Confusion in paper Equation 9

[adam-ce]

I believe there is an error in Equation 9 of the paper. Imo it should say (u, v, 0, 1)^T there.

[hbb1]

The third entry is not used as the third column of H is zero (as we assume zero scaling in the third axis). We opt for (u,v,1,1) because it implies (uw,vw,w,1) where w = 1, similar to the representation of (xz,yz,z,1) for the image plane system.

[adam-ce]

I agree that it's not used in Equation 4 and 5 and I still believe that it is used to get from Eq. 9 to Equation 10.

To my understanding, you would actually have 3 Equations, the third one being for the local z axis. but z=0 on the surface of the 2d Gaussian, so we can set it to 0 => (u, v, z=0, 1) for evaluating the plane equation. using (u, v, 1, 1) would imply for me, that we are intersecting the plane one unit above the actual splat, which gives a different result.

[hbb1]

It does matter what w is since it is a homogenous representation because hu^3=0 and hv^3=0. I understand your confusion. I believe there is no such implication since for the image plane we also use z=1 instead of z=0.

[adam-ce]

Trying to verify and understand Eq 10, I wrote down Eq 9 and solved for u and v. I arrived at something different than Eq 10 (there were some additional factors). So I investigated further. Changing Eq 9 to (u, v, 0, 1) maintains the homogeneous representation (last coordinate = 1) and results in your Eq 10. For me it also makes sense, considering Eqs 9 as plane equations (as mentioned above, you are intersecting 3 planes, and the third is the local tangent plane, homogeneous plane (0, 0, 1, 0) => z = 0).

I use Eq. 10 in my code and it seems correct.

To my understanding, this deals with the local splat coordinate system, not the image plane.

[hbb1]

Ohhhh, I see. Because hu^3 and hv^3 is already zero, which is very implicit that may lead to some confusions. Therefore, using (u,v,1,1) and (u,v,0,1) are mathematically equivalent. The reason behind using (u,v,1,1) instead of (u,v,0,1) is because the equation is from ray-quadratic intersection. And our equation is a degenerate case when the z scale degenerate to zero so I think it would better to retain the (u,v,1,1).

[adam-ce]

"Therefore, using (u,v,1,1) and (u,v,0,1) are mathematically equivalent." => they are in Eq 4/5 for the reasons you said, but
imho not when going from Eq 9 to 10.

[adam-ce]

Ah, now I understand. yeah. I agree. Yeah, I'm using a different H matrix. Thanks for the clarification!

[hbb1]

This is because $h_u = (WH)^Th_x$ where the third column of $H=0$, you will always see $h_u^3 = 0$. This is very implicit and I should have made it more clear for avoiding confusion 🥲.