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20.2. Transport Equation 527
L
i
cos θ
i
Δσ
i
. Thus the BRDF is
ρ =
L
o
L
i
cos θ
i
Δσ
i
.
This form can be useful in some situations. Rearranging terms, we can write down
the part of the radiance that is due to light coming from direction k
i
:
ΔL
o
= ρ(k
i
, k
o
)L
i
cos θ
i
Δσ
i
.
If there is light coming from many directions L
i
(k
i
), we can sum all of them. In
integral form, with notation for surface and field radiance, this is
L
s
(k
o
)=
all k
i
ρ(k
i
, k
o
)L
f
(k
i
)cosθ
i
dσ
i
.
This is often called the rendering equation in computer graphics (Immel et al.,
1986).
Sometimes it is useful to write the transport equation in terms of surface radi-
ances only (Kajiya, 1986). Note, that in a closed environment, the field radiance
L
f
(k
i
) comes from some surface with surface radiance L
s
(−k
i
)=L
f
(k
i
) (Fig-
ure 20.7). The solid angle subtended by the point x
in the figure is given by
Δσ
i
=
ΔA
cos θ
x − x
2
,
where ΔA
the the area we associate with x
. Substituting for Δσ
i
in terms of
Figure 20.7. The
light coming into one point
comes from another point.
ΔA
suggests the following transport equation:
L
s
(x, k
o
)=
all x’ visible to x
ρ(k
i
, k
o
)L
s
(x
, x −x
)cosθ
i
cos θ
x − x
2
dA
.
Note that we are using a non-normalized vector x − x
to indicate the direction
from x
to x. Also note that we are writing L
s
as a function of position and
direction.
The only problem with this new transport equation is that the domain of inte-
gration is awkward. If we introduce a visibility function, we can trade off com-
plexity in the domain with complexity in the integrand:
L
s
(x, k
o
)=
all x’
ρ(k
i
, k
o
)L
s
(x
, x − x
)v(x, x
)cosθ
i
cos θ
x − x
2
dA
,
where
v(x, x
)=
1 if x and x’ are mutually visible,
0 otherwise.
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528 20. Light
20.3 Photometry
For every spectral radiometric quantity there is a related photometric quantity
that measures how much of that quantity is “useful” to a human observer. Given
a spectral radiometric quantity f
r
(λ), the related photometric quantity f
p
is
f
p
= 683
lm
W
800 nm
λ=380 nm
¯y(λ)f
r
(λ) dλ,
where ¯y is the luminous efficiency function of the human visual system. This
function is zero outside the limits of integration above, so the limits could be
0 and ∞ and f
p
would not change. The luminous efficiency function will be
discussed in more detail in Chapter 21, but we discuss its general properties here.
The leading constant is to make the definition consistent with historical absolute
photometric quantities.
The luminous efficiency function is not equally sensitive to all wavelengths
(Figure 20.8). For wavelengths below 380 nm (the ultraviolet range), the light is
not visible to humans and thus has a ¯y value of zero. From 380 nm it gradually
increases until λ = 555 nm where it peaks. This is a pure green light. Then, it
gradually decreases until it reaches the boundary of the infrared region at 800 nm.
Figure 20.8. The lu-
minous efficiency function
versus wavelength (nm).
The photometric quantity that is most commonly used in graphics is lumi-
nance, the photometric analog of radiance:
Y = 683
lm
W
800 nm
λ=380 nm
¯y(λ)L(λ) dλ.
The symbol Y for luminance comes from colorimetry. Most other fields use the
symbol L; we will not follow that convention because it is too confusing to use L
for both luminance and spectral radiance. Luminance gives one a general idea of
how “bright” something is independent of the adaptation of the viewer. Note that
the black paper under noonday sun is subjectively darker than the lower luminance
white paper under moonlight; reading too much into luminance is dangerous, but
it is a very useful quantity for getting a quantitative feel for relative perceivable
light output. The unit lm stands for lu mens. Note that most light bulbs are rated
in terms of the power they consume in watts, and the useful light they produce in
lumens. More efficient bulbs produce more of their light where ¯y is large and thus
produce more lumens per watt. A “perfect” light would convert all power into
555 nm light and would produce 683 lumens per watt. The units of luminance are
thus (lm/W)(W/(m
2
sr)) = lm/(m
2
sr). The quantity one lumen per steradian is
defined to be one candela (cd), so luminance is usually described in units cd/m
2
.
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20.3. Photometry 529
Frequently Asked Questions
• What is “intensity”?
The term intensity is used in a variety of contexts and its use varies with both era
and discipline. In practice, it is no longer meaningful as a specific radiometric
quantity, but it is useful for intuitive discussion. Most papers that use it do so in
place of radiance.
• What is “radiosity”?
The term radiosity is used in place of radiant exitance in some fields. It is also
sometimes used to describe world-space light transport algorithms.
Notes
A common radiometric quantity not described in this chapter is radiant intensity
(I), which is the spectral power per steradian emitted from an infinitesimal point
source. It should usually be avoided in graphics programs because point sources
cause implementation problems. A more rigorous treatment of radiometry can
be found in Analytic Methods for Simulated Light Transport (Arvo, 1995). The
radiometric and photometric terms in this chapter are from the Illumination En-
gineering Society’s standard that is increasingly used by all fields of science and
engineering (American National Standard Institute, 1986). A broader discussion
of radiometric and appearance standards can be found in Principles of Digital
Image Synthesis (Glassner, 1995).
Exercises
1. For a diffuse surface with outgoing radiance L, what is the radiant exitance?
2. What is the total power exiting a diffuse surface with an area of 4m
2
and a
radiance of L?
3. If a fluorescent light and an incandescent light both consume 20 watts of
power, why is the fluorescent light usually preferred?
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