Graphics Reference
In-Depth Information
→
θ
|x x'|
→
n
→
θ '
Figure 2.2
The geometry term for the rendering equation.
the distance to
x
and projective foreshortening. The latter can be expressed as a
geometry term
as suggested by Figure 2.2:
θ
cos
θ
cos
x
)=
G
(
x
,
(2.3)
x
2
x
−
θ
=
·
ω
)
x
)
where cos
(
n
.The
V
(
x
,
term is the visibility function described in
is 1 if
x
and
x
are visible to each other, and 0 otherwise. In
this form, the rendering equation thus becomes
x
)
Chapter 1:
V
(
x
,
dA
(2.4)
where the integral is taken over all surfaces
S
in the environment, and the notation
x
x
→
x
→
x
,
x
→
x
,
x
→
x
,
x
)
x
,
x
)
x
→
x
)
(
)=
(
)+
(
)
(
(
(
L
x
L
e
x
f
r
x
V
G
L
S
y
indicates the direction of light transport is from point
x
to point
y
.
Even for the simplest environments, the rendering equation is too complicated
to solve in the traditional sense of finding a mathematical formula for the radiance
L
. Instead, “solving” the equation means to find a sufficiently accurate numerical
approximation. Integral equations are well studied in mathematics, but computer
graphics presents its own challenges. Numeric accuracy is often less important
than an approximation that is visually plausible, and accurate numerical solutions
can have visually unacceptable artifacts. There are two basic methods for solv-
ing the rendering equation: the radiosity method and Monte Carlo path tracing
(MCPT). Radiosity works by splitting the environment into many small sections,
while Monte Carlo path tracing works by tracing light paths from the scene in
random directions.
→
2.2 The Radiosity Method
The radiosity method employs the
finite element method
to solve the rendering
equation. The basic idea of the finite element method is to divide up a system
