Distance measure for noisy $SE(3)$ transforms I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in R^{3\times3}$. 
If I were to transform some point $p_1 \in R^3$ by $T$, I can easily compute its Euclidean distance to some second point $p_2 \in R^3$ as $\lVert T p_1 - p_2\rVert$. 
What I would like to do instead is incorporate the covariance matrices of my transformation to compute a distance measure similar in function to a Mahalanobis distance.  Is there a closed form way to do this?
2D Example
To help clarify, I've plotted the problem in a two dimensional world below, where points are in $R^2$, and transforms are in $SE(2)$.  Here, a point $p_1$ can be transformed by $T$ to find a point $T p_1$. However, because we know the covariances of the rotation and translation, we can see that random samples of $T p_1$ would be distributed according to the banana shaped distribution shown in red.  What I would like to find then is a distance measure that would will take these covariances into account such that $p_2$ would be significantly "closer" than $p_3$ to $T p_1$.

 A: A modification of the Mahalanobis distance for quaternion data is presented in section 3.5 of Ultramicroscopy 133, 16-25 (2013).
A: Even though this question is a little old, I still find it valuable to update with an appropriate answer.
Let us write the Euclidean distance of the transformed point to the target as: 
$$
\label{eq1}
\tag{1}
d(\mathbf{p}_1,\mathbf{p}_2) = {\lVert \mathbf{R}\mathbf{p}_1 + \mathbf{t} - \mathbf{p}_2 \rVert}_2^2
$$
where $\mathbf{R}$ and $\mathbf{t}$ are rotation matrix and translation vector, respectively. The Mahalonobis distance gives the distance between two points as:
$$
\begin{equation}
\label{eq2}
\tag{2}
d_M(\mathbf{x},\mathbf{y}) = (\mathbf{x}-\mathbf{y})^TC^{-1}(\mathbf{x}-\mathbf{y})
\end{equation}
$$
where $C^{-1}$ is the inverse covariance matrix (or equivalently the conic matrix of the ellipse). We could then plug in $\ref{eq1}$ to $\ref{eq2}$ resulting in :
$$
\begin{equation}
\tag{3}
d_M(\mathbf{p}_1,\mathbf{p}_2) = (\mathbf{R}\mathbf{p}_1 + \mathbf{t} - \mathbf{p}_2)^TC^{-1}(\mathbf{R}\mathbf{p}_1 + \mathbf{t} - \mathbf{p}_2)
\end{equation}
$$
$\mathbf{R}\mathbf{p}_1 + \mathbf{t}$ can also be viewed as $\mathbf{T}\mathbf{p}_1$, by assembling the augmented matrix $\mathbf{T} = [\mathbf{R}|\mathbf{t}]$.
