Mathematical Topics from 3D Graphics - 3D Math Primer for Graphics and Game Development - page 351

ω out ,λ). 5
The value of this function is a scalar that describes the relatively likelihood
that light incident at the point x from direction

Symbolically, we write the BRDF as the function f( x , ω in ,

ω in will be reflected in

the outgoing direction ω out rather than some other outgoing direction. As

indicated by the boldface type and hat, ω might be a unit vector, but

more generally it can be any way of specifying a direction; polar angles are
another obvious choice and are commonly used. Different colors of light are
usually reflected differently; hence the dependence on λ, which is the color
(actually, the wavelength) of the light.
Although we are particularly interested in the incident directions that
come from emissive surfaces and the outgoing directions that point towards
our eye, in general, the entire distribution is relevant. First of all, lights,
eyes, and surfaces can move around, so in the context of creating a surface
description (for example, “red leather”), we don't know which directions
will be important. But even in a particular scene with all the surfaces,
lights, and eyes fixed, light can bounce around multiple times, so we need
to measure light reflections for arbitrary pairs of directions.
Before moving on, it's highly instructive to see how the two intuitive
material properties that were earlier disparaged, color and shininess, can
be expressed precisely in the framework of a BRDF. Consider a green ball.
A green object is green and not blue because it reflects incident light that
is green more strongly than incident light of any other color. 6 For example,
perhaps green light is almost all reflected, with only a small fraction ab-
sorbed, while 95% of the blue and red light is absorbed and only 5% of light
at those wavelengths is reflected in various directions. White light actually
consists of all the different colors of light, so a green object essentially filters
out colors other than green. If a different object responded to green and
red light in the same manner as our green ball, but absorbed 50% of the
blue light and reflected the other 50%, we might perceive the object as teal.
Or if most of the light at all wavelengths was absorbed, except for a small
amount of green light, then we would perceive it as a dark shade of green.
To summarize, a BRDF accounts for the difference in color between two
objects through the dependence on λ: any given wavelength of light has its
own reflectance distribution.
Next, consider the difference between shiny red plastic and diffuse red
construction paper. A shiny surface reflects incident light much more
strongly in one particular direction compared to others, whereas a diffuse
surface scatters light more evenly across all outgoing directions. A perfect
reflector, such as a mirror, would reflect all the light from one incoming

5 Remember that ω and λ are the lowercase Greek letters omega and lambda, respec-
tively.
6 Here and elsewhere, we use the word “color” in a way that's technically a bit dodgy,
but is OK under the assumptions about light and color made in most graphics systems.