Graphics Reference
In-Depth Information
The notion of irradiance appears in many research papers about rendering, and
it's often given the letter
E
. Except for brief mention in our discussion of radiosity
in Chapter 31, we'll have no further use for irradiance, and we will use the letter
E
primarily to denote the eyepoint (or camera) in a rendering algorithm.
The corresponding measure of light leaving a surface in all possible directions is
called
spectral radiant exitance;
the only difference is the direction of the vector
v
o
appearing in
L
:
)=
Exitance
=
M
(
t
,
P
,
λ
L
(
t
,
p
,
v
o
,
λ
)
v
o
·
n
d
v
o
.
(26.66)
v
o
∈
S
+
(
P
)
Once again, we're defining this for reflective-only surfaces. And we can again
extend this to be defined at any point in space: As long as we provide an additional
argument indicating a surface normal, and by integrating over all wavelengths, we
get the
radiant exitance.
The
radiant power
or
radiant flux
Φ
arriving at a surface
M
(whether it's an
actual surface in the scene or some virtual surface like “the surface of a sphere of
radius 1 m surrounding this light source”) is computed by integrating yet again.
Since power is measured in joules per second, we must integrate over a region to
remove the m
2
from the units:
Power
=Φ=
P
∈
M
L
(
t
,
P
,
−
v
i
,
λ
)
v
i
·
n
d
dP
.
(26.67)
v
v
i
∈
S
+
(
P
)
The units of (spectral) power are J s
−
1
nm
−
1
; those of power (arrived at by
integrating out wavelength) are J
sec
,thatis,W.
The meaning of “power” is only well defined when the surface
M
over which
we are integrating is specified (along with the time and the wavelength).
For an imaginary surface in space, like the sphere surrounding the light source
above, the power arriving at one side of the surface and the power leaving the
opposite side are the same; for an actual surface in the scene, the power arriving
at one side of a surface may be large, but for opaque surfaces, no power leaves the
other side, although usually a lot is reflected.
To define radiant flux for a surface that both reflects and transmits, we need to
extend the domain of integration to all of
S
2
, and place absolute values on the dot
product:
Power
=Φ=
P
∈
M
/
L
(
t
,
P
,
−
v
i
,
λ
)
|
v
i
·
n
|
d
dP
.
(26.68)
v
v
i
∈
S
2
)
What is the domain of the “Power” function? Certainly time and wavelength
are still arguments, but what about the surface at which the power is arriving? One
possible answer is that
M
, the thing over which the integral is computed, can be
any measurable subset of any surface in 3-space. (There's no standard name for
the set of all such subsets). Most topics simply ignore the question, and speak
of the “radiant flux
Φ
,” whose domain is ignored. We'll return to this briefly in
Section 26.9.
