# Weighted covariance fusion

I have two covariances on the same parameter.

Let $\xi \in se(3)$ be a 6 dof pose of a body in 3D space with its uncertainty $\Sigma_\xi = \begin{bmatrix} \Sigma_{r}& \Sigma_{t,r}\\ \Sigma_{r,t}& \Sigma_{t} \end{bmatrix}\in R^{6\times6}$, where t and r represent translation and rotation vector respectively.

I have two differnet sensors that can measure the pose $\xi$ indipendently. So, I have two uncertainties $\Sigma_{\xi_1}$, $\Sigma_{\xi_2}$ which are approximated by the Hessian of each constraints.

Now the problem is that $\Sigma_{r}$ from the sensor 1 is more accurate than sensor 2 and $\Sigma_{t}$ from the sensor 2 is more accurate than that of sensor 1.

Currently I am doing the following but I believe there is a better way.

$\Sigma_\xi = \begin{bmatrix} \Sigma_{r_1}& \textbf{0}_{3\times3}\\ \textbf{0}_{3\times3}& \Sigma_{t_2} \end{bmatrix}\in R^{6\times6}$