Maximum Likelihood Estimation

When performing calibration to relate image points to world points, it is reasonable to assume that the world-coordinate point is known with high accuracy, whereas the image-coordinate point is known up to zero-mean Gaussian noise.

The techniques discussed above, particularly DLT, are only approximations of the optimal solution and should be used as the starting point for a nonlinear minimization. To obtain the optimal solution, it is necessary to minimize the sum of the squared errors between the measured noisy position and the position predicted by the model. The maximum-likelihood estimator minimizes an objective function of the form

$$\min_\mathbf{\beta} \Vert \mathbf{p}_i - f(\mathbf{x}_i, \beta) \Vert^2$$ (9.78)

where $\mathbf{x}_i$ is a point in world coordinates and $\mathbf {p}_i$ is the corresponding point in image coordinates, affected by observation noise due to the point-detection algorithm and the spatial pixel quantization applied by every sensor to the optical rays. $\beta$ are the perspective-projection parameters to be estimated, preferably the explicit parameters, including optical distortion.