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

Abstract:

New formula for the colorant mixing is derived. Formula basis on the spectral representation of colors and can be used for predicting the spectral reflectance or concentration of the mixture. Formula has been tested with two types of colorants and results are described in this document.

Introduction

Purpose of this document is to publish testing results of the new mixing formula. Formula is derived by the author in the former research reports where the theory is also covered. In this document applying the formula to real world data is briefly described and the test results are given. Hopefully these results will be motivation for more compherensive tests.

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 type's physical characteristics are equal.

Results

For testing purposes there was two different types of colorant mixtures, oil-based paints and gray pigments. Six mixtures are chosen from the both measured mixtures. Calculated reflectance spectra for the original concentration versus original spectra are presented in results. Metric color differences between the original mixture reflectance spectra and formula reflectance spectra for original concentration are presented in terms of $\Delta E^*_{ab}$, $\Delta C^*_{ab}$ and $\Delta H^*_{ab}$ (most used metrics in colorant industry) [1].

In figures 1 and 2 are reflectance spectras for both pure pigments p1 and p3 or p1 and p4 (solid line), measured reflectance spectras for original concentration (dashed line) and calculated reflectance spectras for original concentration (dotted line). Metric color differences for picture 1 and 2 data are calculated in table 1.

Same characteristics are calculated and plotted for paint measurements in figures 3, 4, 5, 6 and in tables 2 and 3. Left side figures are calculated from model where the same model has used for all different color paint mixtures and right side figures are from models where $\alpha$ is adjusted for the specific colorant pair only. This only shows that same model works for all paints if their physical characteristics are equal.


  


Figure 1: Spectras for gray pigment mixtures (p1 and p3).
\includegraphics[width=\textwidth]{pics/demo1.ps}




Figure 2: Spectras for gray pigment mixtures (p1 and p4).
\includegraphics[width=\textwidth]{pics/demo2.ps}





 
Table 1: L*a*b* metric differences for pigment mixtures.
Mixture $\Delta E^*_{ab}$ $\Delta C^*_{ab}$ $\Delta H^*_{ab}$ $\Delta L^*$ $\Delta a^*$ $\Delta b^*$
p1 p3 1. 5.2769 3.3778 0.0969 -4.0531 -3.3250 0.6029
p1 p3 2. 5.5699 5.5250 0.0338 -0.7053 -5.4323 1.0083
p1 p3 3. 9.5376 6.0874 0.1354 7.3411 -5.9721 1.1872
p1 p4 1. 6.4071 0.2077 1.0607 -6.3153 0.4198 -0.9960
p1 p4 2. 3.3233 0.6501 1.1432 -3.0520 0.3037 -1.2796
p1 p4 3. 17.2085 1.0078 1.1668 17.1393 -0.3317 -1.5057
 


  


Figure 3: Spectras for oil-based paint mixtures (yellow and blue).
\includegraphics[width=\textwidth]{pics/demo4.ps}




Figure 4: Spectras for oil-based paint mixtures (yellow and blue).
\includegraphics[width=\textwidth]{pics/demo5.ps}





 
Table 2: L*a*b* metric differences for paint mixtures.
Mixture $\Delta E^*_{ab}$ $\Delta C^*_{ab}$ $\Delta H^*_{ab}$ $\Delta L^*$ $\Delta a^*$ $\Delta b^*$
miranol1 1. 18.3916 -12.8977 11.8102 -5.6936 -12.6140 -12.1129
miranol1 2. 20.5666 -8.7251 18.3544 -3.1580 -17.5921 -10.1749
miranol1 3. 12.5078 9.7575 5.8115 5.2407 -6.2765 9.4650
miranol1 4. 16.3123 -11.1758 11.1404 -4.1333 -11.7929 -10.4850
miranol1 5. 13.1997 1.0608 12.6763 3.5242 -12.7089 -0.5442
miranol1 6. 11.2469 -0.0082 6.5971 9.1088 2.8440 5.9526
 


  


Figure 5: Spectras for oil-based paint mixtures (white and red).
\includegraphics[width=\textwidth]{pics/demo6.ps}




Figure 6: Spectras for oil-based paint mixtures (white and red).
\includegraphics[width=\textwidth]{pics/demo7.ps}





 
Table 3: L*a*b* metric differences for paint mixtures.
mixture $\Delta E^*_{ab}$ $\Delta C^*_{ab}$ $\Delta H^*_{ab}$ $\Delta L^*$ $\Delta a^*$ $\Delta b^*$
miranol7 1. 11.7249 4.8111 9.0488 -5.6962 10.1594 -1.3467
miranol7 2. 21.3225 16.8358 10.3230 -8.0399 19.7267 -0.9321
miranol7 3. 9.5008 3.2435 6.1955 6.4312 5.4823 -4.3416
miranol7 4. 10.9714 4.7477 8.6117 -4.8652 9.7299 -1.4245
miranol7 5. 16.6628 14.0220 8.5584 -2.7900 16.3404 -1.6897
miranol7 6. 21.5796 -12.3689 12.5526 12.4548 -5.3277 -16.7980
 

Conclusions

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. Still more tests with real world data is needed to evaluate usability of formula and maybe affect some improvement to the formula.

Acknowledgements

All results, documents, data and demos for Matlab are collected to project homepage at Lappeenranta University of Technology:
HTTP://www.lut.fi/~jkamarai/labhtml

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Bibliography

1
ASTM, Philadelphia, Astm standards on color and appearance measurement, 4 ed., 1994.

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Modeling the Mixing Behavior of Colorants: Mixing Formula

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