Not gaussian on mean tangent plane

[RaphGomCar]

Hello,

I was looking at your repo because it looks great but I have some questions. Have you seen in your implementation of "compute_cov_exp" your "y" (tangent of sample without mean) is not coming from a gaussian distribution? Moreover, your "taus" in "visualize_k_motions_exp" is coming from the tangent plane at the identity without a compensation of the mean but we see that this information is coming from a gaussian.

How can that be possible? Don't we need to remove the mean to obtain an ellipsoid on the tangent plane? If not why do we have it on the identity and so why do we calculate a covariance on the mean plane if it stil have information in it?

Have a nice day.

[MikeS96]

Hi @RaphGomCar

Thank you for your interest, I will try to address your questions in the next lines:

1. The variable y is indeed distributed as a zero-mean Gaussian distribution, however, when you transform it back to the group the log map warp the ellipsoids to the banana's shape.
2. In visualize_k_motions_exp the taus are the exponential map of poses in SE2 and what is going on there is that the distributions is being evaluated using the associated "pose" of those taus to obtain the proper pdf. The taus are just used to visualize the distribution in exp coordinates.

If you want to extend more the paper https://www.roboticsproceedings.org/rss08/p34.pdf extend on these concepts very clearly.