Graphics Reference
In-Depth Information
The algorithmic representation of the ray-casting function and the associated
visibility function
—see Exercise 29.1—is the subject of Chapters 36 and 37, but
you've already seen basic examples in Chapter 15.
With our assumptions clearly characterized, we can now analyze the light
transport in our scene.
Consider once again the reflectance equation, Equation 26.80, rewritten without
time or wavelength dependence:
v
o
)=
L
ref
(
P
,
L
(
P
,
−
v
i
)
f
r
(
P
,
v
i
,
v
o
)(
v
i
·
n
P
)
d
v
i
.
(29.3)
v
i
∈
S
+
(
p
)
This expresses the radiance reflected at point
P
in
outward
direction
v
o
in
terms of the light arriving at
P
in other directions.
If
P
happens to be a point of a luminaire, light may also leave
P
in direction
v
o
because it is
emitted
at
P
rather than because it is reflected from there; that is,
v
o
)=
L
e
(
P
,
v
o
)+
L
ref
(
P
,
L
(
P
,
v
o
)
(29.4)
v
o
)+
=
L
e
(
P
,
L
(
P
,
−
v
i
)
f
r
(
P
,
v
o
)(
v
i
·
n
P
)
d
v
i
,
v
i
.
(29.5)
v
i
∈
S
+
(
p
)
This is a basic version (e.g., it handles only reflection) of the
rendering equa-
tion,
which characterizes the function
L
, given the functions
L
e
and
f
r
. It was first
described in computer graphics by Kajiya [Kaj86] and Immel et al. [ICG86], in
slightly differing forms. It is completely analogous to similar equations developed
in subjects like radiative transfer. While we'll concentrate on Kajiya's descrip-
tion and derivation in subsequent chapters, the form presented by Immel et al. is
particularly well suited to the “sampling” needed in Monte Carlo rendering.
You'll notice that the unknown radiance function
L
appears on both sides of
the equation, once under an integral, just as the unknown function
h
appears on
both sides of the differential equation
h
(
x
)=
2
h
(
x
−
1
)
,
(29.6)
once within a derivative. Equation 29.4 is called an
integral equation,
and solving
such an equation is generally more difficult than solving a differential equation.
The next chapter discusses various approaches to finding approximate solutions.
Equation 29.4 expresses the radiance function
L
, considering both the radiance
leaving
the point
P
and the radiance arriving there. Arvo [Arv95] calls the first
of these the
surface radiance
and the second the
field radiance.
The rendering
equation tells us how to compute
surface
radiance from field radiance, because
it's restricted to the case where
0. But to evaluate the right-hand side,
we must know how to compute the
field
radiance as well.
The idea that “closes the loop” in this equation is that any light arriving at a
point
P
, traveling in direction
v
o
·
n
P
>
−
v
i
,musthave
departed
from some other point
∈
M
−
v
i
. The point
Q
must be the point visible from
P
Q
traveling in direction
in direction
v
i
. These observations allow us to write the
transport equation:
L
(
P
,
−
v
i
)=
L
(
R
(
P
,
v
i
)
,
−
v
i
)
,
(29.7)
for any
v
i
with
v
i
·
n
P
>
0.
