How do we choose between combinations of these different categories? Can all materials be represented as a linear combination of these? If so, are there some kind of machine learning techniques to optimize a reflection function to best match real-world images?

It seems like these functions are modeled after the types of reflections that exist in the real world– interestingly, it looks like we can come full circle and model reflections based on functions which were based on the very things we're trying to model! I think the question of if you can represent a reflection as a linear combination of reflection functions is really interesting. My understanding of how this works is that a particular type of surface has a reflection function associated with it, so I'm not sure what adding different functions would even look like.

I found the most interesting of these to be retro-reflective materials because they seem the least natural. After looking more into the examples shown and how they work, it seems we often create retro-reflective material by combining mirrors so that the light bounces back in the same direction it came. An example is a "corner reflector" that is essentially to mirror at a 90 degree angle. Even perhaps more interesting, this occurs in a spherical way with cat eyes - which explains why we often see two bright eyes when shining light at an animal in the night. These examples seem to be based on combinations of idea specular / glossy specular reflections, so I'm also curious if we could do without with model if we dive deeper into the structure that creates them (eg, model the "sphere" for cat eyes, and evaluate how the light reacts). This seems like more work and less efficient, however.

Regarding the linear combinations of reflection functions: I think in general, for constants a_1 to a_m and functions f_1 to f_m from R^n to R^d, the linear combination of these functions is defined as simply a_1 * f_1(x) + ... + a_m * f_m(x) (i.e. the linear combination of the outputs of all the functions). Since reflections can be generally expressed as linear transformations (I assume), this also just amounts to a linear combination of the appropriate matrices (by distributivity). I’m not sure if this operation and the reflections here are sufficient to represent reflections for all materials, but I suspect that there does exist at least some basis for all such reflection functions: the reason for this being that you can probably represent the set of all such reflection functions as a vector space, which can be fully spanned using linear combinations of basis functions.

In the case of an ideal diffuse material, incoming light is reflected with equal probability in all directions. Is this a fair approximation for diffuse materials, though? Because I would feel like there could be slight variations in tending to reflect more in certain directions, and maybe a noisy map could possibly be more effective. Would it make sense to define a varying reflection function for all the points on a surface to account for subtle variations (like the goniometric diagrams described in the previous lecture)? Or is there a more reasonable way to account for these perturbations, by doing something like varying the normals over a surface and using specular reflections, and the variation would be accounted for by the normal map?

THE SURFACE OF THE MOON IS RETROREFLECTIVE??? HOW??? I thought retroreflective materials had to be man-made, like the cubic mirrors in bike retroreflectors. How could this possibly happen in nature?

Oh, I think Ren misspoke in lecture. I looked it up and I think the picture refers to the lunar laser ranging experiment, in which astronauts on the Apollo mission placed retroreflectors on the surface of the moon so that we could measure the rate at which the moon is receding from us...

I also thought the moon might've been referencing the man made retro-reflective material place there, but I looked it up and it seems like the moon's dust actually gives it a retro-reflective effect

Retro-reflective light is actually super key in calculating the distance of the moon from the earth. Since light bounced off retro-reflective sources travels back to the source, scientists are able to fire laser light pulses at retroreflectors that were planted on the moon during the Apollo missions, and, by analyzing the round trip duration, calculate the current lunar distance. More info about this process can be found here: https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment