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problem [Blinn 82]. The degree of scattering is assumed to be small, and only
single scattering is considered. Under these assumptions, all light paths start at
the light source, end at the viewpoint, and lie in a plane. A further simplification,
that the only variable is the depth of light travel in the medium, reduces the prob-
lem to a one-dimensional differential equation that is solved analytically in the
paper.
James Kajiya and Brian P. von Herzen extended Blinn's technique to the 3D
LTE in a 1984 paper entitled “Ray Tracing Volume Densities”, which was aimed
primarily at rendering clouds (Blinn's work had been motivated by rendering the
rings of Saturn) [Kajiya and Von Herzen 84]. In Kajiya's paper, the solution of
the LTE is handled differently if the degree of scattering in the medium is small
or large. In the case of a relatively small degree of scattering, a “low albedo ap-
proximation” is applied, in which in-scattering is only considered from the light;
i.e., only single in-scattering is computed. A voxel grid is constructed that ini-
tially contains only the voxel densities. In the first step, the ray from the light
source to each voxel is traced through the grid, with the radiance attenuated at
each voxel according to Equation (3.5). The next stage traces rays from the view-
point through the voxel, accumulating radiance from each voxel and attenuating
it accordingly.
While the low albedo approximation is likely to be sufficient for many appli-
cations, a cloud rendering that illustrates some obvious shortcomings of the low
albedo approximation is shown in the paper. Accordingly, Kajiya and von Herzen
developed a separate method for solving the LTE for media of high albedo. The
approach relies on an empirical observation known in transport theory: scattering
becomes more isotropic as the degree of scattering increases. By recognizing this,
the authors were able to better approximate in-scattering in high albedo media by
approximating how it varies from true isotropic scattering. The method reformu-
lates the LTE in terms of
spherical harmonics
, which are a collection of basis
functions on the sphere analogous to trigonometric series. Spherical harmonics
work well for low-frequency (slowly changing) functions like the incident light
in clouds, in the sense that only a few terms are needed to produce a good ap-
proximation. In spherical harmonics, the LTE becomes a set of coupled partial
differential equations that can be solved analytically. Although not much atten-
tion was paid to this solution method at the time of its publication, it had a major
influence on multiple scattering work in the latter half of the 1990s.
The early work of Blinn and Kajiya placed significant restrictions on the char-
acteristics of the participating media and the conditions in the environment. This
was partially to simplify the equations involved, but also arose out of necessity,
as computing power was limited at the time. However, the work paved the way
for more general solutions that came later.
