Graphics Reference
In-Depth Information
•
v
i
and
v
o
are respectively the directions from the sample
to
the source and
to the sensor
You can purchase a radiance meter; the light meter used by photographers is
a crude version. The computation done to get readings from the gonioreflec-
tomer, is this: first measure the radiance,
L
1
, arriving from the light source.
Measure the area,
A
, of the light source, and its distance,
r
, from the sample.
The solid angle it subtends is then
A
v
)
pair, place the
/
r
2
. Now for each
(
,
v
−
v
, and the radiance sensor so
that it detects the radiance from the sample in direction
light source so that light arrives in direction
. Call this radiance
value
L
2
. Then the “reading” from the gonioreflectometer is
v
L
1
(
r
2
)
v
·
L
2
n
)
.In
practice, it's best to measure
L
2
, the radiance from the sample when the illu-
minator is off, and
L
2
, the radiance with the illuminator on, and then compute
L
2
=
L
2
−
L
2
; this prevents the too-high readings for grazing angles that can
arise when stray light enters the device, or when there's some offset in the cal-
ibration of the radiance meter, so that even total darkness registers as having
some positive radiance.
We call
f
r
the (spectral)
bidirectional reflectance distribution function,
or
BRDF.
Note that
f
r
has units of sr
−
1
. Because the gonioreflectometer light source
is constant (i.e., the radiance leaving it is constant), the value defining
f
r
is inde-
pendent of time. One can extend the definition of
f
r
to directions
v
o
“on
the wrong side” of the surface by defining it to be zero there. The domain of
f
r
is
then
v
i
and
S
2
S
2
R
+
,
M
×
×
×
(26.79)
where
is the collection of all surfaces in the world. Note that in the definition of
M
f
r
(
P
,
v
i
is a unit vector pointing
toward
the incoming light,
so the incoming light is traveling in direction
v
i
,
v
o
,
λ
)
, the vector
−
v
i
.
With this definition of
f
r
, the correct form for relating the outgoing radiance
L
r
from a surface to the incoming radiance
L
i
is
)=
L
r
(
t
,
P
,
L
i
(
t
,
P
,
v
o
,
λ
−
v
i
,
λ
)
f
r
(
P
,
v
i
,
v
o
,
λ
)
v
i
·
n
(
P
)
d
(26.80)
v
i
v
i
∈
S
+
(
P
)
because the cosine of the colatitude of the incoming direction is just the nega-
tive dot product of that direction vector and the outward surface normal. This
is the
reflectance equation,
the central part of a more general
rendering equa-
tion
[Kaj86, ICG86, NN85], to which we'll return in Chapter 29.
Because of the cosine factor, some topics describe reflectance as the ratio
of outgoing radiance to incoming irradiance. We've instead defined it directly in
terms of
L
i
for clarity.
The BRDF generally has an important symmetry called
Helmholtz reci-
procity:
f
r
(
P
,
v
i
,
v
o
,
λ
)=
f
r
(
P
,
v
o
,
v
i
,
λ
)
.
(26.81)
