Graphics Reference
In-Depth Information
As a final observation about Lambertian surfaces, we note that the argument
above describing the difference between the BRDF and the probability of photon
scattering tells us that if a photon arrives at an ideal, perfectly reflecting (
ρ
=
1)
−
v
i
, it leaves in some other direction
v
o
, which can be thought of as being drawn from some probability distribution
with probability density function
p
. The distribution
p
is given by
Lambertian surface traveling in direction
v
o
)=
1
π
p
(
(
v
o
·
n
)
.
(27.16)
Here are a few common statements about Lambertian reflection, with com-
mentary.
Lambertian reflection scatters light
equally in all directions.
Too vague to be meaningful.
True.
f
r
(
v
o
)
is a constant function
The Lambertian BRDF is constant.
v
i
,
v
o
and
v
i
.
of both
A photon arriving at a Lambertian sur-
face from anywhere is equally likely
to scatter in any direction.
False. The probability of scattering in
direction
v
o
is proportional to
v
o
·
n
.
A photon leaving a Lambertian sur-
face in direction
Half true. If the surface is bathed in
a uniform light field with equal radi-
ance in every direction arriving at the
surface, then this statement is true. If
the surface is illuminated only by a
narrow beam from a laser, then the
incoming light
has
to have come from
that small range of directions.
v
o
is equally likely to
have come from a source in any direc-
tion
v
i
.
You've already seen two forms of the Phong model: the first in Chapter 6, where
light was measured in some ill-defined units of “intensity,” with values rang-
ing from 0 to 1, and the second in Section 14.9.3, where actual physical units
were used, and the constants had been adjusted so that the model was energy-
conservative provided that the sum of the specular and diffuse constants was no
greater than 1. In addition, the latter form eliminated the so-called “ambient” term,
which was an ad hoc construct that was included to simulate the effects of multi-
bounce light transport in a scene.
The general form of the simplified Blinn-Phong BRDF from Chapter 14 is
given by
v
o
)=
k
L
π
+
k
G
8
+
s
8
z
s
, where
f
s
(
v
i
,
(27.17)
π
z
=
max
(
0,
h
·
n
)
and
(27.18)
h
=
v
i
+
v
o
.
(27.19)
2
In Equation 27.17,
h
is called the
half-vector
and
k
L
and
k
G
are the Lambertian
and glossy reflectances, respectively, and may range from 0 to 1. The model is
energy-conservative if
k
L
+
k
G
≤
1.
