Modeling the Mixing Behavior of Colorants: Mixing Formula

J-K Kamarainen
Department of Information Technology,
Lappeenranta University of Technology,
P.O.Box 20, 53851 Lappeenranta, Finland


New formula for the colorant mixing is introduced. Formula basis on the spectral representation of colors and can be used for predicting the spectral reflectance or concentration of the mixture. Formula is three dimensional model and thus can be easily visualized.


Purpose of this document is to describe a new mixing formula. Formula is derived from three dimensional representation of colorant mixtures. Two formula variables are reflectance ratio and concentration and therefore formula is also wavelength independent. Formula gives reasonable predictions of color spectra or concentration of desired mixture. Formula is derivered to work primarily in the visible light area (400-700nm).

Mixing formula

Formula used for the mixture prediction is

\begin{displaymath}d~=~x+(1-x)e^{\alpha c},
\end{displaymath} (1)


where x is reflectance ratio (contrast ratio), c concentration, e Neper's constant and result value d is reflectance factor. $\alpha$ is a colorant type specific value. $\alpha$ should be adjusted for all colorant pairs mixed together or for colorant type if colorant mixtures physical characteristics are equal.

Two colorants A and B are mixed together and concentration for colorant B is cB, spectras for A and B on wavelength $\lambda$are $A_\lambda$ and $B_\lambda$ ( $A_\lambda > B_\lambda$). Mixture spectra is $M_\lambda$. Formula variables can be calculated

d&=\frac{A_\lambda}{S_\lambda} \\
x&=\frac{B_\lambda}{A_\lambda} \\
c&=c_B \\
\end{split}\end{displaymath} (2)

These values are allways between 0 and 1. If $B_\lambda > A_\lambda$

d&=\frac{B_\lambda}{S_\lambda} \\
x&=\frac{A_\lambda}{B_\lambda} \\
c&=1-c_B \\
\end{split}\end{displaymath} (3)

Values are calculated for every wavelength and constant $\alpha$ can be optimized by minimizing meansquare-error between original d and calculated value from formula 2. Measured oil-based paint data converted to (x,c,d)-coordinates in figure 1

Figure 1: Measurement data converted to three dimensional format.





We have introduced a new formula for colorant mixing and thist work showed that the spectral information is suitable to modeling the colorant mixing behavior. Model describes some kind of physical behavior of colorant mixing and is very simple as most physical phenomenons. Previous models of colorant mixing are based on color coordinates [2] or Kubelka-Munk equations [1]. New formula brings improvements comparing to Kubelka-Munk equations and better model of the mixing behavior in physical sense than color-coordinate regression models.


All results, documents, data and demos for Matlab are collected to project homepage at Lappeenranta University of Technology:



Saltzman M. BillMeyer F.W., Jr., Principles of color technology, 2 ed., John Wiley & Sons, Inc, New York, 1981.

Vachon G., Modeling the mixing behavior of inks with polynomials, Color Research and Application 13 (1988), no. 1, 46-49.

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Modeling the Mixing Behavior of Colorants: Mixing Formula

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