Graphics Reference
In-Depth Information
The Torrance-Sparrow model, like the Phong model, combines a diffuse term
with the glossy term, and takes into account the Fresnel equations in adjusting the
reflectivity as a function of incoming-light angle. The distribution of microfacet
slopes is assumed to be exponential: The probability density at slope
α
is propor-
tional to
exp(
2
)
, where the constant
c
is a parameter of the model.
The parameters for the model are the index of refraction (which is wavelength-
dependent), the slope-distribution constant
c
, and the diffuse and glossy constants
k
d
and
k
g
, although they use the ratio
g
=
k
g
/
−
c
2
α
k
d
as well, using a complex-number
version of the index of refraction to represent both the ordinary index of refraction
and the absorption. Torrance and Sparrow report that
c
=
0.05 works well, and
approximately agrees with experimental observations of
c
=
0.035 and
c
=
0.046
for ground glass surfaces. They plot the predicted results for aluminum and mag-
nesium oxide, and show good agreement with the observed data.
Cook and Torrance [CT82] developed an extension of the Torrance-Sparrow
model that explicitly took into account the different nature of diffuse reflection
(usually involving subsurface scattering, or multiple scattering from a sufficiently
rough surface) and specular reflection, especially from metals, which is an almost
entirely surface-based phenomenon. Since specular reflection from microfacets
is again used to model glossy reflection, anything said about specular reflection
here also applies to glossy reflection. The specular-diffuse difference means that
the diffusely reflected and specularly reflected lights from a single surface may
have quite different spectral distributions; plastics, for instance, tend to specularly
reflect light with approximately the same spectral distribution as the illuminant,
while metals (think of copper and gold) tend to have substantial spectral variation
in reflectivity, so the reflected light “takes on the color of the material.”
As with the Phong model, there are three parts: an ambient, a diffuse, and a
specular term. The ambient term is considered to be an average of the diffuse and
specular effects due to light coming from many different directions in the scene,
which is assumed to be uniform. Because of this, the color of the ambient term
is a combination of the diffuse and specular colors (where we are using the term
“color” as a shorthand for “spectral distribution”). The diffuse term is assumed
Lambertian. The complete model, ignoring the ambient term, has the form
f
(
v
i
,
v
o
,
λ
)=
sR
s
(
v
i
,
v
o
,
λ
)+
dR
d
(
λ
)
, where
(27.35)
)=
F
(
v
i
,
λ
)
DG
R
s
(
v
i
,
v
o
,
λ
·
v
o
)
,
(27.36)
π
π
(
n
·
v
i
)(
n
where
s
and
d
are the amounts of specular and diffuse reflectivity, respectively,
R
s
and
R
d
are the (spectral) BRDFs for specular and diffuse reflection, respectively,
F
is the Fresnel term,
D
is the microfacet slope distribution, and
G
is a geomet-
ric attenuation factor that accounts for masking and shadowing of facets; we've
omitted the arguments for both
D
and
G
for now. The entire expression is evalu-
ated at a point
P
of a surface with normal vector
n
(
P
)
, for which we'll write
n
for
simplicity.
It's easiest to express the geometric term in terms of the half-vector
h
=
S
(
v
o
+
v
i
)
.
(27.37)
