\[z_C \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{\mathrm{d}x} & 0 & u_0 \\ 0 & \frac{1}{\mathrm{d}y} & v_0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} R_{11} & R_{12} & R_{13} & \mathbf{T}_x \\ R_{21} & R_{22} & R_{23} & \mathbf{T}_y \\ R_{31} & R_{32} & R_{33} & \mathbf{T}_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_W \\ y_W \\ z_W \\ 1 \end{bmatrix}
\]
\[z_C \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{f}{\mathrm{d}x} & 0 & u_0 & 0 \\ 0 & \frac{f}{\mathrm{d}y} & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} R_{11} & R_{12} & R_{13} & \mathbf{T}_x \\ R_{21} & R_{22} & R_{23} & \mathbf{T}_y \\ R_{31} & R_{32} & R_{33} & \mathbf{T}_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_W \\ y_W \\ z_W \\ 1 \end{bmatrix}
\]
\[z_C \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix}\frac{R_{11} f}{dx} + R_{31} u_{0} & \frac{R_{12} f}{dx} + R_{32} u_{0} & \frac{R_{13} f}{dx} + R_{33} u_{0} & \frac{Tx f}{dx} + Tz u_{0}\\\frac{R_{21} f}{dy} + R_{31} v_{0} & \frac{R_{22} f}{dy} + R_{32} v_{0} & \frac{R_{23} f}{dy} + R_{33} v_{0} & \frac{Ty f}{dy} + Tz v_{0}\\R_{31} & R_{32} & R_{33} & Tz\end{bmatrix} \begin{bmatrix} x_W \\ y_W \\ z_W \\ 1 \end{bmatrix}
\]
\[z_C \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = M_{3\times 4} \begin{bmatrix} x_W \\ y_W \\ z_W \\ 1 \end{bmatrix}
\]
\[z_C \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \begin{bmatrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \end{bmatrix} \begin{bmatrix} x_W \\ y_W \\ z_W \\ 1 \end{bmatrix}
\]
当 世界坐标 和 像素坐标 都已确定，可以求解 M 矩阵