Nonlinear optimization

Finally, using the linear estimates of $R$ , $T$ and $f$ as a starting point one can solve for all the parameters, including the radial lens distortion parameter $k$ which was initialized to 0, by minimizing the image distance

$$\sum_{i=1}^N (x_i - x^p_i)^2 + \sum_{i=1}^N (y_i - y^p_i)^2$$

where $(x_i,y_i)$ are the observed image points and $(x^p_i,y^p_i)$ are the positions predicted by the Tsai model. The nonlinear optimization (over all the parameters) can be carried out by an iterative numerical technique like the Levenberg-Marquardt method.