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206 7. Advanced Shading
Figure 7.3. A cone with a solid angle of one steradian removed from a cutaway view of a
sphere. The shape itself is irrelevant to the measurement; the coverage on the sphere’s
surface is the key.
In the last few paragraphs, we assumed that the light source could be
treated as a point. Light sources that have been approximated in this way
are called point lights. Just as directional lights are a useful abstraction
for lights that are far away compared to the size of the scene, point lights
are a useful abstraction for light sources that are far away compared to
the size of the light source. Intensity is useful for measuring the illumi-
nation of a point light source and how it varies with direction. As we
have seen, the irradiance contribution from a point light varies inversely
to the square of the distance between the point light and the illuminated
surface. When is it reasonable to use this approximation? Lambert’s Pho-
tometria [716] from 1760 presents the “five-times” rule of thumb. If the
distance from the light source is five times or more that of the light’s width,
then the inverse square law is a reasonable approximation and so can be
used.
As seen in Section 5.4, radiance L is what sensors (such as eyes or
cameras) measure, so it is of prime importance for rendering. The purpose
of evaluating a shading equation is to compute the radiance along a given
ray (from the shaded surface point to the camera). Radiance measures the
illumination in a single ray of light—it is flux density with respect to both
area and solid angle (see the left side of Figure 7.4). The metric units of
radiance are watts per square meter per steradian. Radiance is
L =
d
2
Φ
dA
proj
dω
, (7.4)
where dA
proj
refers to dA projected to the plane perpendicular to the ray.
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7.1. Radiometry 207
dA
dω
n
dA
dA
proj
= dAcosθ
Figure 7.4. Radiance is power per unit projected area per unit solid angle.
The relationship between dA and dA
proj
can be seen in the right side of
Figure 7.4, where we see that the projection depends on the cosine of θ,
the angle between the ray and the surface normal n:
dA
proj
= dA cos θ. (7.5)
Note that Equation 7.5 uses the
cos symbol (defined on page 8), which
represents a cosine clamped to zero. Negative cosine values occur when the
projected direction is behind the surface, in which case the projected area
is 0.
A density with respect to two quantities simultaneously is difficult to
visualize. It helps to think of radiance in relation to irradiance or intensity:
L =
dI
dA
proj
=
dE
proj
dω
, (7.6)
where E
proj
refers to irradiance measured in the plane perpendicular to the
ray. Figure 7.5 illustrates these relationships.
The ray exiting the light bulb in the left side of Figure 7.5 has radiance
L. In the middle of the figure we see the relationship between L and the
irradiance E
proj
from the light bulb (measured at a surface perpendicular to
the ray). This irradiance is the flux density per area at a surface point—
it represents photons hitting that point from all parts of the light bulb,
covering a range of directions represented by the solid angle ω. As seen
in the right side of Equation 7.6, the radiance L is the density of E per
solid angle. This is computed for a given direction by looking at only
the irradiance contributed by photons arriving from a small solid angle dω
around that direction and dividing that irradiance by dω.
On the right side of Figure 7.5 we arrive at radiance “the other way,”
starting with intensity I. Intensity represents all photons from the light
bulb going in a certain direction, from all parts of the light bulb, covering a
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208 7. Advanced Shading
ω
E
proj
I
A
proj
LL
L
dA
proj
dω
Figure 7.5. Radiance, radiance computed from irradiance, and radiance from intensity.
projected area A
proj
. As seen in the left side of Equation 7.6, the radiance
L is the density of I per area. This is computed for a given point by
looking at only the intensity contributed by photons passing through a
small area dA
proj
around the given direction and dividing that irradiance by
dA
proj
.
The radiance in an environment can be thought of as a function of
five variables (or six, including wavelength), called the radiance distribu-
tion [288]. Three of the variables specify a location, the other two a di-
rection. This function describes all light traveling anywhere in space. One
way to think of the rendering process is that the eye and screen define a
point and a set of directions (e.g., a ray going through each pixel), and this
function is evaluated at the eye for each direction. Image-based rendering,
discussed in Section 10.4, uses a related concept, called the light field.
An important property of radiance is that it is not affected by distance
(ignoring atmospheric effects such as fog). In other words, a surface will
have the same radiance regardless of its distance from the viewer. At first
blush, this fact is hard to reconcile with the idea that irradiance falls off
with the square of the distance. What is happening physically is that the
solid angle covered by the light’s emitting surface gets smaller as distance
increases. For example, imagine a light source that is square, say some
fluorescent tubes behind a translucent panel. You are sufficiently far away
to use the “five-times” rule. If you double your distance from the light, the
number of photons from the light through a unit surface area goes down
by a factor of four, the square of the distance. However, the radiance from
any point on the light source is the same. It is the fact that the solid angle
of the light source has dropped to about one quarter of its previous value,
proportional to the overall drop in light level.
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7.2. Photometry 209
Figure 7.6. The photometric curve.
7.2 Photometry
Radiometry deals purely with physical quantities, without taking account
of human perception. A related field, photometry, is like radiometry, ex-
cept that it weights everything by the sensitivity of the human eye. The
results of radiometric computations are converted to photometric units by
multiplying by the CIE photometric curve,
3
a bell-shaped curve centered
around 555 nm that represents the eye’s response to various wavelengths
of light [40, 409]. See Figure 7.6. There are many other factors—physical,
psychological, and physiological—that can affect the eye’s response to light,
but photometry does not attempt to address these. Photometry is based
on the average measured response of the human eye when the observer has
adapted to normal indoor lighting conditions. The conversion curve and
the units of measurement are the only difference between the theory of
photometry and the theory of radiometry.
Photometry does not deal with the perception of color itself, but rather
with the perception of brightness from light of various wavelengths. For
example, green light appears considerably brighter to the eye than red or
blue light.
Each radiometric quantity has an equivalent metric photometric quan-
tity. Table 7.2 shows the names and units of each.
3
The full and more accurate name is the “CIE photopic spectral luminous efficiency
curve.” The word “photopic” refers to lighting conditions brighter than 3.4 candelas
per square meter—twilight or brighter. Under these conditions the eye’s cone cells are
active. There is a corresponding “scotopic” CIE curve, centered around 507 nm, that is
for when the eye has become dark-adapted to below 0.034 candelas per square meter—a
moonless night or darker. The eye’s rod cells are active under these conditions.
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210 7. Advanced Shading
Radiometric Quantity: Units Photometric Quantity: Units
radiant energy: joule (J) luminous energy: talbot
radiant flux: watt (W) luminous flux: lumen (lm)
irradiance: W/m
2
illuminance: lux (lx)
radiant intensity: W/sr luminous intensity: candela (cd)
radiance: W/m
2
-sr luminance: cd/m
2
=nit
Table 7.2. Radiometric and photometric quantities and units.
The units all have the expected relationships (lumens are talbots per
second, lux is lumens per square meter, etc.). Although logically the tal-
bot should be the basic unit, historically the candela was defined as a basic
unit and the other units were derived from it. In North America, light-
ing designers measure illuminance using the deprecated Imperial unit of
measurement, called the foot-candle (fc), instead of lux. In either case,
illuminance is what most light meters measure, and it is important in illu-
mination engineering.
Luminance is often used to describe the brightness of flat surfaces. For
example, LCD screens typically range from 150 to 280 nits, and CRT moni-
tors from 50 to 150 nits. In comparison, clear sky has a luminance of about
8000 nits, a 60-watt bulb about 120,000 nits, and the sun at the horizon
600,000 nits [1010].
7.3 Colorimetry
Light is perceived in the visible band, from 380 to 780 nanometers (nm).
Light from a given direction consists of a set of photons in some distribu-
tion of wavelengths. This distribution is called the light’s spectrum.See
Figure 7.7 for an example.
Humans can distinguish about 10 million different colors. For color
perception, the eye works by having three different types of cone receptors
in the retina, with each type of receptor responding differently to various
wavelengths.
4
So for a given spectrum, the brain itself then receives only
three different signals from these receptors. This is why just three numbers
can be used to precisely represent any spectrum seen [1225].
But what three numbers? A set of standard conditions for measuring
color was proposed by the CIE (Commission Internationale d’Eclairage),
4
Other animals can have from two to five different color receptors [418].
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