Graphics Reference
In-Depth Information
Recall from Chapter 29 the division of light in a scene into two categories (see
Figure 31.6): At a point of a surface, light may be arriving from various points in
the distance, a condition called
field radiance.
This light hits the surface and is
scattered; the resultant outgoing radiance is called
surface radiance.
It's also helpful to divide radiance even further: The surface radiance at a point
P
can be divided into the emitted radiance there (nonzero only at luminaires) and
the reflected radiance. These correspond to the two terms on the right-hand side of
the rendering equation. Dually, the field radiance at
P
can be divided into
direct
lighting
,
L
d
(
P
,
)
at
P
(i.e., radiance emitted by luminaires and traveling through
empty space to
P
), and
indirect lighting,
L
i
(
P
,
v
)
at
P
(i.e., radiance from a point
v
Q
to a point
P
along the ray
P
−
Q
, but that was
not
emitted at
Q
). We'll return to
these terms in the next chapter.
Figure 31.6: Top: The surface
radiance consists of all the light
leaving a point of a surface. Bot-
tom: The field radiance consists
of all the incoming light.
Inline Exercise 31.3:
Suppose that
P
and
Q
are mutually visible. How are
the emitted and reflected radiance at
Q
, in the direction
P
−
Q
, related to the
direct and indirect light at
P
, in the direction
P
−
Q
? Express these in terms of
L
d
,
L
i
,
L
r
, and
L
e
, being careful about signs. Use
=
S
(
P
−
Q
)
in expressing
v
your answer.
Writing the rendering equation in the form of Equation 31.27 makes it clear
that the scattering operator transforms one radiance field (
L
) into another (
T
(
L
)
).
Not only does it do so, but it does so
linearly:
If we compute
T
(
L
1
+
L
2
)
, we get
T
(
L
1
)+
T
(
L
2
)
, and
T
(
rL
)=
rT
(
L
)
for any real number
r
, as you can see from
Equation 31.28. This linearity doesn't arise from some cleverness in the formu-
lation of the rendering equation. It's a physically observable property, commonly
called the principle of superposition in physics, and it's extremely fortunate, for
those hoping to solve the rendering equation, that it holds. Later, in Chapter 35,
we'll see this principle of superposition applying to forces and velocities and other
things that arise in physically based animations, and once again it will simplify our
work considerably.
We can rewrite the rendering equation in the form
T
(
L
)=
L
e
L
−
(31.29)
or even
T
)
L
=
L
e
,
(
I
−
(31.30)
where
I
denotes the identity operator: It transforms
L
into
L
, and we've used
TL
to denote the application of the operator
T
to the radiance field
L
.
Much of the remainder of this chapter describes approaches to solving this
equation. Remember as we examine such approaches that
L
e
, the light emitted
by each light source, is given as an input, as is the bidirectional reflectance dis-
tribution function (BRDF) at each surface point, so that the operator
T
can be
computed. The unknown is the radiance field,
L
.
The similarity of this formulation to the way eigenvalue problems are
described in linear algebra is no coincidence. We'll use many of the same tech-
niques that you saw in studying eigenvalues as we look at solving the rendering
equation.
