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642 25. Reflection Models
Figure 25.5. Renderings of polished tiles using coupled model. These images were pro-
duced using a Monte Carlo path tracer. The sampling distribution for the diffuse term is
cos θ/π.
25.4 Smooth Layered Model
Reflection in matte/specular materials, such as plastics or polished woods, is gov-
erned by Fresnel equations at the surface and by scattering within the subsurface.
An example of this reflection can be seen in the tiles in the renderings in Fig-
ure 25.5. Note that the blurring in the specular reflection is mostly vertical due
to the compression of apparent bump spacing in the view direction. This effect
causes the vertically-streaked reflections seen on lakes on windy days; it can either
be modeled using explicit micro-geometry and a simple smooth-surfacereflection
model or by a more general model that accounts for this asymmetry.
We could use the traditional Lambertian-specular model for the tiles, which
linearly mixes specular and Lambertian terms. In standard radiometric terms, this
can be expressed as
ρ(θ, φ, θ
,φ
λ)=
R
d
(λ)
π
+ R
s
ρ
s
(θ, φ, θ
,φ
),
where R
d
(λ) is the hemispherical reflectance of the matte term, R
s
is the specu-
lar reflectance, and ρ
s
is the normalized specular BRDF (a weighted Dirac delta
function on the sphere). This equation is a simplified version of the BRDF where
R
s
is independent of wavelength. The independenceof wavelength causes a high-
light that is the color of the luminaire, so a polished rather than a metal appearance
will be achieved. Ward (G. J. Ward, 1992) suggests to set R
d
(λ)+R
s
≤ 1 in
order to conserve energy. However, such models with constant R
s
fail to show
the increase in specularity for steep viewing angles. This is the key point: in the
real world the relative proportions of matte and specular appearance change with
the viewing angle.
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25.4. Smooth Layered Model 643
One way to simulate the change in the matte appearance is to explicitly dampen
R
d
(λ) as R
s
increases (Shirley, 1991):
ρ(θ, φ, θ
,φ
,λ)=R
f
(θ)ρ
s
(θ, φ, θ
,φ
)+
R
d
(λ)(1 − R
f
(θ))
π
,
where R
f
(θ) is the Fresnel reflectance for a polish-air interface. The problemwith
this equation is that it is not reciprocal, as can been seen by exchanging θ and θ
;
this changes the value of the matte damping factor because of the multiplication
by (1 − R
f
(θ)). The specular term, a scaled Dirac delta function, is reciprocal,
but this does not make up for the non-reciprocity of the matte term. Although this
BRDF works well, its lack of reciprocity can cause some rendering methods to
have ill-defined solutions.
We now present a model that produces the matte/specular tradeoff while re-
maining reciprocal and energy conserving. Because the key feature of the new
model is that it couples the matte and specular scaling coefficients, it is called a
coupled model (Shirley et al., 1997).
Surfaces which have a glossy appearance are often a clear dielectric, such
as polyurethane or oil, with some subsurface structure. The specular (mirror-
like) component of the reflection is caused by the smooth dielectric surface and
is independent of the structure below this surface. The magnitude of this specular
term is governed by the Fresnel equations.
The light that is not reflected specularly at the surface is transmitted through
the surface. There, either it is absorbed by the subsurface, or it is reflected from
a pigment or a subsurface and transmitted back through the surface of the pol-
ish. This transmitted light forms the matte component of reflection. Since the
matte component can only consist of the light that is transmitted, it will naturally
decrease in total magnitude for increasing angle.
To avoid choosing between physically plausible models and models with good
qualitative behavior over a range of incident angles, note that the Fresnel equa-
tions that account for the specular term, R
f
(θ), are derived directly from the
physics of the dielectric-air interface. Therefore, the problem must lie in the
matte term. We could use a full-blown simulation of subsurface scattering as
implemented, but this technique is both costly and requires detailed knowledge
of subsurface structure, which is usually neither known nor easily measurable.
Instead, we can modify the matte term to be a simple approximation that captures
the important qualitative angular behavior shown in Figure 25.4.
Let us assume that the matte term is not Lambertian, but instead is some other
function that depends only on θ, θ
and λ: ρ
m
(θ, θ
,λ). We discard behavior
that depends on φ or φ
in the interest of simplicity. We try to keep the formu-
las reasonably simple because the physics of the matte term is complicated and
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644 25. Reflection Models
sometimes requires unknown parameters. We expect the matte term to be close to
constant, and roughly rotationally symmetric (He et al., 1992).
An obvious candidate for the matte component ρ
m
(θ, θ
,λ) that will be re-
ciprocal is the separable form kR
m
(λ)f(θ)f(θ
) for some constant k and matte
reflectance parameter R
m
(λ). We could merge k and R
m
(λ) into a single term,
but we choose to keep them separated because this makes it more intuitive to set
R
m
(λ)—which must be between 0 and 1 for all wavelengths. Separable BRDFs
have been shown to have several computational advantages, thus we use the sep-
arable model:
ρ(θ, φ, θ
,φ
,λ)=R
f
(θ)ρ
s
(θ, φ, θ
,φ
)+kR
m
(λ)f(θ)f(θ
).
We know that the matte component can only contain energy not reflected in the
surface (specular) component. This means that for R
m
(λ)=1, the incident
and reflected energy are the same, which suggests the following constraint on the
BRDF for each incident θ and λ:
R
f
(θ)+2πkf(θ)
π
2
0
f(θ
)cosθ
sin θ
dθ
=1. (25.1)
We can see that f(θ) must be proportional to (1 − R
f
(θ)). If we assume that
matte components that absorb some energy have the same directional pattern as
this ideal, we get a BRDF of the form
ρ(θ, φ, θ
,φ
,λ)=R
f
(θ)ρ
s
(θ, φ, θ
,φ
)+kR
m
(λ)[1 − R
f
(θ)][1 − R
f
(θ
)].
We could now insert the full form of the Fresnel equations to get R
f
(θ),andthen
use energy conservation to solve for constraints on k. Instead, we will use the
approximation discussed in Section 25.1.1 We find that
f(θ) ∝ (1 − (1 − cos θ)
5
).
Applying Equation (25.1) yields
k =
21
20π(1 − R
0
)
. (25.2)
The full coupled BRDF is then
ρ(θ, φ, θ
,φ
,λ)=
R
0
+(1− cos θ)
5
(1 − R
0
)
ρ
s
(θ, φ, θ
,φ
)+
kR
m
(λ)
1 − (1 −cos θ)
5
1 − (1 −cos θ
)
5
. (25.3)
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25.5. Rough Layered Model 645
The results of running the coupled model is shown in Figure 25.5. Note that
for the high viewpoint, the specular reflection is almost invisible, but it is clearly
visible in the low-angle photograph image, where the matte behavior is less obvi-
ous.
For reasonable values of refractive indices, R
0
is limited to approximately the
range 0.03 to 0.06 (the value R
0
=0.05 was used for Figure 25.5). The value of
R
s
in a traditional Phong model is harder to choose, because it typically must be
tuned for viewpoint in static images and tuned for a particular camera sequence
for animations. Thus, the coupled model is easier to use in a “hands-off” mode.
25.5 Rough Layered Model
The previous model is fine if the surface is smooth. However, if the surface is
not ideal, some spread is needed in the specular component. An extension of the
coupled model to this case is presented here (Ashikhmin & Shirley, 2000). At
a given point on a surface, the BRDF is a function of two directions, one in the
direction towards the light and one in the direction towards the viewer. We would
like to have a BRDF model that works for “common” surfaces, such as metal and
plastic, and has the following characteristics:
1. plausible. As defined by Lewis (R. R. Lewis, 1994), this refers to the
BRDF obeying energy conservation and reciprocity.
2. anisotropy. The material should model simple anisotropy, such as seen on
brushed metals.
3. intuitive parameters. For material, such as plastics, there should be pa-
rameters R
d
for the substrate and R
s
for the normal specular reflectance as
well as two roughness parameters n
u
and n
v
.
4. Fresnel behavior. Specularity should increase as the incident angle de-
creases.
5. non-Lambertian diffuse term. The material should allow for a diffuse
term, but the component should be non-Lambertian to assure energy con-
servation in the presence of Fresnel behavior.
6. Monte Carlo friendliness. There should be some reasonable probability
density function that allows straightforwardMonte Carlo sample generation
for the BRDF.
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646 25. Reflection Models
Figure 25.6. Geometry of reflection. Note that k
1
, k
2
, and h share a plane, which usually
does not include n.
A BRDF with these properties is a Fresnel-weighted Phong-style cosine lobe
model that is anisotropic.
We again decompose the BRDF into a specular component and a diffuse com-
ponent (Figure 25.6). Accordingly, we write our BRDF as the classical sum of
two parts:
ρ(k
1
, k
2
)=ρ
s
(k
1
, k
2
)+ρ
d
(k
1
, k
2
), (25.4)
where the first term accounts for the specular reflection (this will be presented in
the next section). While it is possible to use the Lambertian BRDF for the diffuse
term ρ
d
(k
1
, k
2
) in our model, we will discuss a better solution in Section 25.5.2
and how to implement the model in Section 25.5.3. Readers who just want to
implement the model should skip to that section.
25.5.1 Anisotropic Specular BRDF
To model the specular behavior, we use a Phong-style specular lobe but make this
lobe anisotropic and incorporate Fresnel behavior while attempting to preserve
the simplicity of the initial mode. This BRDF is
ρ(k
1
, k
2
)=
(n
u
+1)(n
v
+1)
8π
(n · h)
n
u
cos
2
φ+n
v
sin
2
φ
(h · k
i
)max(cos θ
i
, cos θ
o
))
F (k
i
·h) . (25.5)
Again we use Schlick’s approximation to the Fresnel equation:
F (k
i
· h)=R
s
+(1− R
s
)(1 − (k
i
· h))
5
, (25.6)
where R
s
is the material’s reflectance for the normal incidence. Because k
i
·h =
k
o
· h, this form is reciprocal. We have an empirical model whose terms are
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