Graphics Reference
In-Depth Information
result. For example, a sample point on one of the stained glass windows in the
church environment of Figure 6.4 might hit a lead separator in the window, but a
slight change in direction gets the value of direct sunlight coming through a pane.
Global illumination computation really needs an average of the radiance coming
from a region of directions on the sphere. The size, or solid angle, of the region
depends on sampling density and also the particular purpose. For example, the
GI computation at a secondary surface intersection in path tracing need not be
particularly precise, so the sum of the average incident radiance over a few larger
averaged regions suffices. Prefiltering the environment map is one way to perform
this kind of local averaging.
The clustering methods described in the previous section can be viewed as a
prefiltering: the sample for each cluster represents the average value of all the
pixels in the cluster region. However, the value is a representative sample, not a
true average. Furthermore, the clusters are fixed for the environment map. What
is needed is a multiresolution representation that provides local averaging at dif-
ferent scales. The image pyramid method used in optical flow and mip mapping
for rendering antialised textures are examples of multiresolution techniques.
An environment map is a radiance function defined on the sphere. An estab-
lished method for approximating functions on the sphere uses
spherical harmon-
ics
. Spherical harmonics are a generalization of linear harmonics to the sphere.
The term “harmonic” arises from music. A musical tone produced by an instru-
ment at a particular frequency, the
fundamental frequency
, actually consists of a
series of vibrations at integer multiples of the fundamental frequency, which are
known as
harmonics
or
overtones
. Any pure tone can be reproduced as a sum of
weighted harmonics. This corresponds to the mathematical notion of a
Fourier
series
of a periodic function.
A function
L
(
θ
,
φ
)
on the unit sphere can be represented in an analogous
manner as sum of spherical harmonic (SH) basis functions:
∞
l
=
0
+
l
m
=
−
l
L
l
,
m
Y
l
L
(
θ
,
φ
)=
(
θ
,
φ
)
,
(7.2)
where each coefficient
L
l
,
m
is computed by integrating
L
(
θ
,
φ
)
against the corre-
sponding basis function
Y
l
,
m
θ
,
φ
)
over the sphere:
π
2π
Y
l
L
l
,
m
=
L
(
θ
,
φ
)
(
θ
,
φ
)
sin
θ
d
φ
d
θ
.
(7.3)
0
0
The same letter “
L
” is used here for both the function and the SH coefficients
to disambiguate coefficient sets for different function, e.g., radiance
L
and irradi-
ance
E
. Figure 7.9 illustrates the coordinate system used for SH expansions. Note
that
θ
is measured from the positive
z
-axis, while
φ
is the “longitude” coordinate.
