Graphics Reference
In-Depth Information
→
Basis function
B
i
(
s
)
→
s
Incidence light direction
v
View direction
V
(
s
) Visibility function
f
(
s
,
v
) Reflectance function (BRDF × cosine)
L
(
s
)
Incident radiance
R
(
v
) Outgoing light toward the viewpoint
(rendering result)
Unoccluded light
→
→
→
→
→
Outgoing light
R
(
v
)
→
→
Light reflected from
object via
f
(
s, v
)
Incoming
light
L
(
s
)
→→
→
Figure 10.4
Geometry for radiance transfer by reflection.
fer function
. The transfer function is necessarily linear, which means that it can
be represented by a matrix. In other words, the SH coefficients of the outgoing
radiance are a linear function of the SH coefficients of the incident radiance.
To make this more concrete, consider the BRDF model of surface reflection.
If
L
is the incident radiance function, the reflected (outgoing) radiance
R
at a
point on the surface is given by
(
s
)
R
(
v
)=
L
(
s
)
V
(
s
)
f
(
s
,
v
)
d
s
,
(10.3)
Ω
4π
where
v
is an outgoing direction (i.e., a viewing direction),
V
(
s
)
is the visibility
function (which is 1 if light coming from direction
s
is visible at the point and
0 otherwise), and
f
is the cosine-weighted BRDF function, i.e., the BRDF
function times the cosine of the incidence angle
(
s
,
v
)
. Figure 10.4 illustrates the
geometry. The integral is performed over then entire sphere; the visibility function
V
θ
s
is zero if the direction
s
lies below the surface.
If the incident radiance
L
(
s
)
is represented as a spherical harmonic expansion,
Equation (10.3) becomes
n
2
i
=
1
i
y
i
(
s
)
V
i
=
1
i
n
2
R
(
v
)=
(
s
)
f
(
s
,
v
)
d
s
=
y
i
(
s
)
V
(
s
)
f
(
s
,
v
)
d
s
.
(10.4)
Ω
4π
Ω
4π
The integral on the right side of Equation (10.4) depends only on the geometry
of the object, and can therefore be precomputed. If
t
i
(
)
denotes the value of the
integral for some SH index
i
and for a specific viewing vector
v
v
, then the reflection
can be expressed as
n
2
i
=
1
i
t
i
(
v
)=
·
t
(
v
)
,
R
(
v
)=
