Graphics Reference
In-Depth Information
specular direction was incorrect when the angle of incidence for the light was small.
Torrance and Sparrow ([TorS67]) worked out a theoretical model that was very accu-
rate. Their model predicted the directional and spectral composition of reflected light.
Phong's model often made objects look plastic, and the new model avoided this plastic
look for metals. Mathematically, the model differs from Phong's by replacing the factor
r
s
with another more complicated one.
The Torrance-Sparrow specular reflectance model
([Blin77]):
Hypothesis.
The surface being modeled consists of collection of many small mirror-
like faces (“micro facets”) oriented in a random manner. The specular component is
assumed to come from the reflection in those faces oriented in the direction
H
.
Conclusion.
DGF
r
s
=
NV
,
where
(9.6)
D is the distribution function of the directions of the small faces,
G is the amount by which these faces shadow and mask each other, and
F is from the Fresnel reflection law.
Here is a more detailed description of the factors:
D:
This term measures the proportionate number of facets oriented at an angle
a from the average normal to the surface. Torrance and Sparrow used a
simple Gaussian distribution and the formula
(
)
2
=
(
)
D
exp
a
2
c
,
where c
2
is the standard deviation of the distribution and a=cos
-1
(
N
•
H
).
The c
2
depends on the surface and plays the role of m in Phong's model.
1/(
N
•
V
):
Since intensity is proportional to the number of facets pointing in the direc-
tion
H
, we see more of the surface if the surface is tilted. The increase is
inversely proportional to the cosine of the tilt, leading to this term.
G:
This term counteracts the previous one. Some facets shadow others. There
are three cases if we assume that facets are V-shaped grooves and angles
are equal but opposite about
N
. See Figure 9.5. See [Blin77] for a mathe-
matical justification for the approximation that
(
)
G
=
min
G
,
G
,
G
,
where
(
)
(
)
(
)
1
2
3
G
(1)
= 1
(case of no interference)
(
NH NV
VH
∑
)
(
∑
)
G
(2)
= 2
(case where the reflected light is blocked)
∑
(
)
(
)
NH NL
VH
∑
∑
G
(3)
= 2
(case where the incident light is blocked)
∑
Note that 0 £ G
(i)
£ 1.
