Graphics Reference
In-Depth Information
→
Emitted
L
e
(
x
,
ω
)
→
S
x
L
i
(
x
,
ω
)
→
Reflected
L
(
x''
→
x'
)
x''
→
L
r
(
x
,
ω
)
L
(
x'
→
x
)
x
x
x'
(a)
(b)
Figure 2.1
Geometry for the rendering equation. (a) Radiance coming off a surface is the sum of the
surface emission (top) and the reflected light (bottom). (b) The reflected light can be taken
from all visible surface points in the scene rather than the hemisphere of directions.
The rendering equation models light transport between object surfaces. The
radiance
L
o
of the light coming off a surface at a point in a particular direction is
the sum of the radiance
L
e
emitted from the surface at the point and the radiance
computed by integrating the incoming radiance over the hemisphere of directions
above the surface point against the BRDF, as in Equation (1.13). The outgoing
radiance is therefore
,
ω
)=
,
ω
)+
,
ω
)
L
o
(
x
L
e
(
x
L
r
(
x
(2.1)
,
ω
,
ω
)
,
ω
)(
·
ω
)
ω
.
,
ω
)+
=
L
e
(
x
f
r
(
x
L
i
(
x
n
d
(2.2)
Ω
The emission is intrinsic to the surface; note that emissive surfaces also reflect
light. Equation (2.2), which is one form of the
rendering equation
, is deceptively
simple in its appearance. The incoming light
L
i
(
ω
may be light emitted from a light source, or reflected from another surface; in
fact, it includes light bouncing back from the surface point
x
itself. In other words,
L
i
(
,
ω
)
x
at
x
from a direction
,
ω
)
,
ω
)
,
ω
)
, ad infinitum.
This complex and intricate interdependence is what makes GI such a difficult
problem.
Because the incoming light comes from other surface points, either by reflec-
tion or direct emission, Equation (2.2) can be reformulated as a summation over
two extra terms: one to encode the visibility of the other surfaces (light from
x
cannot be reflected at
x
if
x
is not visible from
x
), and another to account for
x
depends on
L
r
(
x
, which in turns depends on
L
i
(
x
