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below the surface must equal that of the outgoing flux over the hemisphere above
the surface. In symbols,
F
dr
(
,
ω
)(
ω
·
n
−
)
ω
=
(
,
ω
)(
ω
·
n
+
)
ω
,
L
x
d
L
x
d
(4.8)
Ω
−
Ω
+
where
Ω
−
and
n
−
denote the hemisphere and surface normal below the surface,
respectively;
Ω
+
and
n
+
are those above the surface. Under the diffuse assump-
tion, this is satisfied if
2
A
3
φ
(
x
s
)
−
σ
t
(
n
·
∇
)
φ
(
x
s
)=
0
,
(4.9)
where
x
s
is any point on the surface,
n
is the surface normal,
1
+
F
dr
A
=
(4.10)
1
−
F
dr
and
F
dr
is the diffuse Fresnel reflectance (Section 1.2.6). The two sources have
complementary intensities (the flux of the negative source is the negative of the
flux of the positive source), so it follows from Equation (4.9) that the sources must
cancel each other at the line situated a distance of 2
A
σ
t
)
above the surface.
The fluence function in the dipole approximation is the sum of the two sources
as if they were each in their own infinite medium. Applying Equation (4.7) gives
a formula for the fluence:
/
(
3
e
−
σ
tr
d
r
d
r
σ
t
Φ
4
e
−
σ
tr
d
v
d
v
3
φ
(
x
)=
−
,
(4.11)
π
where
d
r
and
d
v
are the distances from
x
to the positive and negative sources,
respectively, and
is the incident flux. The goal is to find an expression for
the multiple scattering component, which comes from the radiance exiting the
surface (the single scattering component came from the earlier work of Hanrahan
and Krueger). Equation (4.3) relates the fluence, the vector irradiance, and the
first-degree single scattering moment. In the absence of the single scattering term,
Equation (4.3) relates the vector irradiance directly to the fluence:
Φ
1
σ
t
∇φ
(
E
(
x
)=
−
x
)
.
(4.12)
3
Substituting Equation (4.11) into Equation (4.12) produces a formula for the vec-
tor radiant exitance at the surface due to multiple scattering, according to the
dipole model. To see how this applies to the construction of a BSSRDF model,
recall that the BSSRDF relates the incident flux to the outgoing radiance. The
