Maximum Likelihood Estimation

When performing the calibration phase to relate image points to world points, it is easy to assume that the point in world coordinates has high precision, while the value of the point in image coordinates is known only up to Gaussian noise with a mean of zero.

The techniques discussed previously (in particular, the DLT) are mere approximations of the optimal solution and should be used as a starting point for nonlinear minimization. To obtain the optimal solution, it is necessary to minimize the sum of the squared errors between the measured position affected by noise and the position predicted by the model. The maximum likelihood estimator minimizes an objective function of the form

$$\min_\mathbf{\beta} \Vert \mathbf{p}_i - f(\mathbf{x}_i, \beta) \Vert^2$$ (8.68)

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 quantization in pixels that any sensor applies to optical rays. $\beta$ are the parameters of the perspective projection to be estimated, preferably the explicit ones considering the distortion of the optics.