## 2 Materials and Methods
### Metrics
So, the trajectories are first aligned using [PERSON]'s method [82] to find the rigid body transformation \(\mathbf{S}\) to map the estimated trajectory \(\mathbf{P}_{1:n}\) into the ground truth trajectory \(\mathbf{Q}_{1:n}\); hence, the absolute trajectory error for each timestamp \(i\) can be calculated as:  
\[\mathbf{F}_{i}:=\mathbf{Q}_{i}^{-1}\mathbf{S}\mathbf{P}_{i} \tag{4}\]  
In the same way as RPE, the \(RMSE\) for all the timestamps of the translational components is calculated as follows:  
\[RMSE(\mathbf{F}_{ixn}):=\sqrt{\frac{1}{n}{\sum_{i=1}^{n}}{\|trans(\mathbf{F}_{i})\|}^ {2}} \tag{5}\]  
Thus, for the [81] benchmark, the RPE combines the translational and rotational errors elegantly, while the ATE considers only the translational error component. In contrast, for the TUM-Mono benchmark [6], the authors proposed to benefit from large loop sequences. This way, instead of using the complete exploring motion pose information, the TUM-Mono benchmark was built to register the ground truth of each sequence's first and last 10-20 s, using the LSD-SLAM [29] method to track only those segments. In this way, the authors used the accumulated drift for all their metrics, and they demonstrated that the error registered by each evaluated run was not originated in the SLAM method drift used to register the ground truth; instead, it came from the accumulated drift by the algorithm through the entire trajectory. So, any SLAM or VO system can be used to register the start and end segments' ground truth. Consequently, the TUM-Mono benchmark aligns the estimated trajectory with the start and end ground truth segments and measures their differences. Let \(\mathbf{p}_{1}\),..., \(\mathbf{p}_{n}\)\(\in\)\(\mathbb{R}^{3}\) be the estimated tracked positions for the \(1\) to \(n\) frames and \(S\)\(\subset\)\([1;n]\) and \(E\)\(\subset\)\([1;n]\) be the frame indices corresponding to the start and end segments for the groundtruth positions \(\mathbf{\hat{p}}\)\(\in\)\(\mathbb{R}^{3}\). Then, by aligning the estimated trajectory with the groundtruth start and end segments, the two relative transformations can be calculated as:  
\[\mathbf{T}_{s}^{gt}:=\underset{\mathbf{T}\in\text{Sim}(3)}{\text{argmin}}{\sum_{i\in S }}(\mathbf{T}\mathbf{p}_{i}-\mathbf{\hat{p}}_{i})^{2} \tag{6}\]  
\[\mathbf{T}_{e}^{gt}:=\underset{\mathbf{T}\in\text{Sim}(3)}{\text{argmin}}{\sum_{i\in E }}(\mathbf{T}\mathbf{p}_{i}-\mathbf{\hat{p}}_{i})^{2}\] (By these transformations, the accumulated drift can be calculated as:  
\[\mathbf{T}_{drift}:=\mathbf{T}_{e}^{gt}\left(\mathbf{T}_{e}^{gt}\right)^{-1} \tag{8}\]  
The translation, rotation, and scale error components can be extracted as, respectively:  
\[e_{t}:=\left\|translation\Big{(}\mathbf{T}_{drift}\Big{)}\right\|\]  
\[\mathbf{E}_{i}:=\Big{(}\mathbf{Q}_{i}^{-1}\mathbf{Q}_{i+\Delta}\Big{)}^{-1}\Big{(}\mathbf{P}_{i}^ {-1}\mathbf{P}_{i+\Delta}\Big{)} \tag{1}\]0  
\[\mathbf{E}_{i}:=\Big{(}\mathbf{Q}_{i}^{-1}\mathbf{Q}_{i+\Delta}\Big{)}^{-1}\Big{(}\mathbf{P}_{i}^ {-1}\mathbf{P}_{i+\Delta}\Big{)} \tag{1}\]1  
As a result, the authors established the alignment error, which is a metric that equally takes into account the errors produced by the translational, rotational, and scale effects:  
\[\mathbf{E}_{i}:=\Big{(}\mathbf{Q}_{i}^{-1}\mathbf{Q}_{i+\Delta}\Big{)}^{-1}\Big{(}\mathbf{P}_{i}^ {-1}\mathbf{P}_{i+\Delta}\Big{)} \tag{1}\]2  
This metric can be computed individually for the start and end segment, but when it is estimated by combining both intervals, it is equivalent to the translational RMSE when aligned to the ground truth.