I wonder what the relationship here between $L_i$ and $L_o$ is for reflection case. I think it may depend on the property of the surface. For a mirror, the strongest direction of reflection certainly follows the reflection law. However, diffuse reflection surface may result in uniform radiance in all directions. So does the "cos" relation still hold here?
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There are a lot of models describing the relationship between $L_i$ and $L_o$, which can be put into different categories including BRDF(model reflection), BTDF(model transmission) and BSSRDF(model both reflection and transmission). There are models as simple as the Phong reflection model (where the diffuse reflection property for different material is characterized by a constant, $L_o$ = $k_d$$L_i$cos$\theta_i$) and also ones way more complicated. Seems we will have several lectures to discuss these models...
I think in this case we would use a cosine measuring the difference between the outgoing ray in the direction we are computing and the ray reflected across the surface from the incident ray.
The cosine factor is still present regardless of how the surface is reflecting light (mirror, diffuse, etc). It turns out that it's there to account for the reduction in energy arriving at the surface from the light in the first place as the surface's orientation changes w.r.t. the incident light.
Hopefully this will be clear after Tuesday's lecture next week. :-)