# A Method for the Calibration of a LiDAR and Fisheye Camera System
## 3 Results
### Simulation  
Ten pairs of images and point clouds were used in several experiments to obtain the simulation results. In these experiments, the position of the camera remained fixed, while the LiDAR was moved in the x, y, and z axes in a 3D grid with a step of 0.15 m from \(-\)0.45  
Figure 8: CoppeliaSim simulation environment.  
m to 0.45 m with the objective of testing if the method is sensitive to the relative distance between both systems. The resolution of the images is 2160 \(\times\) 1080 pixels. With the aim of speeding up the simulation, instead of using Segment Anything [25], the masks of the rectangles were obtained directly by selecting those pixels that are not black in the image, as shown in Figure 9, while the rest of the process is the same as in the real environment.  
In this simulation, the proposed method is compared with a minimization method that minimizes the reprojection error based on the method proposed in [38]. For this approach, an initial solution is also provided, and the optimization equation is solved using the Scipy library in Python. The reprojection error is calculated as the sum of the distances between the projected LiDAR points and the corners of the rectangle in the camera image. The results of the simulation are shown in Table 1.  
In Table 1, the results of the calibration in the simulation environment are shown. Despite the fact that the minimization method is directly based on the minimization of the mean pixel error, the results are not far from those of the minimization method. The maximum difference in the results appears in the translation error: the error produced by the proposed method is more than twice that of the minimization method. While the minimization method takes more time to solve the optimization equation, the proposed method is much faster and provides similar results, the mean pixel error being the most important metric to evaluate the quality of the calibration.  
In Figure 10, the results for each distance seem to be more dependent on the quality of the extraction of characteristic points (both curves seem to change similarly for every distance). This could be due to the fact that both sensors in the simulation operate in the same manner, likewise obtaining the 3D coordinates of the simulated environment, while the image sensor reprojects them onto an equirectangular image (spherical projection). It  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Method**} & **Mean Rotation** & **Mean Translation** & **MPE** & **Mean Time** \\  & **Error (Degrees)** & **Error (Centimeters)** & **MPE** & **(Seconds)** \\ \hline Proposed & & & & \\ method & 0.0387 & 0.7135 & 0.6516 & 0.0444 \\ \hline Minimization & & & & \\ method & 0.0245 & 0.3233 & 0.5275 & 148.457 \\ \hline \end{tabular}
\end{table}
Table 1: Results from the proposed method and the minimization method in the simulated environment. MPE is the mean pixel error of the selected corners of the rectangles in the point cloud once they are reprojected onto the equirectangular image, which corresponds to the spherical projection.  
Figure 9: Spherical image obtained from the simulation.  
can be concluded that both methods are not very sensitive to the distance between the camera and the LiDAR.