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236 7. Advanced Shading
Insulator R
F
(0
◦
) (Linear) R
F
(0
◦
) (sRGB) Color
Water 0.02,0.02,0.02 0.15,0.15,0.15
Plastic / Glass (Low) 0.03,0.03,0.03 0.21,0.21,0.21
Plastic High 0.05,0.05,0.05 0.24,0.24,0.24
Glass (High) / Ruby 0.08,0.08,0.08 0.31,0.31,0.31
Diamond 0.17,0.17,0.17 0.45,0.45,0.45
Table 7.3. Values of R
F
(0
◦
) for external reflection from various insulators. Various
types of plastic have values ranging from 0.03 to 0.05. There is a greater variance
among different types of glass, ranging from 0.03 to as high as 0.08. Most common
insulators have values close to 0.05—diamond has an unusually high value of R
F
(0
◦
)
and is not typical. Recall that these are specular colors; for example, ruby’s red color
results from absorption inside the substance and is unrelated to its Fresnel reflectance.
Metal R
F
(0
◦
) (Linear) R
F
(0
◦
) (sRGB) Color
Gold 1.00,0.71,0.29 1.00,0.86,0.57
Silver 0.95,0.93,0.88 0.98,0.97,0.95
Copper 0.95,0.64,0.54 0.98,0.82,0.76
Iron 0.56,0.57,0.58 0.77,0.78,0.78
Aluminum 0.91,0.92,0.92 0.96,0.96,0.97
Table 7.4. Values of R
F
(0
◦
) for external reflection from various metals. Note that the
actual value for gold is slightly outside the sRGB gamut; the RGB value shown is after
gamut mapping.
Metals immediately absorb any transmitted light, so they do not exhibit
any subsurface scattering or transparency [192, 193].
Since reflectance depends on the refractive indices of the substances on
both sides of the interface, objects will have lower reflectance when viewed
under water (water has a higher refractive index than air), so they will
appear slightly darker. Other factors also contribute to the darkening of
wet materials both above and under water [605].
Internal Reflection
Although external reflection is the most commonly encountered case in
rendering, internal reflection is sometimes important as well. Internal re-
flection occurs when light is traveling in the interior of a transparent object
and encounters the object’s surface “from the inside” (see Figure 7.21).
The differences between internal and external reflection are due to the
fact that the refractive index for the object’s interior is higher than that
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7.5. BRDF Theory 237
n
n
1
n
2
l
t
-n
r
i
Figure 7.21. Internal reflection at a planar surface.
of air. External reflection is a transition from a low refractive index to a
higher one; internal reflection is the opposite case. Snell’s Law indicates
that in the case of internal reflection that sin θ
t
> sin θ
i
(since they are both
between 0
◦
and 90
◦
, this also implies θ
t
>θ
i
, as seen in Figure 7.21). In
the case of external reflection the opposite is true—compare to Figure 7.18
on page 231. This is key to understanding the differences between internal
and external reflection. In external reflection, a valid (smaller) value of
sin θ
t
exists for every possible value of sin θ
i
between 0 and 1. The same
is not true for internal reflection. For values of θ
i
greater than a critical
angle θ
c
, Snell’s Law implies that sin θ
t
> 1, which is impossible. What
happens in reality is that there is no θ
t
—when θ
i
>θ
c
,notransmission
occurs, and all the incoming light is reflected. This phenomenon is known
as total internal reflection.
The Fresnel equations are symmetrical, in the sense that the incoming
and transmission vectors can be switched and the reflectance remains the
same. In combination with Snell’s Law, this implies that the R
F
(θ
i
)curve
for internal reflection will resemble a “compressed” version of the curve for
external reflection. The value of R
F
(0
◦
) is the same for both cases, and
the internal reflection curve reaches perfect reflectance at θ
c
instead of at
90
◦
(see Figure 7.22).
Figure 7.22 shows that on average, reflectance is higher in the case of
internal reflection. This is why (for example) air bubbles seen underwater
have a highly reflective, silvery appearance.
Since internal reflection only occurs in insulators (metals and semicon-
ductors quickly absorb any light propagating inside them [192, 193]), and
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238 7. Advanced Shading
θ
c
glass (external reflection)
glass (internal reflection)
R
F
0
0.2
0.4
0.6
0.8
0 102030405060708090
Figure 7.22. Comparison of internal and external reflectance curves at a glass-air inter-
face.
insulators have real-valued refractive indices, computation of the critical
angle from the refractive indices or from R
F
(0
◦
) is straightforward:
sin θ
c
=
n
2
n
1
=
1 −
R
F
(0
◦
)
1+
R
F
(0
◦
)
. (7.35)
The Schlick approximation shown in Equation 7.33 is correct for ex-
ternal reflection. It can be used for internal reflection if the transmission
angle θ
t
is substituted for θ
i
. If the transmission direction vector t has
been computed (e.g., for rendering refractions—see Section 9.5), it can be
used for finding θ
t
. Otherwise Snell’s law could be used to compute θ
t
from
θ
i
, but that is expensive and requires the index of refraction, which may
not be available. Another option is to modify the Schlick approximation
for use with internal reflection.
7.5.4 Local Subsurface Scattering
Metals and semiconductors quickly absorb any transmitted light, so body
reflectance is not present, and surface reflectance is sufficient to describe
the interaction of light with the object [192, 193]. For insulators, rendering
needs to take account of body reflectance as well.
If the insulator is homogeneous—with few internal discontinuities to
scatter light—then it is transparent.
12
Common examples are glass, gem-
12
Of course, no real material can be perfectly homogeneous—at the very least, its con-
stituent atoms will cause discontinuities. In practice, any inhomogeneities much smaller
than the smallest visible light wavelength (about 400 nanometers) can be ignored. A
material possessing only such inhomogeneities can be considered to be optically homo-
geneous and will not scatter light traveling through it.
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7.5. BRDF Theory 239
stones, crystals, and clear liquids such as water, oil, and wine. Such sub-
stances can partially absorb light traveling though them, but do not change
its direction.
13
Transmitted light continues on a straight line through such
objects until it undergoes internal reflection and transmission out of the ob-
ject. Techniques for rendering such substances are discussed in Sections 5.7,
9.4, 9.5, and 10.16.
Most insulators (including common substances such as wood, stone,
snow, earth, skin, and opaque plastic—any opaque non-metal) are
heterogeneous—they contain numerous discontinuities, such as air bubbles,
foreign particles, density variations, and structural changes. These will
cause light to scatter inside the substance. As the light travels through
the substance, it may be partially or completely absorbed. Eventually, any
light not absorbed is re-emitted from the surface. To model the reflectance
with a BRDF, it is necessary to assume that the light is re-emitted from
the same point at which it entered. We will call this case local subsurface
scattering. Methods for rendering in cases where this assumption cannot
be made (global subsurface scattering) are discussed in Section 9.7.
The scattering albedo ρ of a heterogeneous insulator is the ratio between
the energy of the light that escapes a surface compared to the energy of the
light entering into the interior of the material. The value of ρ is between
0 (all light is absorbed) and 1 (no light is absorbed) and can depend on
wavelength, so ρ is modeled as an RGB vector for rendering. One way
of thinking about scattering albedo is as the result of a “race” between
absorption and scattering—will the light be absorbed before it has had a
chance to be scattered back out of the object? This is why foam on a
liquid is much brighter than the liquid itself. The process of frothing does
not change the absorptiveness of the liquid, but the addition of numerous
air-liquid interfaces greatly increases the amount of scattering. This causes
most of the incoming light to be scattered before it has been absorbed,
resulting in a high scattering albedo and bright appearance. Fresh snow
is another example of high scattering albedo; there is a lot of scattering
in the interfaces between snow granules and air, but very little absorption,
leading to a scattering albedo of 0.8 or more across the visible spectrum.
White paint is slightly less—about 0.7. Many common substances, such as
concrete, stone, and soil, average between 0.15 and 0.4. Coal is an example
of very low scattering albedo, close to 0.0.
Since insulators transmit most incoming light rather than reflecting it
at the surface, the scattering albedo ρ is usually more visually important
than the Fresnel reflectance R
F
(θ
i
). Since it results from a completely
different physical process than the specular color (absorption in the inte-
13
Although we have seen that light may change its direction when entering or exiting
these substances, it does not change within the substance itself.
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240 7. Advanced Shading
rior instead of Fresnel reflectance at the surface), ρ can have a completely
different spectral distribution (and thus RGB color) than R
F
(θ
i
). For ex-
ample, colored plastic is composed of a clear, transparent substrate with
pigment particles embedded in its interior. Light reflecting specularly will
be uncolored, while light reflecting diffusely will be colored from absorption
by the pigment particles.
Local subsurface scattering is most often modeled as a Lambertian
diffuse term in the BRDF. In this case the directional-hemispherical re-
flectance of the diffuse term R
diff
is set to a constant value, referred to as
the diffuse color c
diff
, yielding the following diffuse BRDF term:
f
diff
(l, v)=
c
diff
π
. (7.36)
What value to choose for c
diff
? Itcouldbesettoρ, but this does not
account for the fact that only light that is not reflected at the surface is
available to undergo subsurface scattering, so there is an energy tradeoff
between the surface (specular) and body (diffuse) reflectance terms. If the
directional-hemispherical reflectance R
spec
of the BRDF’s specular term
happens to have a constant value c
spec
, then it makes sense to set c
diff
=
(1 − c
spec
)ρ.
The Fresnel effect implies that the surface-body reflectance tradeoff
mentioned above changes with incoming light angle θ
i
. As the specular
reflectance increases at glancing angles, the diffuse reflectance will decrease.
A simple way to account for this is to use the following diffuse term [1169]:
f
diff
(l, v)=(1− R
F
(θ
i
))
ρ
π
. (7.37)
Equation 7.37 results in a uniform distribution of outgoing light; the BRDF
value does not depend on the outgoing direction v. This makes some sense,
since light will typically undergo multiple scattering events before it is re-
emitted, so its outgoing direction will be randomized. However, there are
two reasons to suspect that the outgoing light is not distributed perfectly
uniformly. First, since the diffuse BRDF term in Equation 7.37 varies
by incoming direction, Helmholtz reciprocity implies that it must change
by outgoing direction as well. Second, the light must undergo Fresnel
reflectance on the way out (internal reflection and transmission), which
will impose some directional preference on the outgoing light.
Shirley proposed a diffuse term that addresses the Fresnel effect and
the surface-body reflectance tradeoff, while supporting both energy conser-
vation and Helmholtz reciprocity [1170]. The derivation assumes that the
Schlick approximation [1128] (Equation 7.33) is used for Fresnel reflectance:
f
diff
(l, v)=
21
20π(1 − R
F
(0
◦
))
1 − (1 −
cos θ
i
)
5
1 − (1 −
cos θ
o
)
5
ρ.
(7.38)
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