Graphics Reference
In-Depth Information
The LTE cannot generally be solved analytically. Hanrahan and Krueger made
two simplifications in approximating the solution. The first is that the scattering is
independent of surface position, and depends only on the depth of penetration into
each layer. This reduces the LTE to a one-dimensional equation in the direction
perpendicular to the surface. The in-scattering term is then expanded as a gener-
alized geometric series known as a Neumann series. The degree of each term in
the series corresponds to the number of scatter events inside the material. That
is, the linear term corresponds to single scattering, the quadratic term to double
scattering, and higher-degree terms to multiple scattering. However, only the lin-
ear term is included. With this simplification, the LTE can be solved analytically.
The solution amounts to a surface BRDF that includes single scattering inside the
surface.
Although approximating subsurface scattering with only single scattering
might seem mathematically reasonable, mathematical models do not always rep-
resent natural phenomena correctly. Hanrahan and Krueger's paper actually con-
tains a comparison of the single scattering subsurface model to the results of
a Monte Carlo simulation. The simulation works as follows. It begins with a
particle simulation in which a large number of particles are shot from the light
source. The particles are scattered inside the surface according to a scattering
probability function, and the power carried by each scattered particle is stored at
the position of the scattering event. The stored intensity of light is then gathered
during a second rendering phase, which uses Monte Carlo ray tracing.
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The re-
sults of the simulation can then be compared to the single scattering model. The
experiments revealed that the difference increases with the degree of scattering
in the medium. This verified the importance of multiple scattering in materials
having a large scattering albedo.
The layered scattering paper by Hanrahan and Krueger was significant in
that it presented a theoretical solution for single scattering. The authors note that
the same technique can be applied to derive formulas for higher-order scattering,
but left that as future work. The demonstration of the importance of multiple
scattering also had a strong influence on subsequent research.
4.2.2 Modeling Multiple Scattering
The first approach that modeled multiple scattering based on diffusion was pro-
posed by Jos Stam in 1995 [Stam 95]. Much of Stam's research has involved
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This approach is similar to Jensen's photon mapping technique, and in fact predates it. Particle
simulation was not new when photon mapping was developed; its principal novel aspects were the
storage and representation of the particles, and the use of the “nearest neighbors” interpolation method
for photon map lookup.
