小孔模型及畸变总结

参考博客： 最详细、最完整的相机标定讲解

定义 $(X_c,Y_c,Z_c)$表示相机坐标系下的坐标，$(x,y)$表示成像平面坐标系归一化坐标，$(u,v)$表示像素坐标，$(u_0,v_0)$表示像素中心坐标，每个像素占$\frac{1}{k}$x$\frac{1}{l}$物理坐标大小。像素坐标与物理坐标关系 $x=(u-u_0)/fk$ $y=(v-v_0)/fl$小孔模型 $x=\frac{X_c}{Z_c}$ $y=\frac{Y_c}{Z_c}$ 假设$(\bar{x},\bar{y})$为考虑径向畸变的实际成像坐标，则有 $$\left[\begin{array}{c}\bar{x}\\ \bar{y}\end{array}\right]=\left[\begin{array}{c}d(k_1,k_2,r^2)x\\ d(k_1,k_2,r^2)y\end{array}\right],$$其中$d(k_1,k_2,r^2)=1+(k_1+k_2{r}^2),r^2=x^2+y^2$ $$\left[\begin{array}{c}\bar{x}-x\\ \bar{y}-y\end{array}\right]=\left[\begin{array}{c}(k_1{r}^2+k_2{r}^4)x\\ (k_1{r}^2+k_2{r}^4)y\end{array}\right]$$ 两边左乘矩阵 $$\left[\begin{array}{cc}fk& 0\\ 0& fl\end{array}\right]$$等于 $$\left[\begin{array}{c}\bar{u}-u\\ \bar{v}-v\end{array}\right]=\left[\begin{array}{c}(k_1{r}^2+k_2{r}^4)(u-u_0)\\ (k_1{r}^2+k_2{r}^4)(v-v_0)\end{array}\right]$$ 因此 $$\begin{array}{rl}\left[\begin{array}{c}\bar{u}\\ \bar{v}\end{array}\right]& =(1+k_1{r}^2+k_2{r}^4)\left[\begin{array}{c}u-u_0\\ v-v_0\end{array}\right]+\left[\begin{array}{c}{u}_0\\ v_0\end{array}\right]\\ & =(1+k_1{r}^2+k_2{r}^4)\left[\begin{array}{cc}fk& 0\\ 0& fl\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{u}_0\\ v_0\end{array}\right]\end{array}$$