Creating simplified bounds with an inaccuracy threshold

Question

Given a set of non-rotated AABB bounds, I'm hoping to create a simpler set of bounds from the original set, that allows for a specified amount of inaccuracy.

Some examples:

I'm working with this in Unity with Bounds, but it's just basic AABB comparison stuff, nothing Unity-specific. I figure someone must have worked out a system for this at some point in the past, but I had no luck searching around. Encapsulating bounds are easy but this is harder, since you can't just iterate through each bounds one by one. Sometimes a simpler solution can only be seen by looking at the whole thing.

Fast performance isn't critical but would be nice. Inaccuracy is OK in both directions (i.e. the bounds may cover a little less than the actual size or a little more). If it helps, I can expect all bounds in the original set to be connected somewhere - no free-floating pieces in a separate group.

I don't expect anyone to write up a whole system to solve this, I'm more hoping that it's already been solved or that maybe there's an obvious process to achieve it that I haven't thought of yet.

Answer 1

This sounds something that could be handled with Surface Area Heuristics (SAH). SAH is commonly used in ray tracing to build better tree like structures were the triangles are stored. There are multiple sources discussing it more. One good is Wald's thesis chapter 7.3.

The basic idea in the SAH built is to start with the whole space and divide it recursively. Division position is decided by sweeping through all reasonable positions and calculating surface area of both child nodes. The reasonable positions are the positions were any triangle has its upper or lower bound. After sweeping through all the candidates, the division with the smallest total surface area in the children is used.

If SAH is not a good idea for your application, you could use similar sweeping through all candidates, but calculate for example the extra space inside the AABBs.