric that is appropriate to evaluate color under commonly used CIE illuminations, such as A, D50, and D65. This metric is derived from the CIELAB color difference and is critical to evaluating colorimetric precision [33]. The smaller the difference, the greater the precision. Color scientists have developed four commonly used color difference formulas, CIELAB, CMC (l:c), CIE94, and CIEDE2000, for evaluating color differences in the color field. Among them, CIEDE2000 is currently the color difference formula that best approximates the uniformity of human vision. In this paper, CIEDE2000 is employed as it represents the optimal approach for color control and evaluation of printed materials in the present era. The CIEDE2000 formula is shown in Equation 9. Δ E 2000 = Δ L ′ k L S L 2 + Δ C ′ k C S C 2 + Δ H ′ k H S H 2 + R T Δ C ′ k C S C Δ H ′ k H S H 1 2 (9) where K _L , K _C , and K _H are weighting coefficients for lightness, chroma, and hue, respectively. S _L , S _C , and S _H are weighting functions for lightness, chroma, and hue, respectively. R _T is the interaction term. K _L = K _C = K _H = 1, as these specific values are recommended for printed images.

The color difference between the spectral predictions of the deep neural network (DNN) model and the measured spectral values was calculated using the CIEDE2000 formula. To train the model, the loss function employs the CIEDE2000 color difference formula, and the optimal weight coefficients are derived through the use of the improved WOA optimization algorithm, which is based on minimization of the color difference between the prediction and measurement sets.

To reduce the 31 dimensions of the spectral data, PCA was performed to extract the principal components of the spectral dimensions while preserving most of the information from the original variables. It also reduces redundant data resulting from their cross-correlation. Table 2 presents the eigenvalues of the covariance matrix for the seven indicator variables, and Figure 4 illustrates the histogram that displays the principal components’ contribution. According to the analysis of Table 2; Figure 4, selecting 7 principal components results in a cumulative variance contribution rate of 99.99%. This indicates that these seven principal components capture the characteristic information of the 31 original variables. The use of PCA successfully reduces the dimensionality of the variable feature space compared to the 31 original output features of the sample data. The reduced- dimensional data retains most of the information from the original variables while reducing the data redundancy caused by their cross-correlation.

On the basis of the CMYK and spectral reflectance values acquired from the color target measured by the spectrophotometer, a hybrid PCA-IWOA-DNN model is established. The IT8.7/5 (TC1617x) test chart, consisting of 1,617 data samples, was used as training and validation sets. The color differences between the predicted and measured values of the sets were evaluated using two parameters: Root Mean Squared Error (RMSE) and CIEDE2000 (32). Colorimetric values were calculated for the illuminant A, D50, D65 and the CIE 1931 2° standard observer. With hidden layers set to [20 50 30], the prediction results are as shown in Table 3.

Table 3 shows that in five experiments under D50/2° illuminant, the minimum color difference is 0.0146, the optimal average color difference is 0.4757, and the optimal 95th percentile is 1.3684. The experimental results indicate that the overall color difference deviation is lower than the values reported in the literature. These color difference values are much lower than the common color difference requirements for the color reproduction of OK printed products in the printing industry. Figure 5 shows the distribution of the total CIEDE2000 color differences between the training and validation sets under a D50/2° light source in experiment 4. From Figure 5, it can be seen that the maximum color difference at the 95th percentile is 1.8726.

Figure 6 shows the comparison between the chromaticity coordinates of the test set and the chromaticity coordinates predicted by the proposed model. Through their coordinate distributions in the CIE1931xy chromaticity diagram, we can see that the predicted chromaticity coordinates and the actual chromaticity coordinates of the test set have a high degree of overlap, and better prediction results are obtained. The general overview of the mean and minimum color differences is shown in Table 3.

Compared to traditional Clapper-Yule and Williams-Clapper model [1, 11–13, 34], Yule-Nielson [9, 31, 35–38], ANN-BP neural networks [39], the hybrid PCA-IWOA-DNN model significantly improves the accuracy of fitting and the overall distribution of color differences while requiring fewer training samples. In Experiment 5, the prediction results of these models are shown in Table 4.

The method proposed in this study indicates the lowest color difference values under three common light sources, suggesting a potential improvement in color accuracy. Table 4 shows that the hybrid PCA-IWOA-DNN model effectively captures the complex relationships in the data, leading to improved accuracy in color difference prediction. The model uses an improved whale optimization algorithm for optimization and DNN for prediction. As shown in Figure 7, it compares the predicted spectral reflectance of the test set with the measured spectral reflectance.

Based on the results presented in Figure 7, our proposed models appear to outperform conventional models in predicting color reflectance and provide a better match between spectral predictions and actual measurements. The model performs exceptionally well in primary colors, light tones, and dark tones. The figure compares the predicted and actual reflectance values for various colors, including cyan, magenta, yellow, red, green, blue, black, and the corresponding halftone colors. The left side represents the model-predicted spectral reflectance, while the right side represents the actual spectral reflectance measured by the spectrophotometer. By comparing the color differences between the actual measurements and predicted values, the effectiveness of the prediction model can be validated, confirming its ability to predict the spectral reflectance characteristics of different shades of colors. As shown in the figure, the predicted reflectance values in the test set align well with the actual measured spectral reflectance values.