Page 161 - Computational Colour Science Using MATLAB

148 CHARACTERIZATION OF PRINTERS
known for the over-printed area of the secondary colours. So, for example, to
implement where yellow dots are over-printed with cyan dots we need to know
the colour of the over-print area where cyan ink falls on yellow ink.
When inks are printed on top of each other or are mixed together and then
printed, subtractive colour mixing takes place and additivity of reflectance values
is not valid. For subtractive mixing the densities of the inks are approximately
additive, where the density D is related to the reflectance P,

D ¼ log P. ð9:8Þ
10
So, for example, if two inks have reflectance 0.4 and 0.8 at a certain wavelength
and they are mixed together in equal proportions, then the mean of the density
contributions will be 0.0969/2+0.3979/2 ¼ 0.2474 corresponding to a reflectance
of 0.566 (this compares with a value of 0.600 if the reflectances are directly
averaged). Accurate prediction for subtractive mixing often requires application
of the Kubelka–Munk theory of radiation transfer that characterizes each ink or
colorant in terms of its absorption and scattering properties.
The Kubelka–Munk theory (Nobbs, 1985, 1997; McDonald, 1997b) has been
used to predict the reflectance of inks, plastics, paints, textiles and other
materials. The theory characterizes each colorant using the absorption K and
scattering S coefficients that are functions of wavelength and relates these
coefficients to the body reflectance of a sample. The body reflectance is the
reflectance of a surface if the interactions of light at the air/medium interface are
discounted. The body reflectance R is related to the measured reflectance P by
the following equation,

RðlÞ¼½PðlÞ r e Š=½ð1 r e Þð1 r i Þþ r i ðPðlÞ r e ފ ð9:9Þ
for the case where P is measured with a spectrophotometer with the specular
component included. The variables r and r are the external and internal
e
i
reflectance coefficients of the boundary. The inverse of Equation (9.9) is given by
Equation (9.10),
PðlÞ¼ r e þ½ð1 r e Þð1 r i ÞRðlފ=½1 r e Rðlފ. ð9:10Þ

For an opaque sample the body reflectance is related to the K and S coefficients
by Equation (9.11),
2
K=S ¼ð1 RÞ =2R, ð9:11Þ
and the inverse relationship is given by
2 1=2
R ¼ 1 þ K=S ½ð1 þ K=SÞ 1Š . ð9:12Þ
Thus, for opaque samples only the ratio of K to S is required at each wavelength
in order to predict the reflectance R. In the case of dyed textiles the dyes
themselves do not scatter light and the only scattering comes from the textile