Questions About Function `compute_aabb()`

[xbillowy]

Thanks for open sourcing such great work!

I have some question about how the compute_aabb() function works. If I understand correctly, the T = glm::transpose(splat2world) * world2ndc * ndc2pix; calculated in the compute_transmat() function should be the transformation matrix from the 2D Gaussian tangent plane to the 2D image plane. Hence, the T3 extracted in compute_aabb() should be the projection of the center of the Gaussian to the 2D image plane, which I suppose would be the point_image? I'm somewhat confused about the computational logic within compute_aabb(), or is there any underlying mathematical derivation that I might have missed?

Specifically, I do not quite understand what the variable float3 temp_point = {1.0f, 1.0f, -1.0f}; is for. Which coordinate system is it defined under and does its value [1.0, 1.0, -1.0] come with any special meanings? What does the distance derived from it denote, and why is it calculated in this manner? Furthermore, the calculation processes for point_image and extent are both somewhat baffling to me.

[hbb1]

Hi, the named variables are indeed somewhat confusing and I believe the the mathematical calculation is much more clear. Please see here for explanation. You can also check the accuracy of the bbox in our python script.

[xbillowy]

I'm not quite sure about this step in the derivation: $\frac{Ax+By+Cz+D}{\sqrt{A^2+B^2+C^2}}=\frac{h_3}{\sqrt{(h_0)^2+(h_1)^2}}=1$. What do $h_3$, $h_0$, and $h_1$ represent respectively, and why $\sqrt{A^2+B^2+C^2}$ -> $\sqrt{(h_0)^2+(h_1)^2}$?

[hbb1]

This is distance of point-to-plane. The third entry is always zero.

[xbillowy]

So $h_0, h_1, h_2, h_3$ are entries of the transformed plane (now in Gaussian tangent plane, originally defined in the image plane). Since the third entry of the originally defined x- or y-plane is always 0, it remains 0 after transformation.