Mathematical Topics from 3D Graphics - 3D Math Primer for Graphics and Game Development - page 352

direction in a single outgoing direction, whereas a perfectly diffuse surface
would reflect light equally in all outgoing directions, regardless of the di-
rection of incidence. In summary, a BRDF accounts for the difference in
“shininess” of two objects through its dependence on ω out .

More complicated phenomena can be expressed by generalizing the
BRDF. Translucence and light refraction can be easily incorporated by
allowing the direction vectors to point back into the surface. We might call
this mathematical generalization a bidirectional surface scattering distribu-
tion function (BSSDF). Sometimes light strikes an object, bounces around
inside of it, and then exits at a different point. This phenomenon is known
as subsurface scattering and is an important aspect of the appearances of
many common substances, such as skin and milk. This requires splitting
the single reflection point x into x in and x out , which is used by the bidirec-
tional surface scattering distribution function (BSSDF). Even volumetric
effects, such as fog and subsurface scattering, can be expressed, by drop-
ping the words “surface” and defining a bidirectional scattering distribution
function (BSDF) at any point in space, not just on the “surfaces.” Taken
at face value, these might seem like impractical abstractions, but they can
be useful in understanding how to design practical tools.
By the way, there are certain criteria that a BRDF must satisfy in
order to be physically plausible. First, it doesn't make sense for a negative
amount of light to be reflected in any direction. Second, it's not possible
for the total reflected light to be more than the light that was incident,
although the surface may absorb some energy so the reflected light can be
less than the incident light. This rule is usually called the normalization
constraint. A final, less obvious principle obeyed by physical surfaces is
Helmholtz reciprocity: if we pick two arbitrary directions, the same fraction
of light should be reflected, no matter which is the incident direction and
which is the outgoing direction. In other words,

ω in and

f( x , ω 1 , ω 2 ,λ) = f( x , ω 2 , ω 1 ,λ).

Helmholtz reciprocity

Due to Helmholtz reciprocity, some authors don't label the two directions
in the BRDF as “in” and “out” because to be physically plausible the
computation must be symmetric.
The BRDF contains the complete description of an object's appear-
ance at a given point, since it describes how the surface will reflect light
at that point. Clearly, a great deal of thought must be put into the de-
sign of this function. Numerous lighting models have been proposed over
the last several decades, and what is surprising is that one of the earliest
models, Blinn-Phong, is still in widespread use in real-time graphics today.
Although it is not physically accurate (nor plausible: it violates the normal-
ization constraint), we study it because it is a good educational stepping