Q$7.$ Consider the pinhole camera shown in the figure below, which has the intrinsic parameters

matrix:

$K=\left[\begin{array}{ccc}528& 0& 328\\ 0& 530& 241\\ 0& 0& 1\end{array}\right]$

and extrinsic parameters:

$R=Ro{t}_z(45)Ro{t}_x(90),t={\left[\begin{array}{ccc}27& -16& 125\end{array}\right]}^T$

a$)$ Compute the full perspective projection matrix

b$)$ Calculate the $2$D pixel coordinates $p$ obtained by projecting the $3$D points:

$P_1=\left[\begin{array}{c}0\\ 0\\ 500\end{array}\right],P_2=\left[\begin{array}{c}-220\\ -379\\ 831\end{array}\right],P_3=\left[\begin{array}{c}325\\ 72\\ 1061\end{array}\right]$

c$)$ Compute the pixel coordinates p obtained by observing the origin of the robot coordinate

frame.