2.5.1 Sampling

We can encode Gaussian distributions over 3D rotations by representing the mean with an element of $\mathrm{SO(3)}$ and the covariance as a quadratic form over tangent vectors in $\mathrm{so(3)}$ . More precisely, consider a Gaussian distribution given by mean $\mathbf{R\in}\mathrm{SO(3)}$ and covariance $\boldsymbol{\Sigma}\in\mathcal{R}^{3\times3}$ . We can draw a sample rotation $\mathbf{S}$ from the distribution by sampling the zero-mean distribution in the tangent space and left multiplying the mean:

$\displaystyle \boldsymbol{\epsilon}$	$\displaystyle \in$	$\displaystyle \mathcal{N}\left(\mathbf{0},\boldsymbol{\Sigma}\right)$	(38)
$\displaystyle \mathbf{S}$	$\displaystyle =$	$\displaystyle \exp\left(\boldsymbol{\epsilon}\right)\cdot\mathbf{R}$	(39)

Ethan Eade 2012-02-16