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for materials exhibiting a high degree of scattering, the computation and memory
requirements become excessive because of the large number of photon collisions
needed. The result of each collision has to be stored in the photon map.
While working as a postdoctoral researcher at MIT, Jensen joined a project in
the field of medical physics developing skin treatments. Through this project he
became more familiar with existing transport theory literature, as transport theory
is widely used in medical physics. For example, conditions inside a blood vessel
can be estimated from light scattering that occurs around its external wall by ap-
plying transport theory. The method observes the transition of photon densities
in a local region by capturing light scattering in the whole region. This con-
cept, which is totally different from particle simulation, seemed very promising
to Jensen.
4.3.3 The Dipole Model
Of all the publications on the subject of transport theory, a book written by Akira
Ishimaru influenced Stam and Jensen the most [Ishimaru 78]. This paper inspired
the separation into single scattering and multiple scattering and also the use of
an approximation to represent multiple scattering. While Stam was interested
in numerical solutions to the equations, Jensen sought an approximate analytical
solution.
For the simple environment consisting of a single point light source in an in-
finite medium, the diffusion equation (Equation (4.6)) is known to have a closed-
form solution:
σ
t
Φ
4
e
−
σ
tr
r
(
x
)
r
3
φ
(
x
)=
.
(4.7)
π
(
x
)
Here
Φ
is the intensity (flux) of the point light source,
r
(
x
)
is the distance to the
σ
tr
=
3
σ
a
σ
t
. But this solution is not much use for subsurface
scattering—by definition, an infinite medium has no surface! A simple, com-
monly used geometric model for subsurface scattering is a half-space, which is
the collection of points in space that lie on one side of an arbitrary plane. In this
model the surface is a plane, and the material extends infinitely deep below the
surface. Solutions for simple light source arrangements are known for this case,
but they involve infinite series and Jensen was looking for a simple formula. A
hint came from a 1970s medical physics paper by G. Eason et al. concerning the
scattering of light by blood [Eason et al. 78].
The BSSRDF relates the incident flux coming from a particular direction at
an entry point on a surface to the outgoing radiance in another direction at an-
other exit point. When the incident flux is regarded as a thin beam, the resulting
radiance distribution inside the medium up to single scattering is represented by
source to
x
,and
