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be defined for any radiance distribution; e.g.,
Q
0
(
x
)=
Q
(
x
,
ω
)
d
ω
,
Ω
4π
Q
1
(
x
)=
Q
(
x
,
ω
)
ω
d
ω
Ω
4π
.
If the scattering is isotropic, integrating the LTE of Equation (4.1) produces the
expression
are sometimes called the
moments
of degree 0 and 1, respectively, of
Q
(
x
,
ω
)
∇
·
E
(
x
)=
−
σ
a
φ
(
x
)+
Q
0
(
x
)
(4.3)
which relates the vector irradiance, the fluence, and the first-order scattering (or
the emission). Note that Equation (4.3) is not explicitly dependent on direction.
The idea that light is diffused by multiple scattering is formalized by assuming
the diffused radiance
L
d
(
is well approximated by the sum of the fluence and
the surface irradiance in the direction
x
,
ω
)
ω
:
1
4
3
4
E
L
d
(
x
,
ω
)
≈
π
φ
(
x
)+
(
x
)
·
ω
.
(4.4)
π
This is called the
diffusion approximation
. Substituting Equation (4.4) into Equa-
tion (4.2) and integrating over all directions ultimately gives
σ
t
E
)+
Q
1
(
∇φ
(
x
)=
−
3
(
x
x
)
.
(4.5)
σ
s
is known as the
reduced scat-
tering coefficient
;
g
is the “scattering anisotropy” from Equation (3.3.3). As
g
approaches 1 the scattering has a stronger forward bias, which has less influence
on the net flux. Combining Equations (4.3) and (4.5) produces
σ
t
=
σ
s
+
σ
a
,
σ
s
=(
Here
1
−
g
)
σ
s
.Thevalue
)+
∇
·
Q
1
2
σ
t
φ
(
σ
t
Q
0
φ
(
)=
)
−
(
(
)
,
∇
x
3
σ
x
3
x
x
(4.6)
a
which is one form of the
diffusion equation
.
The diffusion equation given in Equation (4.6) describes how particles are
diffused within a medium. It also applies to any situation in which flux can be
defined, such as in fluid simulations. The simplicity of the equation hides the
complexity of the process it governs, as physical equations often do. How easy it
is to solve depends on the boundary of the medium and the conditions imposed
at the boundary. The basic boundary condition for diffusion inside a medium is
that diffuse light cannot come from outside the medium. Stam points out that the
diffusion approximation itself prevents this from being satisfied exactly. Instead,
he uses the condition that the
net
inward flux is zero; i.e., the amount of light en-
tering at a surface is balanced by the amount leaving it. This condition is popular
