Lecture 13: Global Illumination & Path Tracing (42)
brandonlouie
I'm trying to conceptualize the case of ||K|| < 1. Am I correct in thinking that "'energy' of the radiance function decreases after applying K means that after each bounce that the light present in a ray decreases? I think slides 45 - 51 are consistent with my understanding, as the bright red and green walls gradually become less and less bright as K is applied
snowshoes7
@brandonlouie, that feels like correct intuition to me. The light present after each bounce wouldn't be necessarily the same as you identified, so it's like raising a fraction close to but just less than 1 to an arbitrary power--it converges eventually. However, I don't fully feel like I understand the conceptualization of an "operator"--semantically, what's different about it from a scalar function? It looks to me like it's okay to think about them somewhat similarly but I feel like this is the kind of messy conceptual understanding that leads to easy mistakes.
zeddybot
I am also thinking about the mathematical meaning of ||K||<1. Since K here is a linear operator on an infinite-dimensional vector space, what actual choice of norm should we be using. Intuitively, I think it makes sense to use some sort of operator norm, where we consider the maximum amount that K can increase the norm of a function f by. But this just kicks the can down the road to deciding how we quantify what the norm of a light distribution is. Intuitively, it makes sense to see this as the energy of the light in the scene, but how do we describe this mathematically?
0-0-00-0
I tried deducing these equations myself, and was amazed how this math works by just having the properties of inverses and matrix multiplications. ||K||<1 should just mean all eigenvalues of K have absolute values < 1.
I'm trying to conceptualize the case of ||K|| < 1. Am I correct in thinking that "'energy' of the radiance function decreases after applying K means that after each bounce that the light present in a ray decreases? I think slides 45 - 51 are consistent with my understanding, as the bright red and green walls gradually become less and less bright as K is applied
@brandonlouie, that feels like correct intuition to me. The light present after each bounce wouldn't be necessarily the same as you identified, so it's like raising a fraction close to but just less than 1 to an arbitrary power--it converges eventually. However, I don't fully feel like I understand the conceptualization of an "operator"--semantically, what's different about it from a scalar function? It looks to me like it's okay to think about them somewhat similarly but I feel like this is the kind of messy conceptual understanding that leads to easy mistakes.
I am also thinking about the mathematical meaning of ||K||<1. Since K here is a linear operator on an infinite-dimensional vector space, what actual choice of norm should we be using. Intuitively, I think it makes sense to use some sort of operator norm, where we consider the maximum amount that K can increase the norm of a function f by. But this just kicks the can down the road to deciding how we quantify what the norm of a light distribution is. Intuitively, it makes sense to see this as the energy of the light in the scene, but how do we describe this mathematically?
I tried deducing these equations myself, and was amazed how this math works by just having the properties of inverses and matrix multiplications. ||K||<1 should just mean all eigenvalues of K have absolute values < 1.