Mathematical Topics from 3D Graphics - 3D Math Primer for Graphics and Game Development - page 359

10.1.4 The Rendering Equation

Now let's fit the BRDF into the rendering algorithm. In step 2 of our
rendering algorithm (Section 10.1), we're trying to determine the radiance
leaving a particular surface in the direction of our eye. The only way this
can happen is for light to arrive from some direction onto the surface and get
reflected in our direction. With the BRDF, we now have a way to measure
this. Consider all the potential directions that light might be incident upon
the surface, which form a hemisphere centered on x , oriented according to
the local surface normal n . For each potential direction ω in , we measure the

color of light incident from that direction. The BRDF tells us how much of
the radiance from ω out towards our eye (as

opposed to scattered in some other direction or absorbed). By summing
up the radiance reflected towards

ω in is reflected in the direction

ω out over all possible incident directions,
we obtain the total radiance reflected along

ω out into our eye. We add the
reflected light to any light that is being emitted from the surface in our
direction (which is zero for most surfaces), and voila, we have the total
radiance. Writing this in math notation, we have the rendering equation.

The Rendering Equation

L out ( x , ω out ,λ) = L emis ( x , ω out ,λ)

L in ( x , ω in ,λ)f( x , ω in , ω out ,λ)(− ω in n )d ω in . (10.1) +

Ω

As fundamental as Equation (10.1) may be, its development is relatively
recent, having been published in SIGGRAPH in 1986 by Kajiya [37]. Fur-
thermore, it was the result of, rather than the cause of, numerous strategies
for producing realistic images. Graphics researchers pursued the creation of
images through different techniques that seemed to make sense to them be-
fore having a framework to describe the problem they were trying to solve.
And for many years after that, most of us in the video game industry were
unaware that the problem we were trying to solve had finally been clearly
defined. (Many still are.)
Now let's convert this equation into English and see what the heck it
means. First of all, notice that x and λ appear in each function. The whole
equation governs a balance of radiance at a single surface point x for a
single wavelength (“color channel”) λ. So this balance equation applies to
each color channel individually, at all surface points simultaneously.