Graphics Reference
In-Depth Information
where
t
n
2
.Inotherwords,
the reflection for a particular direction reduces to a dot product of the light vector
and a
transfer vector
that represents the reflection computation.
If the reflection is diffuse, the result is independent of the viewing direction
(
v
)
denotes the vector of the
t
i
values for
i
=
1
,
2
,...,
v
,
and the entire reflection computation therefore reduces to the vector dot product
R
=
l
·
t
. If the reflection is not diffuse, then it depends on the viewing direc-
tion. The reflected radiance
R
(
v
)
can itself be expressed as a spherical harmonic
expansion
m
2
j
=
0
r
j
y
j
(
v
)
where each coefficient
r
j
is computed by integrating
R
against the SH basis func-
tion
y
j
.Thatis,
R
(
v
)
≈
i
=
1
i
n
2
r
j
=
y
j
(
v
)
y
i
(
s
)
V
(
s
)
f
(
s
,
v
)
d
sd
v
(10.5)
Ω
4π
Ω
4π
i
=
1
i
n
2
=
(
)
(
)
(
)
(
,
)
,
y
j
v
y
i
s
V
s
f
s
v
d
sd
v
(10.6)
Ω
4π
Ω
4π
and because the double integral depends only on the geometry, it can be precom-
puted. If
t
ij
denotes the integral evaluated for a particular
i
and
j
,then
n
2
i
=
1
t
ij
i
.
r
j
=
(10.7)
Regarding the values of
t
ij
as elements of a matrix
T
and the SH coefficients
r
j
as a light vector
r
of the reflected light, the reflection computation amounts to the
matrix multiplication
T
l
=
.
r
This formalizes the notion that the outgoing light vector is a linear function of the
incident light vector. The matrix
T
is the
transfer matrix
, a particular representa-
tion of a general
transfer function
.
The derivation of the transfer matrix above assumes a simple BRDF
representation of surface reflectance. It also includes shadowing by way of the
visibility function
V
. More effects can be included. For example, the transfer
function can account for interreflection between nearby parts of the object, i.e.,
any light that reflects from the surface at the point hits another nearby point on the
object, and bounces back to be reflected in another direction. Multiple bounces
can also be included. In general, these effects of occlusion and interreflection are
known as
self-transfer
.
