Graphics Reference
In-Depth Information
z
L
(
θ
,
φ
)
Y lm
(
θ
,
φ
)
Ylm
(
θ
,
φ
)
θ
y
φ
x
Basis function
Environment map
Figure 7.9
Approximation of an environment map using spherical harmonic basis functions.
The spherical harmonic basis functions
Y
l
, which are defined below, are com-
plex functions on the sphere. For each
l
=
0
,
1
,...
there is a basis function for each
m
=
−
l
,...,
l
. The SH basis functions are an
orthonormal basis
, which means
1f
l
π
2π
l
and
m
m
,
=
=
∗
Y
m
Y
l
(
θ
,
φ
)
(
θ
,
φ
)
sin
θ
d
φ
d
θ
=
(7.4)
l
0oh rw e
,
0
0
where the
∗
superscript means complex conjugation. The formula for the coeffi-
cients
L
l
,
m
given in Equation (7.3) depends on this property.
3
Spherical harmonic basis functions.
The spherical harmonic basis func-
tions
Y
l
=
,
,...
=
−
,−
+
,...,
are defined for each
l
0
1
and for each
m
l
l
1
l
by
K
l
P
|
m
|
Y
l
(
θ
,
φ
)=
(
cos
θ
)(
cos
m
φ
+
i
sin
m
φ
)
,
−
l
≤
m
≤
l
l
3
Functions on the sphere form a general vector space; i.e., each function on the sphere is a “vector”
in this space. Integration of one function against another on the sphere is the “dot product” for this
space. The SH basis functions are analogous to the unit direction vectors in 3D space. Equation (7.4)
essentially states that the SH basis functions are mutually orthogonal (the dot product of two vectors
is zero if, and only if, the vectors are orthogonal). The fact that the dot product of a basis function
with itself is 1 means that the basis functions are “unit vectors.”
