In the paper "Are we ready for Autonomous Driving? The KITTI Vision Benchmark suite" by Lenz and Urtasun (https://www.cvlibs.net/publications/Geiger2012CVPR.pdf) \
The translation error metric is defined as $E_{trans}(\mathcal{F}) = \frac{1}{|\mathcal{F}|} \sum_{(i,j) \in \mathcal{F}}^{}\left \| (\mathbf{\hat{p_j}} \ominus \mathbf{\hat{p_i}}) \ominus (\mathbf{p_j} \ominus \mathbf{p_i}) \right \|_2\ ...$ Eqn (1). Here the boldfaced symbols are 4x4 matrices.
The paper defined $\ominus$ as the inverse compositional operator. The authors referred this symbol to the paper titled "On measuring the accuracy of SLAM algorithms" by Keummerle et al.
My interpretation of Equation 1, in a linear algebraic sense is as follows
$E_{trans}(\mathcal{F}) = \frac{1}{|\mathcal{F}|} \sum_{(i,j) \in \mathcal{F}}^{}\left \| (\mathbf{\hat{p_j}}^{-1} \mathbf{\hat{p_i}})^{-1} (\mathbf{p_j}^{-1} \mathbf{p_i}) \right \|_2 $
Is my interpretation correct?
Based on the information found here https://github.com/MichaelGrupp/evo/blob/master/notebooks/metrics.py_API_Documentation.ipynb
$E_{trans}(\mathcal{F}) = \frac{1}{|\mathcal{F}|} \sum_{(i,j) \in \mathcal{F}}^{}\left \| (\mathbf{p_i}^{-1} \mathbf{p_j})^{-1} (\mathbf{\hat{p_i}}^{-1} \mathbf{\hat{p_j}}) \right \|_2$
I had the variables in wrong order.