# A Survey of Visual SLAM Based on RGB-D Images Using Deep Learning and Comparative Study for VOE
## 5 Comparative Study for VOE
### Preparing GT Trajectory for Evaluating VOE  
To evaluate the results of the VOE model, GT data are very important. We prepared GT data of the camera trajectory according to the predefined coordinate system in a real-world space, as shown in Figure 5 (the X axis is red, the Y axis is green, and the Z axis is blue). Four points on the RGB image were marked with a self-developed tool in the Python 3.9 programming language, as shown in Figure 8. The coordinates (x,y) of the four marked points were also taken on the depth image. This was based on the data acquisition process of capturing a pair of RGB-D images. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Data Acquisition Times** & **Direction** &
\begin{tabular}{c} **Number of RGB-D** \\ **Frames** \\ \end{tabular} \\ \hline \multirow{2}{*}{1 ST} & FO-D & 21,333 \\ \cline{2-3}  & OF-D & 22,948 \\ \hline \multirow{2}{*}{2 ND} & FO-D & 19,992 \\ \cline{2-3}  & OF-D & 21,116 \\ \hline \multirow{2}{*}{3 RD} & FO-D & 17,995 \\ \cline{2-3}  & OF-D & 20,814 \\ \hline \multirow{2}{*}{4 TH} & FO-D & 17,885 \\ \cline{2-3}  & OF-D & 18,548 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: The number of frames of four data acquisitions of the TQU-SLAM benchmark dataset. Figure 6: The structure of an Intel RealSense D435 sensor includes a color (RGB) camera to collect color image data and two cameras to collect stereo depth (D) and depth images based on the infrared projector. The depth image contains the distance value from the camera to the object; this value is collected based on the reflection signal of infrared light on the object’s surface. Figure 7: Illustration of a moving vehicle to collect data from the environment using devices such as an Intel RealSense D435 sensor and a computer mounted on the vehicle. The 3D point cloud data of four points were generated based on the camera's intrinsic parameters via Formula (13). \[ \begin{bmatrix}fx&0&cx\\ 0&fy&cy\\ 0&0&1\end{bmatrix} = \begin{bmatrix}525.0&0&319.5\\ 0&525.0&239.5\\ 0&0&1\end{bmatrix}  \tag{13}\]  
where \(fx,fy,cx,\) and \(cy\) are the intrinsic parameters of the camera. For each marker point with coordinates \((x_{d},y_{d})\) and depth value \(d_{a}\) on the depth image, the coordinates of point \(Ma\)\((x_{m},y_{m},z_{m})\) are calculated via Formula (14). \[ \begin{split} x_{m}=&\frac{(x_{d}-cx)\times d_{a}}{fx}\\ y_{m}=&\frac{(y_{d}-cy)\times d_{a}}{fy}\\ z_{m}=& d_{a}\end{split}  \tag{14}\]  
Figure 8 shows the coordinates of four points marked according to the camera coordinate system, with the original coordinate system being the center of the camera. Therefore, to find the coordinates of the four points marked in the real-world coordinate system, it is necessary to find the rotation and translation matrix (transformation matrix) to transform four points from the camera coordinate system to the real-world coordinate system. The result of a transformation from \(M(x,y,z)\) in the camera coordinate system to \(M^{\prime}(x^{\prime},y^{\prime},z^{\prime})\) in to the real-world coordinate system was calculated according to Formula (15). \[ \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\\ 1\end{bmatrix} = \begin{bmatrix}Ro_{11}&Ro_{12}&Ro_{13}&Tr_{1}\\ Ro_{21}&Ro_{22}&Ro_{23}&Tr_{2}\\ Ro_{31}&Ro_{32}&Ro_{33}&Tr_{3}\\ 0&0&0&1\end{bmatrix} * \begin{bmatrix}x\\ y\\ z\\ 1\end{bmatrix}  \tag{15}\]  
where \(Ro_{11},Ro_{12},Ro_{13},Ro_{21},Ro_{22},Ro_{23},Ro_{31},Ro_{32}\), and \(Ro_{33}\) are the components of the rotation matrix from the camera coordinate system to the real-world coordinate system. \(Tr_{1},Tr_{2},\) and \(Tr_{3}\) are the components of the translation matrix from the camera coordinate system to the real-world coordinate system. The transformation result of point \(M^{\prime}\) is shown in Formula (16). \[ \begin{split}\begin{cases}x^{\prime}&=&Ro_{11}x&+Ro_{12}y&+Ro_{13 2}+Tr_{1}\\ y^{\prime}&=&Ro_{21}x&+Ro_{22}y&+Ro_{23}z+Tr_{2}\\ z^{\prime}&=&Ro_{31}x&+Ro_{32}y&+Ro_{33}z+Tr_{3}\end{cases}\end{split}