Graphics Reference
In-Depth Information
passes through the pinhole.
•
P
is within the part of the image plane associated to pixel
ij
.
In this case, the radiance
L
in
(
P
,
•Theay
t
→
P
+
t
v
−
v
)
arriving at
P
is multiplied by the sensor
response associated to this ray.
For a real physical sensor, the sensor response is called
flux responsivity
and
has units of W
−
1
. Since the radiance is in Wm
−
2
sr
−
1
, and we integrate out square
meters and steradians, this makes the measurement
m
ij
unitless.
For a typical sensor,
M
ij
has a form that's independent of the pixel
ij
.For
example, it might have the constant value 1 on a small rectangle representing the
pixel, and zero elsewhere. While the region on which it takes the value 1 changes
with
ij
, the form of the function doesn't change.
When we consider the space of all paths along which light might travel in a
scene, from the point of view of rendering, some are more important than others.
In particular, the paths that end up entering the virtual camera matter more to us
than do those that end up absorbed in some distant and invisible part of the scene.
Thus, the function
M
ij
can help us decide which paths might be worth examining.
Because of this, it's been called the
importance function
[Vea96].
In the introduction, we talked (broadly) about two types of scattering. The first is
mirrorlike: Radiance arriving from some direction
v
i
leaves in a single direction
v
o
, perhaps after attenuation by some factor 0
1. The two main examples
of mirrorlike reflectance are (a) mirrors, and (b) Snell's-law refraction. The second
type of scattering is diffuselike scattering, in which radiance arriving in some
direction is scattered over a whole solid angle of directions (perhaps uniformly
with respect to angle—the Lambertian case—or perhaps nonuniformly). In this
second kind of scattering, the outgoing radiance in a direction
≤
c
≤
v
o
due to radiance
along a single ray in direction
−
v
i
is infinitesimal: To get a nonzero outgoing
radiance, we must sum radiance scattered from a whole solid angle of incoming
directions; the integral in the rendering equation expresses exactly this. For the
integral formulation to apply to the
first
part requires the fiction of “infinite values”
for
f
s
akin to delta functions.
We can treat the process of converting incoming radiance to outgoing radi-
ance as an operation
K
that takes the incoming radiance and scattering functions
as input and produces the outgoing radiance
L
out
=
K
(
L
in
,
f
s
)
. What we've said
in the previous paragraph is that
f
s
should be written as a sum
f
s
+
f
s
, the first
being the “finite part” and the second being the “impulse part”; the rule for com-
bining the field radiance with the finite part to get the surface radiance can then be
legitimately expressed as an integral:
v
o
)=
S
2
K
(
L
in
,
f
s
)(
P
,
f
s
(
P
,
v
o
)
L
in
(
P
,
v
i
,
−
v
i
)
|
v
i
·
n
|
d
v
i
.
(29.16)
In the opaque (i.e., reflection-only) case, the integral would be over a hemisphere,
and we'd write
f
r
instead of
f
s
.
The rule for combining the incoming radiance with the impulse part has the
form
v
o
)=
v
i
∈
H
(
v
o
)
K
(
L
in
,
f
s
)(
P
,
f
s
(
v
o
)
L
in
(
P
,
v
i
,
−
v
i
)
,
(29.17)
