Modeling the Mixing Behavior of Colorants

J-K Kamarainen
Department of Computer and Information Sciences,
University of Delaware

Abstract:

This document describes two new methods for the colorant mixing and color production. New methods base on the spectral color representation. Methods are developed and tested with oil-based paint data, which is measured in the Information Technology Lab in Lappeenranta University of Technology. Two introduced methods are neural approach method (MLP neural network) and new colorant mixing formula. The mixing formula is produced by the regression analysis on paint measurements. Methods can be applied to the color production system instead of widely used Kubelka-Munk method.

1. Introduction

Purpose of this document is to describe the work and results of the colorant mixing research project. Project consists of colorant mixture measurements and developing and evaluating modeling methods for the colorant mixing behavior. Colorant mixture measurements were made on oil-based paints. Mixture measurements were made by Minolta CM2002 spectrophotometer (measuring reflectance) and Mettler PM34 weighing-machine (measuring concentration). Three included components in every measurement are reflectances of two or more mixed colorants, reflectance of the mixture of colorants and concentrations of colorants in the mixture.

Measurement data was used as a learning data in the multilayer perceptron (MLP) neural network and same data was also modeled by nonlinear regression analysis. Motivation for the regression analysis found from the MLP purposes preprocessed representation of the measurement data, because data shaped a smooth surface in three dimension.

Allready exists well-known Kubelka-Munk [2] method for the subtractive color mixing (producing desired color). Though, the Kubelka-Munk is a very rough approximation and new methods are needed.

2. Measurements

In the figure 1 is an example of the one measurement set. Top left shows reflectance curves of the each mixture, top right shows the increasing concentration of the blended colorant (blue) and in two additional figures are reflectance values converted to xy- and a*b*-color-coordinates.

In the Figure 1 can be seen that the exponential increase in the concentration causes almost linear change in the mixture reflectance (mixture reflectance curve and a*b*-representation changes are almost constant, linear).


  
Figure 1: Measurement data for mixture of yellow substrate colorant and blue colorant.

\includegraphics[width=5cm]{pics/miranol1_eng.ps}




\includegraphics[width=5cm]{pics/miranol1_osuus_eng.ps}




\includegraphics[width=5cm]{pics/miranol1xy.ps}




\includegraphics[width=5cm]{pics/miranol1ab.ps}



3. Modeling with MLP neural network

Problem was to predict the concentration of two colorants, when the reflectance of both, used colorants and target mixture, are known. The problem was siplified for the MLP network learning. Simplifying was made by an assumption, that the mixture reflectance is wavelength independent and the mixture reflectance on the specific wavelength depends only on the ratio between mixed colorants' reflectances and the concentration of the colorants. This assumption leads to a factor, which reflects the change in the more intense reflectance value (decreases that by the value of the factor). Derived model is


factor = f(x,c), (1)

 

where x is the reflectance ratio and c is concentration. Reflectance ratio in model is a scalar between 0 and 1 and the concentration is also a scalar between 0 and 1. The factor of the mixture reflectance must be calculated wavelength by wavelength. Ratio x is always lower intensity reflectance value divided by the higher intensity reflectance value and the concentration c is concentration for the lower intensity colorant. Some instances use a name contrast ratio for reflectance ratio.

Colorants A and B are blended. Reflectances on wavelength $\lambda$ are $A_\lambda$ and $B_\lambda$ ( $A_\lambda > B_\lambda$). Factor on this wavelength is calculated from 3


\begin{displaymath}factor~=~f(\dfrac{B_\lambda}{A_\lambda}, c_B).
\end{displaymath} (2)

Now mixture's reflectance on this wavelength is $factor * A_\lambda$.

There was 2 input neurons (x, c) and 3 hidden layer and 1 output neuron (factor) in the used multilayer perceptron (MLP) network [1].

3.1 Results

Test set mean error for the MLP neural network is shown in the table 1. Also some callculated spectras are compared with the original mixture spectras in the figure 2.


 
Table 1: Results for the MLP neural network.
Learning set size Number of the hidden layer neurons Test set mean error
100 3 4.6946
 


  
Figure 2: Sample of the original spectras and produced spectras (dotted line).

\includegraphics[width=4cm]{pics/mlp059.ps}




\includegraphics[width=4cm]{pics/mlp0511.ps}



\includegraphics[width=4cm]{pics/mlp0515.ps}




\includegraphics[width=4cm]{pics/mlp0517.ps}



4. Regression analysis for preprocessed data

When the measurement data is preprocessed into previously defined format (ratio, concentration and factor), it can be represented in the three dimension. All measurements are converted to this new format and shown in the different view angle in the figure 3. Figure shows that the measurement data shapes simple smooth surface and the formula for the colorant mixing can be derived from the shape of this surface


 
d = x+(1-x)eac, (3)

where d is the factor, x reflectance ratio, c concentration, e Neper's constant and a color substance dependent constant.

Paint data fits into formula 3 with constat a =-4.9230and the error distribution (deviation) between produced spectral data and all the original measurement data is shown in the histogram in the figure 4.


  
Figure 3: All measurement data converted to three dimensional format.

\includegraphics[width=4cm]{pics/plotx_eng.ps}




\includegraphics[width=4cm]{pics/ploty_eng.ps}




\includegraphics[width=4cm]{pics/plotz_eng.ps}




  
Figure 4: Calculated spectral data deviation from the original measurement data.
\includegraphics[width=6cm]{pics/model1errhist.ps}

5. Conclusions

Work showed, that the spectral color representation based color production is possible and easily implemented by the neural networks. This work offers results and methods for implementing a color mixture producing system, based on the spectral color representation. Also a new formula for colorant mixing behavior was introduced and this formula can replace the Kubelka-Munk method.

Bibliography

1
S. Haykin, Neural Networks A Compherensive Foundation, New York:John Wiley, 1994.

2
K. McLaren, The Colour Science of Dyes and Pigments, Bristol:J W Arrowsmith Ltd, 1983.

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Modeling the Mixing Behavior of Colorants

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