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heavy for high-degree expansions. On the other hand, limiting the number of
terms also limits the accuracy of the expansion. Ramamoorthi and Hanrahan
found one application in which a remarkably low-degree SH representation is
sufficient. As they demonstrate in their paper “An Efficient Representation for
Irradiance Environment Maps” nine SH terms are sufficient for a useful approxi-
mate the irradiance produced by an environment map [Ramamoorthi and Hanra-
han 01a].
As described in Chapter 1, light incident on a Lambertian (diffusely reflecting)
surface is reflected equally in all directions. Reflected radiance from a Lambertian
surface is therefore dependent only on the surface irradiance—the direction of
incident illumination is unimportant. (This property of diffuse reflection is often
exploited in rendering, as discussed in Chapter 2.) Ramamoorthi and Hanrahan
looked to spherical harmonics to approximate surface irradiance. The surface
irradiance at a point is the integral of the cosine-weighted incident radiance. From
Equation (1.6),
E
(
n
)=
L
(
ω
)
cos
θ
d
ω
(7.5)
Ω
(
n
)
where
n
is the surface normal, and
Ω
(
n
)
is the hemisphere above the surface; it is
expressed as a function of
n
to emphasize that it depends on the surface normal.
Now consider the SH expansion of the incident radiance
+
l
m
=
−
l
L
l
,
m
Y
l
,
m
(
θ
,
φ
)
.
If
L
comes from an environment map, then the coefficients
L
l
,
m
can be precom-
puted. The goal is to find the coefficients
E
l
,
m
for the expansion of the surface
irradiance. This is complicated by the surface itself—radiance from below the
surface does not contribute to the irradiance. Assuming for the moment that
E
and
L
are expanded in the same spherical coordinate system, Equation (7.5) can
be extended to an integral over the entire sphere by adding a geometric “clipping”
term
A
∞
l
=
0
L
(
θ
,
φ
)=
(
θ
)=
max
(
cos
θ
,
0
)
:
(
ω
)
(
θ
)
d
ω
.
E
(
n
)=
L
A
(7.6)
Ω
4π
Substituting the SH expansions of
L
and
A
into Equation (7.6) produces an expres-
sion that can be integrated to obtain the SH coefficients
E
l
,
m
for the irradiance.
Were it not for the orthonormal property of the SH coefficients, the nested
double infinite summation that results from the substitution would be of little
practical use. However, expanding the sums results in a collection of terms of the
