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
 
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