Camera Calibration

Recall from camera basics that a world point ${\bar{P}}^w=[x_w{y}_w{z}_{w}1{]}^T$ projects to image pixels via

$$\lambda \overline{p}=K\left[\begin{array}{c}{R}_w^c|t_w^c\end{array}\right]{\bar{P}}^w$$

Out of these, we need to calibrate:

• Intrinsic parameters $K\in {\mathbb{R}}^{3x3}$, i.e. (${\alpha }_u,{\alpha }_v,u_0,v_0$)
• Extrinsic paramaters: $[R_w^c|t_w^c]\in {\mathbb{R}}^{3\times 4}$

Additionally, we need to calibrate radial distortion, which cause lines to bend.

The common correction model for this is:

$$\left[\begin{array}{c}{u}_d\\ v_d\end{array}\right]=(1+k_1{r}^2)\left[\begin{array}{c}u-u_0\\ v-v_0\end{array}\right]+\left[\begin{array}{c}{u}_0\\ v_0\end{array}\right]$$

where

$$r=\sqrt{(u-u_0{)}^2+(v-v_0{)}^2}$$

We also may need to calibrate for misalignment of $u$ and $v$ axes, as some sensors do not have perfectly square pixels or perfectly orthogonal pixel grid axes.

Typically, we calibrate using a known pattern like a checkerboard. We detect the corners in multiple images, and use those known 3D→2D corresponds, and then solve for intrinsic, extrinsic, and distortion parameters.