M-SAC

The RANSAC strategy is to return, among all generated hypotheses, the one with the smallest number of elements outside a fixed threshold. This strategy can be viewed as an M-estimator that minimizes a loss function of the form

$$\rho = \left\{ \begin{array}{ll} 0 \quad & \vert e\vert< \tau \\ 1 \quad & \vert e\vert> \tau \\ \end{array}\right.$$ (4.130)

that is, one that assigns a score of 1 to all elements farther from the evaluated model than the threshold, and 0 to elements within the threshold $\tau$.

This concept can therefore be generalized in M-SAC techniques (M-Estimator Sample and Consensus), in which the RANSAC loss function is modified.

As noted in the previous section, data noise can be viewed in part as Gaussian noise affecting the inliers, combined with a uniform distribution of outliers. The negative Maximum Likelihood is in fact the theoretically correct loss function and forms the basis of MLESAC methods, but it is computationally expensive.

A good approximation, characteristic of M-SAC techniques, is to use the following loss function:

$$\rho = \left\{ \begin{array}{ll} e^2 \quad & \vert e\vert<... ...\\ \tau^2 \quad & \vert e\vert> \tau \\ \end{array}\right.$$ (4.131)

This loss function models reasonably well the case of inliers affected by zero-mean Gaussian error and uniformly distributed outliers.