IPM逆透视变换

逆透视变换矩阵

Image --> Ground:

$$T_{ig}=h\cdot \left[\begin{array}{cccc}-\frac{1}{f_x}\cos (\gamma )& \frac{1}{f_y}\sin (\theta )\sin (\gamma )& \frac{c_x}{f_x}\cos (\gamma )-\frac{c_y}{f_y}\sin (\theta )\sin (\gamma )-\cos (\theta )\sin (\gamma )& 0\\ \frac{1}{f_x}\sin (\gamma )& \frac{1}{f_y}\sin (\theta )\cos (\gamma )& -\frac{c_x}{f_x}\sin (\gamma )-\frac{c_y}{f_y}\sin (\theta )\cos (\gamma )-\cos (\theta )\cos (\gamma )& 0\\ 0& \frac{1}{f_y}\cos (\theta )& -\frac{c_y}{f_y}\cos (\theta )+\sin (\theta )& 0\\ 0& -\frac{1}{h{f}_y}\cos (\theta )& \frac{c_y}{h{f}_y}\cos (\theta )-\frac{1}{h}\sin (\theta )& 0\end{array}\right]$$

Ground --> Image:

$$T_{gi}=\left[\begin{array}{cccc}{f}_x\cos (\gamma )+c_x\cos (\theta )\sin (\gamma )& -f_x\sin (\gamma )+c_x\cos (\theta )\cos (\gamma )& -c_x\sin (\theta )& 0\\ -f_y\sin (\theta )\sin (\gamma )+c_y\cos (\theta )\sin (\gamma )& -f_y\sin (\theta )\cos (\gamma )+c_y\cos (\theta )\cos (\gamma )& -f_y\cos (\theta )-c_y\sin (\theta )& 0\\ \cos (\theta )\sin (\gamma )& \cos (\theta )\cos (\gamma )& -\sin (\theta )& 0\\ \cos (\theta )\sin (\gamma )& \cos (\theta )\cos (\gamma )& -\sin (\theta )& 0\end{array}\right]$$