Graphics Reference
In-Depth Information
the only source of incoming light is the region
R
, this simplifies to an integral
over directions
that point from
Q
to some location in
R
; we'll denote this set of
directions
Ω
Q
. The density is then
d
(
Q
)=
v
∈
Ω
Q
v
L
(
Q
,
−
v
)(
v
·
n
(
Q
))
d
.
(27.10)
v
We know the radiance arriving at
Q
from the inline exercise above. Furthermore,
since the disk
R
appears very small as viewed from
Q
, the vectors
all point in
v
approximately the same direction (namely,
S
(
P
Q
)
, where
P
is the center of
the region
R
, and hence the center of the sphere as well). This means that we can
approximate
d
(
Q
)
well by
−
d
(
Q
)=
m
(Ω
Q
)
(
ρ/π
)cos(
θ
)
m
(Ω)(
v
·
n
(
Q
))
.
(27.11)
The last dot product is approximately 1, because
points almost exactly toward
the center of the large sphere on which
Q
lies. We can rewrite the measure of
Ω
Q
as
(
A
v
r
2
)cos(
θ
)
, where
θ
is the angle between
n
and
Q
/
−
P
, that is, the outgoing
angle, to get
d
(
Q
)=
A
θ
)
.
r
2
(
ρ/π
)cos(
θ
)
m
(Ω)cos(
(27.12)
Everything in this expression is constant (as a function of
Q
) except the final
cosine. When we integrate this power density over the entire hemisphere, to get
the total power arriving at the hemisphere, the result is
Arriving power
=
A
)
m
(Ω)
θ
)
r
2
(
ρ/π
)cos(
θ
cos(
(27.13)
S
+
(
r
)
=
A
r
2
r
2
(
ρ/π
)cos(
θ
)
m
(Ω)
π
(27.14)
=
A
ρ
cos(
θ
)
m
(Ω)
,
(27.15)
which is exactly
times the power that arrived at our Lambertian surface. So the
scattering conserves power only if
ρ
1.
The computational model for a Lambertian surface consists of a normal vector
and a per-wavelength (or per-primary) reflectance value.
The number
ρ ≤
is called the Lambertian reflectance value; it also happens to be
the cosine-weighted integral of the Lambertian BRDF over the upper hemisphere,
and it represents the fraction of arriving power that's reflected by the surface.
While such a notion might be useful for other kinds of scattering as well, for
a general BRDF
f
r
the ratio of leaving power to arriving power depends on the
distribution of arriving power, so reflectance becomes a function of both the BRDF
and the light field. We'll have no use for this more general notion.
ρ
By the way, one explanation of Lambertian reflectance is that it arises in part
from lots of subsurface scattering; in fact, a standard material used as a cali-
bration tool for optics (it has 99% reflectivity over the visible spectrum, with a
very nearly exactly Lambertian BRDF, when measured at a scale of a millime-
ter or so) is
Spectralon
. These reflectance properties are due to its physical
structure: It's a porous thermoplastic that generates many subsurface reflec-
tions in the first few tenths of a millimeter. Thus, what appears macroscopically
to be a Lambertian “surface” reflector is really a complex
sub
surface reflector.
