2.5.3 Bayesian combination of rotation estimates

The information from the two Gaussians can be combined in a Bayesian manner to yield $\left(\mathbf{R}_{c},\boldsymbol{\Sigma}_{c}\right)$ by first finding the deviation between the two means in the tangent space, and then weighting by the information of the two estimates. The information (inverse covariance) adds, as usual:

$\displaystyle \boldsymbol{\Sigma}_{c}$	$\displaystyle =$	$\displaystyle \left(\boldsymbol{\Sigma}_{0}^{-1}+\boldsymbol{\Sigma}_{1}^{-1}\right)^{-1}$	(41)
	$\displaystyle =$	$\displaystyle \boldsymbol{\Sigma}_{0}-\boldsymbol{\Sigma}_{0}\left(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1}\right)^{-1}\boldsymbol{\Sigma}_{0}$	(42)
$\displaystyle \mathbf{v}$	$\displaystyle \equiv$	$\displaystyle \mathbf{R}_{1}\ominus\mathbf{R}_{0}$	(43)
	$\displaystyle =$	$\displaystyle \ln\left(\mathbf{R}_{1}\cdot\mathbf{R}_{0}^{-1}\right)$	(44)
$\displaystyle \mathbf{R}_{c}$	$\displaystyle =$	$\displaystyle \exp\left(\boldsymbol{\Sigma}_{c}\cdot\boldsymbol{\Sigma}_{1}^{-1}\cdot\mathbf{v}\right)\cdot\mathbf{R}_{0}$	(45)