Graphics Reference
In-Depth Information
nonmatching indices vanish in the integration. This property was utilized in the
irradiance environment map technique of Ramamoorthi and Hanrahan described
in Section 7.2.5.
Rotation of SH expansions.
The SH coefficients of a rotated function can
be computed from a linear transformation of the SH coefficients of the original
function. To make this more precise, suppose
R
is a general 3D rotation matrix.
The function
g
can be called a
rotation
of the function
f
by
R
.Both
f
and
g
have SH expansions:
(
s
)=
f
(
R
s
)
n
−
1
l
=
0
l
m
=
−
l
n
−
1
l
=
0
l
m
=
−
l
g
l
Y
l
,
m
(
s
)
.
f
f
l
Y
l
,
m
(
(
s
)=
s
)
,
g
(
s
)=
For each degree
l
, the SH coefficient set
g
l
l
)of
g
can be computed
from a linear transformation of the SH coefficient set
f
l
of
f
. (The actual trans-
formation depends on the matrix
R
, and is rather involved.) The SH expansion of
a rotated function can thus be computed by transforming the SH coefficients. This
property is very useful in PRT, because it means that rotating an SH expansion in
global coordinates to or from a local surface coordinate system can be effected by
matrix multiplications.
(
m
=
−
l
,...,
10.2.2 BRDF representation in spherical harmonics
In terms of the BRDF, the reflection in a particular direction at a surface point is
the integral of the product of the cosine-weighted BRDF and the incident radiance
function, normally over the hemisphere of directions above the surface point. As
was done in the irradiance environment map method of Chapter 7, the integral can
be extended to the entire sphere by introducing a clipping term:
R
(
v
)=
L
(
s
)
f
r
(
s
,
v
)
max
(
cos
θ
,
0
)
d
s
,
Ω
4π
b
(
s
,
v
)
where
f
r
is the BRDF function,
θ
is the angle
s
makes with the surface normal,
and
zeros all
values below the surface and thereby restricts the integral to incident directions
above the surface. The function
b
v
is a (fixed) viewing direction. The clipping term max
(
cos
θ
,
0
)
(
s
,
v
)
thus represents the BRDF, the cosine, and
the surface clipping function.
If both
b
are expanded in spher-
ical harmonics, then by virtue of the SH product integration formula of Equa-
tion (10.8), the computation of reflectance reduces to a multiplication of the SH
coefficients. This major simplification results in very efficient rendering; how-
ever, there are some complications. One problem involves differing coordinate
(
s
,
v
)
and the incident radiance function
L
(
s
)
