Modeling the Mixing Behavior of Colorants
JK 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 oilbased 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 KubelkaMunk method.
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 oilbased paints.
Mixture measurements were made by Minolta CM2002 spectrophotometer
(measuring reflectance) and Mettler PM34 weighingmachine (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 wellknown KubelkaMunk [2] method for the
subtractive color mixing (producing desired color). Though,
the KubelkaMunk is a very rough approximation and new methods are needed.
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*colorcoordinates.
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.

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
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
are
and
(
).
Factor on this wavelength is calculated from 3

(2) 
Now mixture's reflectance on this wavelength is
.
There was 2 input neurons (x, c) and 3 hidden layer and 1 output neuron
(factor) in the used multilayer perceptron (MLP) network [1].
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).

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
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.

Figure 4:
Calculated spectral data deviation from the original measurement data.

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 KubelkaMunk method.
 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.
Modeling the Mixing Behavior of Colorants
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