# An Accurate, Efficient, and Stable Perspective-n-Point Algorithm in 3D Space
## 3 Experiments and Analysis
### Materials and Experimental Protocol for Physical Experiments  
Based on synthetic data, the above section compared the performance of the Hidden PnP algorithm with the other eight PnP algorithms, and the Hidden PnP algorithm proved to have better accuracy, stability, anti-noise ability, and calculation speed. In this section, the authors applied the Hidden PnP algorithm and the other eight algorithms to physical objects, so as to reveal their actual performances.  
The experimental materials employed included a high-resolution camera and high-precision checkerboard calibration board. In addition, the experiment adopted a Rmoncam G180 camera, featuring a distortionless angle of 120 (Figure 10a), a resolution of 1920 \(\times\) 1080, and a pixel size of 3.01 \(\upmu\)m. [PERSON]'s camera calibration method [25] was also employed to calibrate the internal parameters of the Rmoncam G180 camera. As to the calibration board, a customized black and white checkerboard was utilized (Figure 10b), in which the size of each checkerboard was 30 mm \(\times\) 30 mm, and the machining accuracy error was 0.01 mm. Table 3 depicts the internal parameters of the Rmoncam G180 camera's calibration.  
Figure 11 illustrates the specific experimental process. The cyan circle '\(\circ\)' in Figure 12 refers to the position of all corner points within the extraction range of corner points on the calibration plate. Figure 13 reveals the 10 images of the calibration plate under different attitudes, where blue '\(\circ\)' stands for the sub-pixel corner point randomly extracted by various PnP algorithms, while the red '\(+\)' represents the reprojection corner point calculated through the PnP algorithms. The loss function to evaluate the performance of the PnP algorithms was constructed as:  
\[\bar{\delta}=\frac{1}{100}\sum_{i=1}^{100}(u_{i}-\bar{u}_{i})^{2}+(v_{i}-\bar{ v}_{i})^{2} \tag{27}\]  
The following can be observed from the reprojection error test data in Table 4: the DLS algorithm had the largest error, followed by the LHM algorithm. Except for the Hidden PnP algorithm, the error accuracy of the remaining six algorithms did not differ greatly. In  
Figure 10: (**a**) Rmoncam G180 camera. (**b**) High-precision checkerboard calibration board.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Specific Data** \\ \hline Focal length & \(f_{i}=735.725581,f_{i}=735.80416\) \\ Principal point & \((CC_{i},CC_{j})=(598.40811,381.51785)\) \\ Radial distortion coefficient & \(k_{1}=0.06689,k_{2}=-0.08019\) \\ Tangential distortion coefficient & \(p_{1}=-0.00088,p_{2}=-0.00196\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Internal parameters of Rmoncam G180 camera.  
general, the mean and median errors of reprojection coordinate points of the Hidden PnP algorithm were basically equal to, or even slightly superior to, the optimized algorithms, including OPnP, SRPnP, and ASPnP. Nonetheless, the solution method of the Hidden PnP performed with the best efficiency. The experiment on physical objects in this section also revealed that the Hidden PnP could match the best existing PnP method in solution accuracy, which supported the cost performance of the algorithm in solving PnP problems.  
Figure 11: Flow chart of physical objects reprojection experiment.  
Figure 12: Extraction range of corner points on calibration board.