Graphics Reference
In-Depth Information
into a collection of discrete cells or “elements” and then deal with the interaction
of these elements. For global illumination, each surface of an environment is
split into small
patches
, and the solution is computed by modeling the effect of
light transfer between these patches. The finite element method was developed in
the middle of the twentieth century to solve a variety of differential and integral
equations, and continues to be used in a diverse range of engineering problems.
Its use in computer graphics extends beyond radiosity to the simulation of fluids
and deformable bodies.
In the basic radiosity method, the surface are assumed to be Lambertian: all
reflection is assumed to be perfectly diffuse. Each surface BRDF is then constant,
so Equation (2.4) simplifies to
x
)
x
,
x
)
x
)
dA
B
(
x
)=
B
e
(
x
)+
f
r
,
d
(
x
)
B
(
V
(
G
(
x
,
(2.5)
S
)+
ρ
d
(
x
)
x
)
x
)
x
)
dA
=
B
e
(
x
B
(
V
(
x
,
G
(
x
,
(2.6)
π
S
where
ρ
d
is the surface albedo at
x
, and the radiance
L
is replaced by the radiant
exitance (radiosity)
B
. Each patch is assumed to have constant illumination
(and emission, in the case of a light source) and therefore the radiant exitance
of each patch is the albedo of the patch times the irradiance, plus any emission.
The value of the integral in Equation (2.6) is therefore constant for each pair of
patches. This allows the rendering equation to be
discretized
into a system of
linear equations with one equation for each surface patch. The radiant exitance of
patch
i
is
(
x
)
N
1
B
j
F
ij
.
=
+
ρ
d
B
i
B
e
,
i
(2.7)
The constant value
F
ij
is the fraction of the light leaving patch
i
that arrives at
patch
j
;itiscalledthe
form factor
between the patches.
Because the radiant exitance is constant on each surface patch, so is the out-
going radiance. An image rendered directly with the radiosity solution comes out
looking like the scene is made from patchwork quilts. The discretization is visu-
ally unacceptable, even though the values may be numerically close to the correct
solution. In practice, rendering an image from a radiosity solution requires an
extra processing step known as
radiosity reconstruction
. A simple reconstruction
method is to interpolate or otherwise smooth out the radiosity values between
patches. Another approach is to perform a
final gather
at each point by sam-
pling the hemisphere of directions using ray tracing: the radiance in each ray
comes directly from the radiant exitance of the surface patch it hits. Radiosity
