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2 5
2 5
Reflection Models
As we discussed in Chapter 20, the reflective properties of a surface can be sum-
marized using the BRDF (Nicodemus et al., 1977; Cook & Torrance, 1982). In
this chapter, we discuss some of the most visually important aspects of material
properties and a few fairly simple models that are useful in capturing these prop-
erties. There are many BRDF models in use in graphics, and the models presented
here are meant to give just an idea of non-diffuse BRDFs.
25.1 Real-World Materials
Many real materials have a visible structure at normal viewing distances. For ex-
ample, most carpets have easily visible pile that contributes to appearance. For
our purposes, such structure is not part of the material property but is, instead, part
of the geometric model. Structure whose details are invisible at normal viewing
distances, but which do determine macroscopic material appearance, are part of
the material property. For example, the fibers in paper have a complex appearance
under magnification, but they are blurred together into an homogeneous appear-
ance when viewed at arm’s length. This distinction between microstructure that
is folded into BRDF is somewhat arbitrary and depends on what one defines as
“normal” viewing distance and visual acuity, but the distinction has proven quite
useful in practice.
In this section we define some categories of materials. Later in the chapter,
we present reflection models that target each type of material. In the notes at the
end of the chapter some models that account for more exotic materials are also
discussed.
637
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638 25. Reflection Models
25.1.1 Smooth Dielectrics and Metals
Dielectrics are clear materials that refract light; their basic properties were sum-
marized in Chapter 4. Metals reflect and refract light much like dielectrics, but
they absorb light very, very quickly. Thus, only very thin metal sheets are trans-
parent at all, e.g., the thin gold plating on some glass objects. For a smooth
material, there are only two important properties:
1. How much light is reflected at each incident angle and wavelength.
2. What fraction of light is absorbed as it travels through the material for a
given distance and wavelength.
Figure 25.1. The amount
of light reflected and trans-
mitted by glass varies with
the angle.
The amount of light transmitted is whatever is not reflected (a result of energy
conservation). For a metal, in practice, we can assume all the light is immediately
absorbed. For a dielectric, the fraction is determined by the constant used in
Beer’s Law as discussed in Chapter 4.
Figure 25.2. Light is re-
peatedly reflected and re-
fracted by glass, with the
fractions of energy shown.
The amount of light reflected is determined by the Fresnel equations as dis-
cussed in Chapter 4. These equations are straightforward, but cumbersome. The
main effect of the Fresnel Equations is to increase the reflectance as the incident
angle increases, particularly near grazing angles. This effect works for transmitted
light as well. These ideas are shown diagrammatically in Figure 25.1. Note that
the light is repeatedly reflected and refracted as shown in Figure 25.2. Usually
only one or two of the reflected images is easily visible.
25.1.2 Rough Surfaces
If a metal or dielectric is roughened to a small degree, but not so small that diffrac-
tion occurs, then we can think of it as a surface with microfacets (Cook & Tor-
rance, 1982). Such surfaces behave specularly at a closer distance, but viewed
at a further distance seem to spread the light out in a distribution. For a metal,
an example of this rough surface might be brushed steel, or the “cloudy” side of
most aluminum foil.
For dielectrics, such as a sheet of glass, scratches or other irregular surface
features make the glass blur the reflected and transmitted images that we can
normally see clearly. If the surface is heavily scratched, we call it translucent
rather than transparent. This is a somewhat arbitrary distinction, but it is usually
clear whether we would consider a glass translucent or transparent.
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25.2. Implementing Reflection Models 639
25.1.3 Diffuse Materials
A material is diffuse if it is matte, i.e., not shiny. Many surfaces we see are diffuse,
such as most stones, paper, and unfinished wood. To a first approximation, diffuse
surfaces can be approximated with a Lambertian (constant) BRDF. Real diffuse
materials usually become somewhat specular for grazing angles. This is a subtle
effect, but can be important for realism.
25.1.4 Translucent Materials
Many thin objects, such as leaves and paper, both transmit and reflect light dif-
fusely. For all practical purposes no clear image is transmitted by these objects.
These surfaces can add a hue shift to the transmitted light. For example, red paper
is red because it filters out non-red light for light that penetrates a short distance
into the paper, and then scatters back out. The paper also transmits light with a
red hue because the same mechanisms apply, but the transmitted light makes it all
the way through the paper. One implication of this property is that the transmitted
coefficient should be the same in both directions.
25.1.5 Layered Materials
Figure 25.3. Light hit-
ting a layered surface can
be reflected specularly, or it
can be transmitted and then
scatter diffusely off the sub-
strate.
Many surfaces are composed of “layers” or are dielectrics with embedded parti-
cles that give the surface a diffuse property (Phong, 1975). The surface of such
materials reflects specularly as shown in Figure 25.3, and thus obeys the Fresnel
equations. The light that is transmitted is either absorbed or scattered back up
to the dielectric surface where it may or may not be transmitted. That light that
is transmitted, scattered, and then retransmitted in the opposite direction forms a
diffuse “reflection” component.
Note that the diffuse component also is attenuatedwith the degree of the angle,
because the Fresnel equations cause reflection back into the surface as the angle
increases as shown in Figure 25.4. Thus instead of a constant diffuse BRDF, one
that vanishes near the grazing angle is more appropriate.
Figure 25.4. The light
scattered by the substrate
is less and less likely to
make it out of the surface as
the angle relative to the sur-
face normal increases.
25.2 Implementing Reflection Models
A BRDF model, as described in Section 20.1.6, will produce a rendering which
is more physically based than the rendering we get from point light sources and
Phong-like models. Unfortunately, real BRDFs are typically quite complicated
and cannot be deduced from first principles. Instead, they must either be measured
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640 25. Reflection Models
and directly approximated from raw data, or they must be crudely approximated
in an empirical fashion. The latter empirical strategy is what is usually done, and
the development of such approximate models is still an area of research. This
section discusses several desirable properties of such empirical models.
First, physical constraints imply two properties of a BRDF model. The first
constraint is energy conservation:
for all k
i
, R(k
i
)=
all k
o
ρ(k
i
, k
o
)cosθ
o
dσ
o
≤ 1.
If you send a beam of light at a surface from any direction k
i
, then the total
amount of light reflected over all directions will be at most the incident amount.
The second physical property we expect all BRDFs to have is reciprocity:
for all k
i
, k
o
, ρ(k
i
, k
o
)=ρ(k
o
, k
i
).
Second, we want a clear separation between diffuse and specular components.
The reason for this is that, although there is a mathematically-clean delta function
formulation for ideal specular components, delta functions must be implemented
as special cases in practice. Such special cases are only practical if the BRDF
model clearly indicates what is specular and what is diffuse.
Third, we would like intuitive parameters. For example, one reason the Phong
model has enjoyed such longevity is that its diffuse constant and exponent are
both clearly related to the intuitive properties of the surface, namely surface color
and highlight size.
Finally, we would like the BRDF function to be amenable to Monte Carlo
sampling. Recall from Chapter 14 that an integral can be sampled by N random
points x
i
∼ p where p is defined with the same measure as the integral:
f(x)dμ ≈
1
N
N
j=1
f(x
j
)
p(x
j
)
.
Recall from Section 20.2 that the surface radiance in direction k
o
isgivenbya
transport equation:
L
s
(k
o
)=
all k
i
ρ(k
i
, k
o
)L
f
(k
i
)cosθ
i
dσ
i
.
If we sample directions with pdf p(k
i
) as discussed in Chapter 24, then we can
approximate the surface radiance with samples:
L
s
(k
o
) ≈
1
N
N
j=1
ρ(k
j
, k
o
)L
f
(k
j
)cosθ
j
p(k
j
)
.
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25.3. Specular Reflection Models 641
This approximation will converge for any p that is non-zero where the integrand
is non-zero. However, it will only converge well if the integrand is not very large
relative to p. Ideally, p(k) should be approximately shaped like the integrand
ρ(k
j
, k
o
)L
f
(k
j
)cosθ
j
. In practice, L
f
is complicated, and the best we can ac-
complish is to have p(k) shaped somewhat like ρ(k, k
o
)L
f
(k)cosθ.
For example, if the BRDF is Lambertian, then it is constant and the “ideal”
p(k) is proportional to cos θ. Because the integral of p must be one, we can
deduce the leading constant:
all k with θ<π/2
C cos θdσ =1.
This implies that C =1/π,sowehave
p(k)=
1
π
cos θ.
An acceptably efficient implementation will result as long as p doesn’t get too
small when the integrand is non-zero. Thus, the constant pdf will also suffice:
p(k)=
1
2π
.
This emphasizes that many pdfs may be acceptable for a given BRDF model.
25.3 Specular Reflection Models
For a metal, we typically specify the reflectance at normal incidence R
0
(λ).The
reflectance should vary according to the Fresnel equations, and a good approxi-
mation is given by (Schlick, 1994a)
R(θ, λ)=R
0
(λ)+(1− R
0
(λ)) (1 − cos θ)
5
.
This approximation allows us to just set the normal reflectance of the metal either
from data or by eye.
For a dielectric, the same formula works for reflectance. However, we can set
R
0
(λ) in terms of the refractive index n(λ):
R
0
(λ)=
n(λ) − 1
n(λ)+1
2
.
Typically, n does not vary with wavelength, but for applications where dispersion
is important, n can vary. The refractive indices that are often useful include water
(n =1.33), glass (n =1.4 to n =1.7), and diamond (n =2.4).
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