Modeling the Mixing Behavior of Colorants: Mixing Formula
JK Kamarainen
Department of Information Technology,
Lappeenranta University of Technology,
P.O.Box 20, 53851 Lappeenranta, Finland
Abstract:
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 (400700nm).
Formula used for the mixture prediction is

(1) 
where x is reflectance ratio (contrast ratio), c concentration,
e Neper's constant and result value d is reflectance factor.
is a colorant type specific value.
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 c_{B}, spectras for A and B on wavelength are
and
(
). Mixture
spectra is .
Formula variables can be calculated

(2) 
These values are allways between 0 and 1. If

(3) 
Values are calculated for every wavelength and constant
can
be optimized by minimizing meansquareerror between original d and
calculated value from formula 2. Measured oilbased
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 KubelkaMunk equations [1].
New formula brings improvements comparing to KubelkaMunk equations and better
model of the mixing behavior in physical sense than colorcoordinate regression
models.
All results, documents, data and demos for
Matlab are collected to project homepage
at Lappeenranta University of Technology:
HTTP://www.lut.fi/~jkamarai/labhtml
to3em
 1

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

Vachon G., Modeling the mixing behavior of inks with polynomials, Color
Research and Application 13 (1988), no. 1, 4649.
Modeling the Mixing Behavior of Colorants: Mixing Formula
This document was generated using the
LaTeX2HTML translator Version 98.1p1 release (March 2nd, 1998)
Copyright © 1993, 1994, 1995, 1996, 1997,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.
The command line arguments were:
latex2html no_navigation split 0 formulareport.tex.
The translation was initiated by Joni Kämäräinen on 19980714
Joni Kämäräinen
19980714