I was going through the camera calibration lecture by professor Shree Nayar.

At 4:49, he is :

Scaling the projection matrix implies simultaneously scalign the world and the camera, which does not change the image. Thus, we can set the scale of the projection matrix arbitrarily. Given following 3D to 2D mapping: $$ \underbrace{\left[\begin{array}{c} u^{(i)} \\ v^{(i)} \\ 1 \end{array}\right]}_{\text{known image coordinates}} \equiv \underbrace{\left[\begin{array}{llll} p_{11} & p_{12} & p_{13} & p_{14} \\ p_{21} & p_{22} & p_{23} & p_{24} \\ p_{31} & p_{32} & p_{33} & p_{34} \end{array}\right]}_{\text{unknown projection matrix}} \underbrace{\left[\begin{array}{c} x_w^{(i)} \\ y_w^{(i)} \\ z_w^{(i)} \\ 1 \end{array}\right]}_{\text{known world coordinates}} $$ we cane two ways to scale projection matrix:

  1. Set the scale so that $p_{34}=1$
  2. Set the scale so that $\Vert p\Vert^2=1$

I understand given homogenous coordinates of a point $$P_h=\left[\begin{array}{c} u \\ v \\ w \end{array}\right]$$ we can obtain the non-homogenous coordinates as: $$P=\left[\begin{array}{c} u/w \\ v/w \\ w/w \end{array}\right]$$

But my stupid eyes are not able to make clear sense of two stated ways of scaling the projection matrix. Exactly how ensuring just the last element $p_{34}=1$ or norm of matrix $\Vert p\Vert^2=1$ is enough to scale the whole projection matrix?



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