Graphics Reference
In-Depth Information
where
H
(
v
o
)
is the (finite) set of directions
v
i
that undergo mirrorlike scattering
v
o
, and
f
s
at
P
to direction
v
o
)
denotes the constant by which incoming radi-
ance is scaled to produce outgoing radiance, which we've previously called the
magnitude of the impulse.
There's a slight subtlety here. The form given in Equation 29.17 is only
valid if
L
in
is continuous at
(
P
,
(
v
i
,
−
v
i
)
. Otherwise, the value must be determined by
a limit. As far as we know, this detail is generally ignored in graphics, perhaps
because almost all physical processes involve convolution, and convolution tends
to produce continuous functions. That is to say, perhaps our pure mathematically
modeled
L
in
has a discontinuity, but if we were in the real world, things like vol-
umetric scattering would tend to make it actually continuous. Any picture whose
appearance
depended
on the discontinuity of
L
in
would be nonphysical anyhow!
30% mirror
n
P
50% Lambertian
Consider the situation in Figure 29.3. The surface is 50% Lambertian and 30%
a mirror reflector (so 20% of arriving light is absorbed). We'll compute the light
reflected from the point
P
=(
0, 0, 0
)
under two different lighting conditions.
P
Figure 29.3: Scattering exam-
ple: a 50% matte reflector that's
also 30% mirror-reflective, but
not transmissive.
1. The surface is bathed in light from all points of the postive-
x
hemisphere
(see Figure 29.4). The radiance
L
in
(
P
,
−
v
i
)
is 6 W m
−
2
sr
−
1
for all
v
i
v
i
·
100
T
n
P
with
≥
0.
2. The surface is illuminated by a uniformly radiating sphere (see Fig-
ure 29.5) of radius
r
P
<
1 at position
Q
=(
1, 1, 0
)
; the total power of
the luminaire is 10 W.
We'll examine the behavior of the second case as
r
0 as well.
In each case, we'll compute the reflected radiance in the direction
→
Figure 29.4: Light arrives from
everywhere
v
o
=
S
(
−
110
T
)
.
in
the
right
half-
space.
We start with situation 1. Let's begin by computing the diffusely reflected
light. This is
5
Q
(1, 1, 0)
v
o
)=
S
+
(
P
)
L
ref, 0
(
P
,
f
s
(
P
,
v
o
,
v
i
)
L
(
P
,
−
v
i
)(
v
i
·
n
P
)
d
v
i
.
(29.18)
n
P
Rewriting
v
i
in polar coordinates,
(
x
,
y
,
z
) = (cos
θ
sin
φ
,
cos
φ
,
sin
θ
sin
φ
)
,the
radiance field
L
(
P
,
−
v
i
)
is zero unless
x
>
0, that is,
cos
θ>
0, so we can restrict
P
−
2
≤ θ ≤
2
. Similarly, since we're only considering reflectance, we can
restrict to 0
to
Figure 29.5: The point P is illu-
minated by a tiny, uniformly emit-
ting,
≤ φ ≤
2
. Thus, our integral becomes
spherical
lamp
at
loca-
v
o
)=
π/
2
−π/
2
π/
2
tion Q.
L
ref, 0
(
P
,
f
s
(
P
,
v
o
,
v
i
)
L
(
P
,
−
v
i
)(
v
i
·
n
P
)sin
φ
d
φ
d
θ
,
0
(29.19)
where the factor of
sin
comes from the change to polar coordinates. Within this
restricted domain, the value of
L
is the constant 6, so the integral becomes
φ
v
o
)=
π/
2
−π/
2
π/
2
L
ref, 0
(
P
,
f
s
(
P
,
v
o
,
v
i
)
6
cos
φ
sin
φ
d
φ
d
θ
,
(29.20)
0
