In a paper,the pinhole camera model is defined as follows:
$$ w\small \left(\begin{array}{c} u\\ v \\ 1 \end{array}\right)=\left( \begin{array}{ccc} f_{u} & 0 & u_{c}\\ 0 & f_{v} & v_{c}\\ 0 & 0 & 1 \end{array} \right) \left(\begin{array}{cc} \mathbf{R}_{c}^{T} &-\mathbf{R}_{c}^{T}\mathbf{t}_{c} \end{array}\right) \left(\begin{array}{c} \mathbf{ p}\\ 1 \end{array}\right)= \mathbf{K} \mathbf{P}_{c}\left(\begin{array}{c} \mathbf{ p}\\ 1 \end{array}\right). $$ Where $f_{u}$ and $f_{v}$ are the focal lengths in $u$ and $v$ directions, the principal point offset $(u_{c}, v_{c})$. $t_{c}$ the position of the camera center in a world frame and $R_{c}$ the rotation from the camera back to the world frame. $p$ is a D point in the world frame. If $u_{i}$ and $v_{i}$ denote the measurements of point $p_{i}$, the reprojection error is as follows: $$ \epsilon_{i}= \small \left(\begin{array}{cc} u_{i}\\ v_{i} \end{array}\right)-\left[ \begin{array}{ccc} \left(\begin{array}{ccc} 0& 0 &1\end{array}\right) & \mathbf{K} \mathbf{P}_{c} & \left(\begin{array}{c} \mathbf{p}_{i}\\ 1\end{array}\right) \end{array} \right]^{-1}\left(\begin{array}{ccc} 1 & 0 &0 \\ 0 &1 &0 \end{array}\right) \mathbf{K} \mathbf{P}_{c} \left(\begin{array}{c} \mathbf{ p_{i}}\\ 1 \end{array}\right) $$
I don't understand how the expression of the reprojection error is derived.
The square is measuring the third coordinate entry, which is the homogenization coordinate $w$. By multiplying with it's inverse, i.e. dividing by $w$, you are dehomogenizing, ensuring a $1$ in the third coordinate and also dropping that coordinate thanks to the $2\times3$ matrix. The result will be a 2d Euclidean distance vector between projected and actual point in the plane.