M-SAC

The RANSAC policy is to return, among all the generated hypotheses, the one that has the smallest number of outliers beyond a fixed threshold. This policy can be viewed as an M-estimator that minimizes a loss function of the type

$$\rho = \left\{ \begin{array}{ll} 0 \quad & \vert e\vert< \tau \\ 1 \quad & \vert e\vert> \tau \\ \end{array}\right.$$ (3.113)

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

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

As noted in the previous section, the noise in the data can be partially viewed as Gaussian noise on the inliers associated with a uniform distribution of outliers. The negative Maximum Likelihood is, in fact, the theoretically correct loss function, which underlies the MLESAC methods, but it is quite computationally intensive.

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

$$\rho = \left\{ \begin{array}{ll} e^2 \quad & \vert e\vert<... ...\\ \tau^2 \quad & \vert e\vert> \tau \\ \end{array}\right.$$ (3.114)

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