Page 162 - Computational Colour Science Using MATLAB

CHARACTERIZATION OF HALF-TONE PRINTERS 149
fibres to which the dyes are applied. The application of the Kubelka–Munk
theory to opaque dyed textiles is therefore often referred to as the one-constant
version of the theory. For many pigmented surface coatings, such as paints, the
pigments both absorb and scatter and the two-constant theory is required.
However, Equation (9.12) can still be used to predict the reflectance of the
surface coating if it is applied at a thickness that achieves opacity. For
translucent printing inks, however, the reflectance of the paper upon which the
ink is printed makes a contribution to the reflectance of the system and therefore
Equation (9.12) must be replaced by
R ¼½ðR g R 1 Þ=R 1 ðR 1 R g 1Þ expfð1=R 1 R 1 ÞSxgŠ=
½ðR g R 1 Þ ðR g 1=R 1 Þ expfð1=R 1 R 1 ÞSxgŠ, ð9:13Þ

where R is the reflectance of the substrate and R 1 is the reflectance [as defined
g
by Equation (9.12)] of an opaque layer of the pigmented layer. The scattering
coefficient S is defined for a unit thickness of the layer and x is the thickness of
the layer. According to the theory the values of K and S should be linearly
related to the pigment volume concentration in the layer and to the thickness of
the layer. However, in practice severe departures from linearity can occur
(Nobbs, 1997). In order to predict the reflectance for a mixture of colorants or
inks the K and S contributions are determined for each component and then
assumed to be additive in order to allow the computation of K and S for the layer
and thus, via Equation (9.13), the reflectance R. In all cases, once the body
reflectance is known Equation (9.11) can be used to yield a prediction of the
reflectance P.
The Kubelka–Munk theory is routinely used for the prediction of reflectance
for systems of printing inks (for example, in lithography) and forms the basis of
computer match-prediction systems. However, one of the difficulties in applying
the theory to the characterization of printers is in determining the values of K
and S for the individual inks. One method to determine K and S is to print each
colorant over two different substrates or papers (for example, a white and a
black) and then to use Equation (9.13) to set up a system of two simultaneous
equations with two variables (K and S). For many printing systems it is difficult
to obtain these samples, especially since it is required that the surface properties
(roughness, etc.) of the two substrates must be identical. An alternative approach
is to treat the Kubelka–Munk coefficients as free parameters and to derive their
values based on an optimization routine and a set of samples of known
reflectance (Bala, 2003).
The Kubelka–Munk model could be used to predict the overlap areas in a
half-tone printing process and Neugebauer-type models could then be used to
predict the reflectance of a given area. The traditional Kubelka–Munk theory
assumes that the printed layer is homogeneous, however, whereas for half-tone
printing one ink is printed on top of another to generate a more layered