Digital image processing requires signi cant amount of calculations for characterization and pro le making. Moreover, enhancing the computing precision does not always lead to better results, and the gamut might describe less part of color space than could do. Instead of expanding the existing methods and color prediction models, we o er a simple technique for spectral re ection prediction using an arti cial neural network in Matlab. The proposed method is fast and easy-to-operate. The experimental veri cation showed its good performance based on minimization of the color di erence CIE Lab dE Lab dE.
Introduction In order to create prints with accurately reproduced colors on a given reproduction system, it is essential to specify the color response that the system provides for a given substrate, ink's type, and with given amounts of inks during the process of characterization and pro le making.
In general, due to the printing process, the deposited ink surface coverage
is larger than the nominal one resulting in a physical dot gain responsible for
the ink spreading, which depends on the inks, on the substrate, and so on [
Empirical surface models only take into account superpositions of ink halftones, which the re ected light is supposed to be a function of the e ective ink surface coverage. These models do not deal with the light propagation within the print and only demonstrate the relationship between the re ected light and surface coverages by ink.
Physically inspired models engage a more detailed analysis of light-print interaction based on mathematical prediction of how the light goes within a halftone print and what the resulting fade is.
Ink spreading models describe the physical dot gain as a di erence between the e ective and the nominal surface coverages. They show how much an ink dot spreads out in all ink superposition conditions and rely on ink spreading curves mapping the nominal surface coverages to the e ective surface ones.
Spectral re ection prediction models (SRPM) are helpful in studying the
impact of di erent factors such as inks, substrate, the illumination conditions, and
the halftones in uencing the range of printable colors and in creating printer
characterization pro les for the purpose of color management [
One of the most cited one is Kubelka{Munk model that is widely used to
predict the properties of multiple layers of ink overlaid at a given location and
given information about each constituent ink's re ectance and opacity [
= S( ) 2R1( ) ; where K is the absorption and S is the scattering coe cients, R is the re ectance of an in nitely thick sample and the prediction of S and K from re ectance is made at the given wavelength .
Formula (1) allows predicting the combined K and S coe cients for multiple inks l K ( ) =KB ( ) + X ciKi( ); i=1 where B refers to the substrate, l is the number of ink layers, ci is the concentration, and Ki is the absorption coe cient of the i -th layer. The value S ( ) is computed analogously.
Another famous CPM is the Neugebauer model, which predicts the CIE XYZ
tristimulus values of a color halftone patch as the sum of the tristimulus values
of their individual colorants [
Today, the most applicable model is the Yule{Nielsen modi ed spectral
Neugebauer model (YNSN) where the Yule{Nielsen relationship is applied to the
spec(1)
(2)
tral Neugebauer equations [
The Enhanced YNSN accounts the ink spreading connected with the
respective physical dot-gains in di erent conditions (from 1 to 4 colorants
superposition). The model engages multiple ink spreading (tone reproduction) curves to
characterize the physical dot-gain [
Current CPMs accounting the physical dot-gain and are able to predict
reectance spectra as a function of ink surface coverage for 3{4 inks [
All mentioned approaches have their advantages and drawbacks. Majority of
models are too complicated to be embedded in a real digital image processing
work ow without substantial development and adjustment that takes time. For
instance, the YNSN-based models are extremely critical to selection of the
parameter n that is usually tted by brute force, and is utterly laborious except
some approaches [
Gradation scales are known as an unaltered attribute of contemporary digital
image processing systems [11, pp. 88{89]. At the same time, the authors express
doubts about the rational use of its features. The main problem is the fact that
using the gradation curves in conventional 2D embodiment signi cantly reduces
the quantity and quality of information extracted from them. In work [
Nevertheless, precise color prediction is still the issue that should be resolved.
One of the promising way to build the color prediction model is not model
variation at all. This is the arti cial neural network (ANN) approach. Works
on this topic had started at early 1990th. Most of researchers were focusing
on application of ANN to the Kubelka-Munk approach [13{15]. The work [
This work is devoted to further development of the ANN technique applied to color prediction. We work with an ANN training algorithm, de ne the applicability of the technique for spectral prediction, and guess how preliminary linearization a ects the accuracy.
For the experiment, we use the 4-color (CMYK) wide-format ink-jet printer Mimaki CJV30-160BS. Print mode: 720 720 dpi, variable dot. Substrate: a vinyl banner fabric as a weak-absorbent substrate. The measurement tools: spectrophotometer x-Rite iOne iSis + x-Rite Pro leMaker package. Charts generation is made in the ArgyllCMS package.
We describe the approach as follows: print a specially developed test chart !
Measure CIE Lab coordinates of the patches ! Build ANN ! Predict spectral
re ectance ( ) by ANN using test chart patches recipes as ANN inputs !
Assess the quality of prediction by the color di erence formula dE 94 [
For the ANN development, we use Matlab 16 package. The ANN type is
multilayer perceptron with one hidden layer, and 10 hidden neurons. The training
techniques are the Levenberg{Marquardt (L-M) method and bayesian
regularization [
We predict spectral re ectance by test chart patches recipes. A preliminary experiment stage consists of training the ANN by the training data set (training chart). The training set contains 2448 patches, which recipes are obtained from a discrete sequence f0; 16.8; 33.3; 50.2; 66.7; 83.1; 100g for each color in all possible combinations (so-called a `multicube'). For ANN prediction, the test chart with 2496 patches is developed in the ArgyllCMS. The patches recipes take random values evenly distributing the elds in the color space of the ideal CMYK-device. This data set acts as input for the ANN. The ANN output is the predicted spectrum. For each spectra, the CIE Lab coordinates are calculated. The test chart is then measured and the actual values of the CIE Lab coordinates are established. The color di erence between predicted and measured values for each patch of the test chart is calculated by the dE 94 formula.
The preliminary stage revealed that 6{7% of predicted spectra have negative values of individual spectral components, which is physically impossible. This occurs in cases where the actual value of the re ection coe cient is close to zero. Under the absence of restrictions, the network selects patterns in this way. However, we do not yet have grounds for introducing restrictions.
We study several options of this problem solution. The simplest case is ignoring of the negative values and removing them from further consideration. However, this can steep the results of the prediction.
The other option may be an increase in the samples sizes for both training and test sets. The sample volumes are arti cially increased by a three-fold measurement of the scales. The results of measurement are placed in a single protocol. This option also did not bring signi cant results, except for a signi cant increase in the training time of the network. Similar results are also obtained in attempts to use the averaged values for training and predicting.
A variant of the solution of the problem is the transition from the spectral re ection coe cients i( ) to the spectral optical density Di( )(4) (4) where i is the counter of patches, is the wavelength, = f380, 390, . . . , 730g nm.
The nal experiment is the following. The training set consists of the results of triple measurements of the training scale. Thus, each patch is included in the training set three times with the same composition of predictors, but with a statistically di erent composition of spectral components. The additional bene t of this approach is the ability to train the network on statistically blurred data.
Such, we train the ANN with spectral density D and predict also its values. After prediction, the spectral density D values are recalculated back into the spectral re ectance i ( ) =10 Di( ); 8 i; : (5) The predicting quality is assessed by the color di erence dE 94. The list of performed experiments and their brief description are given in Table 1.
For comparison of the results obtained in each experiment, arrays of dE 94 are tted by di erent distributions (see Table 2 and Fig. 1). Goodness of t is assessed by the 2 criterion.
The distribution that t dE 94best in most cases is lognormal. We use analytical distribution only for the ability to evaluate median and 95% quantile. Median shows the mean value of the color di erence in the test set. The quantile is the upper border of the color di erence with 95% probability. Analysis of plots in Fig. 1 con rms validity of such estimate.
We also build the dependencies of the dE 94 from the total ink parameter (see Fig. 2). The plots allow assessing where the ANN predicts better, in \lights" or in \shades".
As it can be seen from Fig. 2a, the ANN badly predicts spectral re ectance. In some cases, the dE 94 exceeds 6 that is completely unacceptable in real print production. Moreover, color di erence is distributed unevenly in relation with the total ink parameter: the prediction error increases in the \shades". The determination coe cient and trend in the gure accentuate such increase. Nevertheless, as Table 2 shows, the mean value of the color di erence is just about 2 and 95% quantile is less than 5 that is quite good. At the same time, existence of the negative spectral components in some predictions does not allow us to consider this experiment successful. In further experiments, we replace the prediction of spectral re ectance i( ) with one of spectral density Di( ) without changes in the ANN.
Figure 2b shows the results of predictions by the ANN trained with nonlinearized printer sample. Table 2 reveals the awesome results of the third experiment where 95% values of color di erence are less or equal 1.5. Notwithstanding, in this case, we also observe the dependence of the color di erence and the total ink parameter. \Lights" are predicted much worse than \shades". This might be explained with the assumption that a non-linearized printer exceeds the ink supply for highly saturated tones. The reason why the network predicts these formulations better is the excessive number of dark patches in the training set.
Figures 2c and 2d show the results of prediction for the ANN trained by data from the linearized printer. The determination coe cients in these cases are signi cantly lower than previous ones. This can be interpreted as the complete absence of the dependence of the color di erence on the total ink parameter. Suchwise, the ANN predicts the spectrum of any patch recipe with equal success. The only di erence between Figures 2c and 2d is the algorithm of ANN training: Experiment 4 uses the Levenberg-Marquardt method while Experiment 5 applies the Bayesian regularization. ExperimentA brief experiment description number 1
The test set is printed on a linearized printer and measured three times. 7488 dE 94 values are calculated. dE 94 of each patch from the triple average is obtained. The lowest border of prediction accuracy is evaluated. The test set is printed on a linearized printer and measured three times. The ANN is trained to predict ( ) value according to the recipe of the patch. The L-M algorithm is applied. 7059 recipes are predicted without negative values of ( ). Estimation of direct prediction of ( ) is done. The test set is printed on a non-linearized printer and measured once. The ANN is trained to predict the spectral D value according to the patch recipes. The L-M algorithm is applied. Estimation of indirect prediction of ( ) is done. We compare the prediction results of a linearized and non-linearized printing system.
The test set is printed on a linearized printer and measured three times. The ANN is trained to predict the spectral D value according to the patch recipes. The L-M algorithm is applied. Estimation of indirect prediction of ( ) is done. We compare the prediction results of a linearized and non-linearized printing system.
The test set is printed on a linearized printer and measured three times. The ANNis trained to predict the spectral D value according to the patch recipes. The Bayesian regularization algorithm is applied. We compare network learning algorithms.
As it can be seen from Table 2, the spreading of color di erences dE 94 in Experiment 4 is not tted well by both lognormal and normal distributions.
At this stage, the correlation with the total ink parameter is not high, but 1.4 times higher than in the case of Experiment 5. Moreover, there is an inexplicable border around dE 94=3 (see Fig. 2c). Consequently, it can be argued that, in general, for all experiments, the Levenberg-Marquard training algorithm shows unsatisfactory results. The lowest correlation with the total ink and even distribution of low dE 94are obtained in Experiment 5 with the Bayesian regularization for the ANN training method.
We o er a technique for spectral re ection prediction using an arti cial neural network. The proposed method is easy-to-operate and does not involve sophisticated color prediction models. The experimental veri cation showed its good performance based on minimization of the color di erence CIE Lab dE 94.
Application of arti cial neural networks for solving the problem of color prediction by its recipe shows the excellent result subject to certain conditions.
First, the color reproduction system must be linearized prior to the prediction. Next, we strongly recommend training the network not for prediction the spectral re ectance but for the spectral density, since there are probability of appearance of negative values during the ANN prediction.
Additional bene ts of our study are the following: we rst use the uniformity of the dE distribution from the total ink as a criterion for prediction quality assessment. Our approach solve the problem of the Black (K) channel generation automatically as we use the CMYK patches as inputs and outputs while common color prediction models operate in CMY recipes only, which requires additional e orts for the CMY-CMYK converting.
The results obtained are preliminary. Some issues remain unsolved. We would expect that the prediction could be even more precise when the appropriate training algorithm and sample volume were selected.