I think the main part that confused me was the fact that we are summing (or integrating) over all of the differential outgoing radiance values (each of which have their infinitesimal incoming radiance that altogether compose the incoming irradiance). Do I have that right?
Yes, that’s right. Each differential incoming irradiance contributes a little to the total outgoing radiance. We simply sum up all (differential) contributions.
I noticed that in the integral we are solving for the total outgoing radiance along a given direction. It is a function that requires the BRDF to solve. It seems like somewhat of a circular definition that the BRDF includes the (diffferential) outgoing radiance as a variable, which is what we really want! I'm assuming that in practical implementations, the BRDF is a known equation of some kind that specifies the reflection ratios?
This is exactly why I don’t like much the ratio explanation of what the BRDF is. Formally, the BRDF is defined as the ratio between differential outgoing radiance and differential incoming irradiance (see Matt's post before):
In practice we want to know the outgoing radiance, thus we rearrange to
Now, it’s obvious that the BRDF must be something given to us. The BRDF now "converts" the differential incoming irradiance to the differential outgoing radiance.
I know Blinn Phong shading is a horrible example (because it lacks conservation of energy, and it's not physically accurate in any way), but is the specular component where we take the dot product between the half vector and normal technically the "BRDF" portion of the lighting equation?
You can normalize the Blinn-Phong model to make it energy conserving. Then, it's not sooo bad anymore. ;)
And yes, you're right. Let me wrap things up:
First of, the dot product between normal and light is not contained in the BRDF. It is already contained in the incoming irradiance. The differential incoming irradiance is the product of the dot product between light vector and normal and the differential incoming(!) radiance. (See Matt's post.)
The simple Phong BRDF without any normalization is:
are the material’s diffuse and specular color (in other words the amount of flux reflected per color channel) and
is the angle between eye vector and reflection vector (or in blinn-phong the half vector and the normal).
With the normalization the Blinn-Phong BRDF is:
(The factors are individually obtained by integrating the BRDF weighted by the dot product of normal and light over the hemisphere.)
Well, and for the classic Phong BRDF it is:
Hope this helped a little.