Graphics Reference
In-Depth Information
As described in earlier chapters, the computation of
L
gd
[
involves inte-
grating of the direct (one-bounce) radiance
L
d
over the hemisphere of directions
above the surface of patch
i
. This integration averages out high frequency direct
illumination over the scene patches, as illustrated in Figure 8.41. The integral
over the partially lit patches has approximately the same value as the integral over
the patches with the partial illumination spread evenly across the patches, i.e., as
if the full illumination were reduced by the coverage factor
c
,
i
]
α
.Thatis,
L
gd
[
c
,
i
]=
α
L
gd
[
c
,
i
]
,
(8.12)
and a similar argument shows the same approximation holds for the multiple
bounce radiance, and therefore
L
g
=
α
L
g
. In other words, the global component
of a partially illuminated image is approximately equal to the global
component of the fully illuminated image scaled by the fraction of the partial
illumination.
Separation formulas.
If pixel
i
is directly lit in the first partially illuminated
image, then
L
d
[
c
,
i
]=
L
d
[
c
,
i
]
and it follows that
L
+
[
c
,
i
]=
L
d
[
c
,
i
]+
α
L
g
[
c
,
i
]
,
(8.13)
L
−
[
,
]=(
−
α
)
[
,
]
.
c
i
1
L
g
c
i
(8.14)
The left sides of these equations are the pixel values in the partially illuminated
images, so the system can be solved for the two unknowns
L
d
[
,
which are the direct and global components of the illumination, respectively. Be-
cause of physical limitations of the projector, deactivated pixels do allow some
light through. If this light has a fraction
b
of the power emitted by the active
pixels, and
c
,
i
]
and
L
g
[
c
,
i
]
1
α
=
2
, Equations (8.13) and (8.14) become
L
g
[
2
,
c
,
i
L
+
[
c
,
i
]=
L
d
[
c
,
i
]+(
1
+
b
)
L
g
[
2
,
c
,
i
L
−
[
c
,
i
]=
bL
d
[
c
,
i
]+(
1
+
b
)
which can likewise be solved for the direct radiance
L
d
[
c
,
i
]
and indirect (global)
radiance
L
g
[
c
,
i
]
. The simplicity of these formulas is truly remarkable.
Conditions for separation.
The averaging assumption of indirect illumina-
tion depends on the high frequency nature of the lighting, and also assumes the
reflectance is not extremely specular. If any of the surfaces were mirrors, a point
on patch
i
might see a direct reflection of one of the checkerboard parts of the
light, while a nearby point on the same patch did not. The separation theorem is
