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underlying assumption is that multiply scattered light loses its directional de-
pendence. Consequently, the radiance from multiple subsurface scattering does
not depend on direction. It is proportional to the radiant exitance
M
at each
surface point, which is the irradiance at the surface due to light coming from be-
neath the surface. This is where Equation (4.12) comes in. The irradiance from
inside the surface is the scalar product of the vector irradiance with the inward sur-
face normal:
M
(
x
)
)=
E
n
. The ratio of the radiant exitance to the (differential)
incident flux therefore comes from combining Equations (4.11) and (4.12):
(
x
(
x
)
·
z
r
e
−
σ
tr
d
r
d
r
+
z
v
e
−
σ
tr
d
v
d
v
)=
α
4
1
d
r
1
d
v
R
d
(
r
σ
tr
+
σ
tr
+
(4.13)
π
σ
tr
=
3
,and
d
r
=
r
2
z
r
and
d
v
=
r
2
σ
a
σ
t
as above,
r
where
=
x
i
−
x
o
+
+
z
v
are the distances from
x
o
to the virtual sources. The value of
α
=
σ
s
σ
t
provides a measure of the relative importance of scattering compared to absorp-
tion. The function
R
d
(
r
)
is sometimes called the diffuse
reflectance profile
of the
surface.
Jensen and his coauthors refer to
R
d
as the
diffuse BSSRDF
, although this is
somewhat imprecise because the actual diffuse (multiple scattering) component of
the BSSRDF has to be multiplied by the Fresnel transmittance terms for incoming
and outgoing light. The true multiple scattering BSSRDF component is thus
1
π
S
d
(
r
,
θ
i
,
θ
o
)=
F
t
(
η
,
θ
i
)
R
d
(
r
)
F
t
(
η
,
θ
o
)
where
θ
o
are the angles of incidence and exitance measured from the sur-
face normal. The factor of 1
θ
i
and
comes from converting radiant exitance to radi-
ance. The Fresnel terms do introduce a directional dependence, but
S
d
remains
rotationally invariant. It is also “separable” as the product of three functions of
single independent variables. The final BSSRDF model is the sum of
S
d
and the
single scattering BSSRDF model developed by Hanrahan and Krueger.
/
π
Verification of the Dipole BSSRDF model.
The multiple scattering dipole
model ultimately depends only on two material properties: the absorption coeffi-
cient
σ
s
. These values can be measured, approx-
imately, for a real surface. The physical accuracy of the model can therefore be
verified by comparing values computed from the dipole model to measurements
from an illuminated surface sample. The authors developed a method for do-
ing this based on photographs of the sample under controlled lighting conditions.
σ
a
and the scattering coefficient
