Graphics Reference
In-Depth Information
The spectral radiance
L
characterizes the light energy flowing, at each instant, at
each point in the world, in each possible direction. Formally its domain is
R
3
S
2
R
+
,
R
×
×
×
where
S
2
denotes the unit sphere in 3-space (the set of all possible directions in
which light can flow) and
R
+
is used for the set of all possible wavelengths. In
practice,
R
+
may be replaced by the range of wavelengths that are visible. The
codomain of
L
is
R
.
Starting from
L
, we can describe, via integration, all of the terms convention-
ally used in
radiometry,
the science of the measurement of radiant energy. An
alternative approach is to start from energy or power, and define all the terms by
differentiation. We discuss this approach briefly in Section 26.9.
L
(
t
,
P
,
,
λ
)
d
λ
,
(26.42)
v
0
S
2
. In engineering, the letter
L
is usually
used for this quantity, with
L
λ
being reserved for spectral radiance; in graphics,
however, the spectral radiance is often denoted by
L
. Because for us the symbol
R
3
which is defined for
(
t
,
P
,
)
∈
R
×
×
v
λ
actually is one of the arguments to the function, it's a bad choice for a subscript.
We'll therefore carry out the remainder of this discussion in the spectral case
(keeping
as an argument), and discuss the nonspectral case at the end. Until
then, when we speak of radiance we'll be speaking of spectral radiance; when we
speak of irradiance we'll mean spectral irradiance, etc.
The most interesting thing about radiance, from a computer graphics point
of view, is that in a steady-state situation, that is, one in which
L
is independent
of
t
, radiance is constant along rays in empty space (assuming, for the moment,
that there are no point light sources; see Exercise 26.3). In mathematical terms,
this means that the function
L
cannot be just any function. We also know, from
physical considerations, that
L
can never be negative.
λ
Why is
L
constant along rays in empty space? Try an experiment (see
Figure 26.25): Look through a narrow cardboard tube at a tiny region of a well-lit
latex-painted wall. You'll see a small disk of light at the end of the tube, outlined
in red in the figure. Now move twice as far away from the wall, and look again at
the same region. Again, you'll see a small disk of light (outlined by the larger blue
circle), and it will appear
equally bright
(assuming that the wall is about equally
well lit over the region where you're looking). There's an easy explanation for
this: When at first you were at distance
r
from the wall, light leaving the wall
spread out to illuminate a hemisphere of radius
r
; when you move to distance 2
r
,
it's illuminating a hemisphere of radius 2
r
, whose area is four times as great. But
as you look through your tube, you see four times as large a region of the wall.
Hence the total energy coming down the tube toward your eye is constant. In each
case, the light energy passing through the eye end of the tube is approximately the
integral of the radiance over the region of the tube end. Because we're assuming
the wall is uniformly lit, this is just the (approximately constant) radiance times
Figure 26.25: A radiance mea-
surement tool.
