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form
L l , m A l , m Y l , m ( θ , φ )
Y l , m ( θ , φ ) ,
m .Further-
more, the nonzero terms reduce to just L l , m A l , m . In other words, integration of a
product of SH expansions amounts to summing the products of their coefficients.
This property is used extensively in the precomputed radiance transfer methods
described in Chapter 10. For the surface irradiance problem, the SH coefficients
for the clipping term A
l and m
=
=
which, by virtue of Equation (7.4), are zero unless l
( θ )
, which does not depend on
φ
, are only nonzero if
m
=
0. Consequently,
l = 0 A l Y l , m ( θ , φ )
A
( θ )=
( SHlm
( θ , φ )
does not actually depend on
φ
). The SH expansion of surface irra-
diance actually becomes
4
l = 0
l
m = l
+
π
E
( θ , φ )=
1 A l L l , m Y l , m ( θ , φ ) .
(7.7)
2 l
+
This is indeed a remarkable result, as it reduces integration of the environment
map to a direct multiplication of SH coefficients.
The L l , m coefficients in Equation (7.7) can be precomputed for a particular
environment map. The A l coefficients depend only on l ; Ramamoorthi and Han-
rahan derived a direct formula for A l in terms of l . Figure 7.10 shows a plot of
the A l as a function of l . Remarkably, the values drop to zero very quickly—only
three terms are needed for a reasonable approximation. This means l can be lim-
ited to 2 in Equation (7.7), and the resulting irradiance approximation consists of
only nine terms.
The derivation above assumes the irradiance is computed in the same global
spherical coordinates
( θ , φ )
, and the surface normal is implicitly assumed to have
θ =
0. In other words, the irradiance is computed for a specific surface, the
“equator” plane of the global spherical coordinate system. Surfaces in the scene
to be rendered have arbitrary surface normals. Suppose a scene surface has (unit)
normal
n . The derivation above still applies, but in a local spherical coordinate
system for which
refers to
this local coordinate system, so the values of A l are unchanged. However, the
radiance coefficients L l , m have to be recomputed for this local coordinate system,
which would normally require filtering the environment map again. Doing this
for each surface normal direction defeats the purpose.
Fortunately, there is a better way. The spherical coordinate systems each cor-
respond to a rectangular coordinate system. This is illustrated in Figure 7.11, in
n is the “north pole.” The clipping function A
( θ )
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