## 3 Results
### Comparative Analysis
\begin{split}\mathbf{R}=\begin{pmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{pmatrix}\\ \\ q_{x}=\begin{cases}\frac{1}{2}\sqrt{1+r_{11}+r_{22}+r_{33}},&if\ r_{11}+r_{22}+r_{33}> \eta\\ \frac{1}{2}\sqrt{\frac{\left(r_{32}-r_{23}\right)^{2}+\left(r_{13}-r_{31} \right)^{2}+\left(r_{21}-r_{12}\right)^{2}}{3-r_{11}-r_{22}-r_{33}}},&otherwise \end{cases}\\ q_{y}=\begin{cases}\frac{1}{2}\sqrt{1+r_{11}-r_{22}-r_{33}},&if\ r_{11}-r_{22}-r_{33}> \eta\\ \frac{1}{2}\sqrt{\frac{\left(r_{32}-r_{23}\right)^{2}+\left(r_{12}+r_{21} \right)^{2}+\left(r_{31}+r_{13}\right)^{2}}{3-r_{11}+r_{22}+r_{33}}},&otherwise \end{cases}\\ q_{z}=\begin{cases}\frac{1}{2}\sqrt{1-r_{11}+r_{22}-r_{33}},&if-r_{11}+r_{22}-r_ {33}>\eta\\ \frac{1}{2}\sqrt{\frac{\left(r_{13}-r_{31}\right)^{2}+\left(r_{12}+r_{21} \right)^{2}+\left(r_{23}+r_{32}\right)^{2}}{3+r_{11}-r_{22}+r_{33}}},&otherwise \end{cases}\\ q_{z}=\begin{cases}\frac{1}{2}\sqrt{1-r_{11}-r_{22}+r_{33}},&if-r_{11}-r_{22}+r_ {33}>\eta\\ \frac{1}{2}\sqrt{\frac{\left(r_{21}-r_{12}\right)^{2}+\left(r_{31}+r_{13} \right)^{2}+\left(r_{32}+r_{23}\right)^{2}}{3+r_{11}+r_{22}-r_{33}}},&otherwise \end{cases}\\ \end{cases}\end{split}  \tag{14}\]  
As reported in [83], the best results that outperformed [PERSON]'s rotation to the quaternion method were achieved for \(\eta=0\); so, we set this value to build the trajectory files for those methods that did not match the evaluation format. In addition, the ORB-SLAM2, DF-ORB-SLAM, DynaSLAM, SVO, CNN-SVO, and DSM required a different calibration camera model than that the provided by the benchmark that includes full photometric data considering the geometric intrinsic calibration, photometric calibration, and the nonparametric vignette calibration, while the rest of the methods used an ATAN camera model based on the FOV distortion model of [84] provided in the official PTAM repository [85]. We used the ROS calibration package [86] to estimate three radial and two tangential distortion coefficients \(\mathbf{d}_{coeff}=\left(k_{1}\ k_{2}\ p_{1}\ p_{2}\ k_{3}\right)\), following the formulation of Equations (6) and (7). The results were also tested and compared with the OpenCV Camera calibration package [87]. For each undistorted pixel at \(\left(x_{u},y_{u}\right)\) coordinates, its position in the distorted image is \(\left(x_{d},y_{d}\right)\):  
\[ \begin{split} x_{u}&=x_{d}\big{(}1+k_{1}r^{2}+k_{2}r^{4}+k _{3}r^{6}\big{)}\\ y_{u}&=y_{d}\big{(}1+k_{1}r^{2}+k_{2}r^{4}+k_{3}r^{6} \big{)}\end{split}  \tag{16}\]  
\[ \begin{split} x_{u}&=x_{d}+\big{[}2p_{1}x_{d}y_{d}+p_{ 2}\big{(}r^{2}+2x_{d}^{2}\big{)}\big{]}\\ y_{u}&=y_{d}+\big{[}p_{1}\big{(}r^{2}+2y_{d}^{2} \big{)}+2p_{2}x_{d}y_{d}\big{]},\end{split}  \tag{17}\]  
where \(r\) is the distorted radius \(r_{d}=\sqrt{x_{d}^{2}+y_{d}^{2}}\). As suggested in [6], to make a fair comparison based on the accumulated drift over the aligned start and end sequences, we disabled the loop closure for the SLAM methods ORB-SLAM2, DF-ORB-SLAM, DynaSLAM, LDSO, and DSM. Figure 4 presents each algorithm's cumulative error plots for the translational, rotational, and scale errors. These graphs depict the number of runs for each error type below a certain x-value. Hence, the methods close to the top left corner were better because they reached a determined error value after more executions. As can be seen in Figure 4, the sparse-direct methods (DSO, LDSO, and DSM) achieved the best overall performance, followed by the sparse-indirect method (ORB-SLAM2), the dense-indirect method (DF-ORB-SLAM), and the hybrid method (SVO); the dense-direct method (LDS-SLAM) showed the worst performance. The ORB-SLAM2 and SVO CNN versions showed an important improvement over their classic versions. At the same time, the CNN-DSO did not outperform the DSO in accumulated translation, rotation, and scale metrics but remained close to the performance of the DSO.