Gaussian Mixture Midway-Merge for Object SLAM with Pose Ambiguity

Authors: Jae Hyung Jung

Year: 2022

Notes:

Merge Gaussian Mixture (GM) on Lie Groups, Object SLAM with symmetric objects
Gaussian Mixture can be used to model multiple hypothesis
How to define GM on Lie Groups? Proposes to use the midway point technique
We note $X \in G$ an element in a Lie Group
Gaussian distribution on Lie Groups: $X = \exp(-\xi^\wedge)\hat{X}$ with $\hat{X}$ the mean and $\xi \sim N(0,P)$, we note it $X \sim N_G(\hat{X}, P)$
GM on lie Group: $\sum w_i N_G(\hat{X}_i, P_i)$ with $\sum w_i = 1$

Gaussian Mixture Merge:

IV - A : Express a given Gaussian mixture distribution N_G(\hat{X}_i, P_i) but in the lie algebra of a common point $\hat{X}_c$
IV - B : Compute a midway point $\hat{X}_f = (\hat{X}_2\hat{X}_1^{-1})^{w_2}\hat{X}_1$, express and merge the two gaussian distribution $X_1$ and $X_2$ on the Lie Algebra of the midway point
V : Numerical experiments with $SO(3)$

SLAM problem: