[Ch7, 7.7.3] Questions

[nicolasrosa]

Hello @gaoxiang12,

Sometimes I get very confused about the notations used. For example, in topic 7.7.3, I believe that for being consistent with those presented in chapter 5, we can do the following notation equivalences:

Am I correct?

1. When you say "3D space points P", are you referring to Pw, correct?

2. About R, t, in this topic, is Rcw and tcw, correct?

3. In equation 7.35, T should be T*, right?

4. In equation 7.43, why there is a negative sign? Have you considered the following?
   

   If yes, should the equation 7.36 be reversed? Numerically, it shouldn't change because the resulting value is squared in the least-squares minimization, but I would change it for didactic reasons since until then you haven't presented the residual definition.

UPDATE: It doesn't need to answer question 4. At the end of the reading, you said the definition of the error.

[gaoxiang12]

Hi @nicolasrosa ,

1. Yes, and since we don't explicitly distinguish the homogeneous/non-homogeneous coordinates, we use a simple notation Pw in the text. Sometimes we multiply it with R and it is a 3D vector, and sometimes we multiply it with T so it is a 4D vector at that time. I think it makes the text more fluent. Otherwise, we may add a lot of superscripts to the equations.
2. Yes. We use world-to-camera transform as default. But if we consider a two-view problem, we may use T12 or T21 as the estimated variable.
3. Not sure. What do you mean by T*? I basically use the star superscript for the optimal solution like (7.36).

[nicolasrosa]

1. Nice, this confirms what I was suspecting. The only problem is that, so far in the book, P = [X, Y, Z]^T has been used to represent the 3D space points described in the {camera} frame, like in the topic 7.3.1 Epipolar Geometry on page 157.

   

   Considering your next answer, it becomes clear that for the PnP problem, P now represents the 3D space points in {world} frame.

2. This answer clarified everything for me!

3. Yes, I just said that because without the * we cannot differentiate the following equations, right?

   

Thank you, your answers really helped me!