Graphics Reference
In-Depth Information
version is the thing that's easy to work with mathematically, just as it's possible to
measure the distance a car travels over some small time interval, but we work with
instantaneous velocity when we're studying the mathematics of motion. What are
the units of
L
? Taking as a model “piece of surface” a rectangle in the
xy
-plane, and
assuming that the light flow is in a set of directions
Ω
all of which are essentially
perpendicular to the
xy
-plane, we know that
t
1
x
1
y
1
λ
1
energy
≈
L
(
t
,
(
x
,
y
,0
)
,
−
v
,
λ
)
d
λ
d
dy dx dt
.
(26.39)
v
v
∈
Ω
t
0
x
0
y
0
λ
0
Note that in the integrand above,
L
has four arguments:
t
, the point
(
x
,
y
,0
)
,
−
v
points
out
of the surface, but we want to
sum up the light coming
in
to the surface.
Using MKS units for the surface and time, but nanometers for wavelength
(which follows long-standing convention), we find that
L
must have the units of
joules per second per square meter per nanometer per steradian. One joule per sec-
ond is one watt, so we can also say “watts per square-meter nanometer steradian.”
What happens if the direction
, and
λ
.The
is negated because
v
v
along which light arrives is
not
parallel to the
surface normal? Then the amount of light energy arriving at the surface, per unit
area, is smaller than if it
were
parallel, by the Tilting principle.
Thus, the more general and exact formula for the energy arriving at that small
region of the
xy
-plane from directions in the solid angle
Ω
, in the given time inter-
val and wavelength interval, is
v
energy
=
t
1
t
0
x
1
y
1
λ
1
L
(
t
,
(
x
,
y
,0
)
,
−
v
,
λ
)
v
·
e
3
d
λ
d
dy dx dt
.
v
x
0
y
0
v
∈
Ω
λ
0
(26.40)
For a region
R
of an arbitrary plane, with normal vector
n
, the energy arriving
at
R
in the interval
t
0
≤
t
≤
t
1
, at wavelengths
λ
0
≤ λ ≤ λ
1
, in directions opposite
those in a solid angle
Ω
,is
energy
=
t
1
t
0
λ
1
L
(
t
,
P
,
−
v
,
λ
)
|
v
·
n
|
d
dP d
λ
dt
,
(26.41)
v
λ
0
P
∈
R
v
∈
Ω
where
P
∈
R
...
dP
is an area integral over the area
R
.
When we are concerned with overall light energy, rather than caring about
how much is transported at each different wavelength, we can integrate
L
over
all
wavelengths
, giving us a new function depending on time, position, and direc-
tion, with units of watts per square-meter steradian. This new function is called the
radiance.
This relationship between spectral radiance and radiance is quite gen-
eral: For any photometric quantity, the spectral version has the wavelength
λ
as a
parameter, while the version without the adjective “spectral” has been integrated
over all possible wavelengths.
The function
L
, defined for all times and points and all directions (and possibly
for all wavelengths) describes fully the way light flows around the scene. We'll call
L
(
t
,
P
,
λ
)
the “radiance” or “spectral radiance” at time
t
, location
P
, etc. But the function
L
, considered as a whole, is sometimes also called the
plenoptic function,
particularly in computer vision.
)
or
L
(
t
,
P
,
,
λ
v
v
