I am implementing the ATLAS SLAM framework for a ground robot, using EKF Slam for local maps and using line segment features. The line segment features can be abstracted to their respective lines
α represent the distance and angle in the distance-angle representation of lines.
In the given framework, there is a local map matching step where lines of the local maps will be matched, and there is a need for a distance metric between 2 lines. The mahalanobis distance is suggested in the literature, however strictly a mahalanobis distance is between a single measurement and a distribution and not between 2 distributions.
How do I find the mahalanobis distance between line 1
[d1,α1] with covariance matrix
S1 and line 2
[d2,α2] with covariance matrix
In the EKF Algorithm from the book Probabilistic Robotics by Sebastian Thrun, there is a computation during the feature update step, where it looks like the covariances (of a new measurement and an existing measurement) are multiplied to give a resultant covariance matrix, and then the inverse is used in the Mahalanobis distance computation.
That would be similar to
Mahalanobis_Distance = [d2-d1,α2-α1] * Inverse(S1*S2) * [d2-d1,α2-α1]'
Is that correct?