## 2 Materials and Methods
### Camera Calibration and Accuracy Verification
#### 2.4.2 Accuracy Verification  
Accuracy verification is an indispensable part of photogrammetry, and it directly affects the measurement results. Therefore, the reliability of photogrammetry must be verified before the test. Accuracy verification includes measuring the spacing between plane CT points in the air, the spacing between curved surface CT points in the air, and the spacing between CT points in the glass cover.  
1. Measurement of the spacing between plane CT points in the air  
As shown in Figure 9, the plane spacing of CT points was measured using a vernier caliper and photogrammetry. The measurement results are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Caliper** & **Photogrammetry** & **Absolute Error** & **Relative Error** \\  & **/mm** & **/mm** & **/mm** & **/\%** \\ \hline
1-2 & 6.68 & 6.686 & 0.006 & 0.090 \\
1-4 & 20.53 & 20.539 & 0.009 & 0.044 \\
1-6 & 33.86 & 33.871 & 0.011 & 0.032 \\
1-8 & 46.32 & 46.328 & 0.008 & 0.017 \\ Average & & & 0.009 & 0.046 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparative analysis of vernier caliper and photogrammetry planes.  
Figure 8: Image acquisition device (Nikon D5600).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Correction Parameters** & **Before Correction** & **After Correction** \\ \hline Focal length f (mm) & 4.1500 & 4.4616 \\ Total number of pixels in the x direction of the sensor M (pixel) & 3264 & 3264 \\ Total number of pixels in the y direction of the sensor N (pixel) & 2448 & 2448 \\ Sensor size F (mm) & 5.1517 & 5.1439 \\ Sensor size Fy (mm) & 3.8638 & 3.8638 \\ Image principal point coordinates Px (mm) & 2.5759 & 2.5680 \\ Image principal point coordinates Py (mm) & 1.9320 & 1.9187 \\ Radial distortion parameter K1 (10-3) & \(-\)3.591 & 0 \\ Radial distortion parameter K2 (10-4) & 2.721 & 0 \\ Tangential distortion parameter P1 (10-4) & 1.962 & 0 \\ Tangential distortion parameter P2 (10-5) & \(-\)2.712 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of camera parameters before and after calibration.  
From the data in Table 4, we can see that the photogrammetry method's average relative error is 0.046%, and the absolute error is less than 0.011 mm. Moreover, the relative error decreases with increasing spacing because the number of CT points identified by photogrammetry increases, thereby improving the measurement accuracy. The test results show that the photogrammetry method meets the plane accuracy requirements.  
2. Measurement of CT point spacing on curved surfaces in the air  
The measurement object is a rigid cylinder with CT points attached to its surface (Figure 10). The cylinder's axial height and radial length were measured using a vernier caliper as the true value. Subsequently, images of the rigid cylinder were captured from multiple angles using a verified camera, ensuring that each encoding point appeared in at least three photos. By importing these captured images into the PhotoModeler (PM) software, a three-dimensional stereo model of the cylinder was reconstructed. The encoding point coordinates determined the axial height, and the radial length was obtained by fitting the MATLAB program 23.2.0.2365128 (R2023b). The measurement results are shown in Table 2.  
From the data in Table 5, we can see that the axial absolute error does not exceed 0.11 mm, the average value is 0.011 mm, and the average relative error is 0.012%; the radial absolute error does not exceed 0.0.7 mm, the average value is 0.03 mm, and the average relative error is 0.088%. The results show that the photogrammetry method meets the surface accuracy requirements.  
Figure 10: Rigid cylinder with CT points attached.  
Figure 9: Measurement of the spacing between plane CT points in air.