# A Method for the Calibration of a LiDAR and Fisheye Camera System
## 3 Results
### Real Environment  
For the real environment experiments, a set of 11 images and point clouds was used. The resolution of the fisheye images is 2600 \(\times\) 2160 pixels, so the resolution of the spherical image, where the error in pixels is measured, is 4320 \(\times\) 2160 pixels, which is double the size along each axis compared to the spherical image used in the simulation. The positions of the LiDAR and the camera during the experiments remained fixed, while the calibration rectangles were placed at different positions and orientations. The position of the LiDAR is close to the camera, and the rotation between both systems is assumed to be minimal but never 0, and, as stated before, the ground truth transformation is not known. Two rectangular polygons with different dimensions are used in the experiments in such a way that they can cover the maximum space possible in the images and point clouds, making sure that the most distorted parts of the fisheye image are covered. The dimensions are exactly the same as the ones used in the simulation, one being 0.59 m \(\times\) 0.41 m and the other 1.89 m \(\times\) 1.70 m, and they are positioned at different distances from the camera and LiDAR, as can be seen in Figure 11b. In these experiments, the results cannot be compared with the actual transformation, but the quality of the calibration can be evaluated by calculating the reprojection error, quantified as the MPE (mean pixel error). Figure 11 shows sample images of the real environment experiment with the two rectangles of different sizes. Initially, Figure 11a is an undistorted pinhole image with a field of view of 90\({}^{\circ}\), while Figure 11b is a fisheye-distorted image where the extra field of view is clearly visible. Figure 11a is shown just for comparative purposes. In the present work, only fisheye images are used. Table 2 shows the results of the calibration in the real environment. The results of the calibration using the proposed method and the minimization method are shown. The MPE is the mean pixel error of the selected corners of the rectangles from every point cloud once  
Figure 10: Rotation, translation, and mean pixel error versus distance between camera and LiDAR. they are reprojected onto the equirectangular image, which corresponds to the spherical projection. The standard deviation is also shown. The mean pixel error and the standard deviation cannot be compared directly with the results from Table 1, where the image used to measure them was half the resolution in both axes. The first row from Table 2 shows the results from the proposed method where the Kabsch algorithm was used. The rotation for every axis is close to 0, which is expected because the LiDAR and the camera are placed in a similar orientation. The translation is \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{5}{c}{**Axis**} & \multirow{2}{*}{
\begin{tabular}{c} **MPE** \\ **(Pixels)** \\ \end{tabular} } & \multicolumn{2}{c}{**Standard Deviation**} & \multicolumn{1}{c}{**Time**} \\ \cline{3-8}  & \(\mathbf{x}\) & \(\mathbf{y}\) & \(\mathbf{z}\) & \multicolumn{1}{c}{**(Pixels)**} & \multicolumn{1}{c}{**(Pixels)**} & \multicolumn{1}{c}{**(Pixels)**} & \multicolumn{1}{c}{**(s)**} \\ \hline \multirow{3}{*}{Proposed method} & rotation & 0.012 & \(-\)0.638 & \(-\)0.131 & \multirow{3}{*}{4.421} & \multirow{3}{*}{3.754} & \multirow{3}{*}{0.3004} \\  & (degrees) & & & & & \\ \cline{2-8}  & translation & \(-\)5.97 & 0.93 & 14.40 & & & \\ \hline \multirow{3}{*}{Minimization} & rotation & 0.093 & \(-\)0.735 & \(-\)0.279 & \multirow{3}{*}{3.691} & \multirow{3}{*}{2.434} & \multirow{3}{*}{597.48} \\  & (degrees) & & & & & \\ \cline{2-8}  & translation & \(-\)7.318 & 1.337 & 13.526 & & & \\ \cline{1-1} \cline{2-8}  & (cm) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results from the proposed method and the minimization method in the real environment.