Graphics Reference
In-Depth Information
Specular direction
→
n
→
r
θ
θ
→
→
l
v
α
Viewing direction
p
Figure 8.2
Geometry for the Phong model and general cosine lobe specular models. The specular
value depends on the angle α between the view direction and the reflected light direction.
light source, which is denoted by
l
, the direction of specular reflection from the
light source is denoted by
r
,and
v
denotes the viewing direction. The angle
α
is
v
. (In BRDF notation,
l
and
the angle between
ω
r
,
respectively.) Phong's original model included an ambient term and was not cast
in the BRDF framework. The actual Phong model is more properly called a “sur-
face appearance model,” or a “shading model.” Expressed as a BRDF, Phong's
model is the sum of a diffuse and a specular term:
r
and
v
correspond to
ω
i
and
c
d
π
+
f
Phong
(
l
n
,
v
)=
c
s
(
cos
α
)
.
The specular reflection is thus controlled by two parameters: the
specular coeffi-
cient c
s
and the
specular exponent n
. In the original model,
c
d
/
π
and
c
s
had to
sum to less than one (minus the ambient term) so that the computed pixel value
stays in the proper range. As a general BRDF this is not necessary; however, the
value of the specular coefficient
c
s
has to be adjusted in order to conserve energy.
Applying the Phong model to a particular surface therefore amounts to selecting
the diffuse and specular coefficients, which is done for each color channel inde-
pendently; normally the specular term has the same value for all the channels, so
the highlights appear white.
One way of visualizing a BRDF model is to plot
f
r
(
θ
i
,
φ
i
,
θ
r
,
φ
r
)
as a function
of
θ
r
only, i.e., with the other three variables fixed. Normally such a diagram
has
so that the incoming and outgoing directions are in the same
plane, which is known as the
plane of incidence
. Of course, this only shows a 1D
slice of the BRDF, but it is useful nonetheless to get an idea of the shape of the
function. Figure 8.3 illustrates such plots; Figure 8.3(a) shows the specular part
of the Phong model for various exponent values. The teardrop-shaped curves are
known as cosine
lobes
. Notice that the lobe becomes narrower as the exponent
n
increases. As
n
φ
r
=
φ
i
+
π
→
∞
the reflection approaches perfect mirror reflection. A