Graphics Reference
In-Depth Information
. That is, a half-angle lobe looks very
much like a cosine lobe when the angles of incidence and reflection are small.
However, cos
Near normal incidence, cos
θ
h
≈
cos
α
θ
h
gets narrower in the azimuthal direction (i.e., as a function of
φ
) toward grazing in a way that better matches the apparent foreshortening of a
surface viewed at an oblique angle. Another advantage is that cos
θ
h
is always
positive. A drawback, though, is that the computation of cos
θ
h
requires normal-
izing the half-vector
h
, and this results in an instability near grazing reflection,
where
l
v
can get arbitrarily small. The half-angle model is often used in place
of the specular deviation angle
+
, because it does a better job simulating real sur-
face reflectance. Many BRDF models that have what is described as a “Phong”
term actually depend on cos
α
θ
h
instead of cos
α
; the Ashikhmin-Shirley model is
one example.
5
Phong and the cosine lobe models, regardless of whether they use cos
α
or
θ
h
, are really just heuristic approximations; they do not arise from any real
physical properties of surfaces. Some observed reflection phenomena that have
a known physical basis cannot be captured by Phong and Phong-like models.
However, cosine lobes are suited to importance sampling because they are eas-
ily inverted. A number of extensions and modifications have been proposed over
the years, some of which improve the physical basis of the model, others have
concentrated on efficiency of computation. The Ashikhmin-Shirley model de-
scribed above is an example of the former. An example of the latter is the Schlick
model, which replaces the exponentiated cosine with rational functions that are
faster to evaluate [Schlick 93]. Nevertheless, the basic Phong model remains in
widespread use because it can successfully express realistic appearances using in-
tuitive parameters. The notion of quantifying specularity in terms of a “specular
exponent” has become permanently ingrained in at least two generations of CG
professionals.
cos
8.1.3 Off-Specular Reflection
The Phong model assumes that light is reflected the most strongly in the direction
of the specular reflection. However, in the case of rough surfaces, the peak in
the distribution of reflected light is not always in the specular direction. This is
known as an
off-specular peak
, and is one example of the general phenomenon
of
off-specular reflection
. Observed off-specular peaks become especially strong
near grazing angles. Fresnel reflection is a primary cause of off-specular peaks
5
Some authors described the Phong model with cos
θ
h
as the “Blinn-Phong”
model, but “Phong” is often applied to both. Blinn himself described the “Phong shading function”
in terms of cos θ
h
as if it were part of Phong's original model; apparently he considered raising the
cosine to an integer power the salient part of the Phong model.
α
replaced by cos
