[mip-nerf 360] Derivation for Distortion Loss: from Eq 14 to Eq 15

[Xiaoming-Zhao]

Hi, thanks a lot for the amazing work of mip-nerf 360. I am trying to derive Eq. (15) from Eq. (14) with the assumption that $w_s$ is constant on each interval. I find the derivation is non-trivial so that I want to make sure my understanding is correct.

Let's assume $u \in [s_i, s_{i + 1})$ and $v \in [s_j, s_{j + 1})$. Essentially, there are six situations when considering the relationship between $[s_i, s_{i + 1})$ and $[s_j, s_{j + 1})$:

1. $s_{i + 1} < s_j$
2. $s_i < s_j , s_j < s_{i + 1} < s_{j + 1} $
3. $s_i > s_j, s_{i + 1} < s_{j + 1} $
4. $s_i > s_j, s_{i + 1} > s_{j + 1} $
5. $s_i < s_j, s_{i + 1} > s_{j + 1} $
6. $s_{i} > s_{j + 1} $

Within each situation, there are also some sub-situations we need to take into account to complete the integration. May I know whether this is the right direction to go? Are there some easier or more elegant ways to do this?

It would also be much appreciated if the derivation can be made public.

Thanks a lot in advance.