Graphics Reference
In-Depth Information
The model proposed in the paper “Non-Linear Approximation of Reflectance
Functions” by Eric P. F. Lafortune, Sing-Choong Foo, Kenneth E. Torrance, and
Donald P. Greenberg is an example of this approach [Lafortune et al. 97]. This
model, which has come to be known as the
Lafortune BRDF model
, combines
multiple cosine lobes with different directions, sizes, and widths. It handles many
characteristics of reflection.
A Lafortune BRDF consists of a sum of generalized cosine lobes, each of
which is given by the matrix expression
⎤
⎡
⎤
⎡
⎤
⎞
⎛
⎝
⎡
⎣
n
T
ω
C
x
ω
rx
ω
ix
⎦
⎣
⎦
⎣
⎦
⎠
(
ω
,
ω
)=
.
S
C
y
ω
(8.7)
i
r
ry
ω
rz
iy
ω
iz
C
z
The diagonal elements
C
x
,
C
y
,and
C
z
are parameters that control the shape of the
lobe; the value of the exponent
n
controls the specularity, just as in the Phong
model. The actual Lafortune BRDF model uses a weighted sum of lobes having
the form of Equation (8.7), with varying weights and exponent.
If
C
denotes the diagonal matrix, Equation (8.7) can be expressed as
i
C
ω
C
T
(
ω
,
ω
)=
ω
=(
ω
)
·
ω
,
S
i
r
r
i
r
(
C
is symmetric, so
C
T
C
). In other words, a Lafortune lobe can be expressed
as the dot product of the incoming direction
=
ω
i
, transformed by
C
, with the out-
going vector
ω
r
.If
C
is the matrix that effects a reflection in the surface normal,
C
T
C
T
as in the
Phong model. In this sense, a Lafortune lobe is a proper generalization of a Phong
C
z
specify a more general specular direction. With
C
x
=
then
(
ω
i
)
is the specular direction, and therefore
(
ω
i
)
·
ω
r
=
cos
α
C
y
=
−
1and
C
z
=
1, the
Width of the lobe :
n
Axial direction
(
C
x
ix
C
y
iy
C
z
iz
)
,
,
→
→
i
r
(
weight
)
Figure 8.14
A specular lobe in the Lafortune. The lobe is defined in terms of the angle that the outgoing
direction ω
r
makes with a general linear transformation of the incoming direction ω
i
.A
Phong lobe is a Lafortune lobe where the transformation is just the reflection in the surface
normal.
Length of the lobe :
ρ
s
