pigmented colorants: dependence on media and time - Cornell ...

effects or are too computationally expensive to do so interactively.
As contrasted to the previous approaches, we can work at the macroscopic level
and simply model the aggregate behavior of the paint with respect to incident light.
This approach was taken by German scientists Paul Kubelka and Franz Munk, who
developed a simple set of differential equations to describe the transport of light in
<strong>pigmented</strong> materials [KM31]. The model is very effective for <strong>pigmented</strong> materials,
is not very complicated and is very efficient. The original paper in 1931 is based on
the assumption of a homogeneous material of a medium that is infinite in extent.
The model describes the material properties of a <strong>pigmented</strong> material in terms of
only two wavelength-dependent parameters: an absorption constant, K(λ), and a
scattering constant, S(λ):
106
R∞ =
1
1+ K
S +
1+
K 2
− 1 S
(4.9)
Equation 4.9 represents the solution to the most basic Kubelka-Munk differen-
tial equations as they were originally presented [KM31]. The diffuse reflectance,
R∞, of a paint sample of complete hiding (a paint of a thickness such that the sub-
strate cannot been seen underneath) is a function of the absorption and scattering
coefficients. Note that the <strong>dependence</strong> on wavelength is omitted for clarity and the
equation is computed over the visible spectrum. The derivation of the solutions of
Kubelka and Munk’s differential equations and other improvements to the theory
over the years are described in Appendix A.
The resulting solutions to the differential equations have found wide scientific
utility in areas as diverse as the study of paint, paper, textiles, and skin, as well as
art conservation and planetary science. In computer graphics, the equations have
been used in a variety of rendering contexts for materials that exhibit subsurface