## 4 Influencing Factors and Error Analysis
### Effect of Unit Pixel Size on Measurement Error  
The measurement result for defects is obtained by multiplying the pixel distance in the corrected image by the real distance represented by a unit pixel. Therefore, factors that affect the measurement error include both the circumferential arc length \(i\) and the axial distance \(j\), which are represented by a single pixel. \(I\) is determined by the aperture size \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(r_{\text{max}}\)} & \multirow{2}{*}{**Image Center**} & \multirow{2}{*}{**Mean/Pixel**} & \multirow{2}{*}{**Standard**} & \multirow{2}{*}{**Unit Pixel**} & \multirow{2}{*}{**Positive**} & \multirow{2}{*}{**Negative**} & \multirow{2}{*}{**Deviation**} \\  & & & **Deviation** & & **Deviation** & & **Range** \\ \hline \multirow{2}{*}{\(\varphi\)12} & (295, 386) & 503.68 & 12.75 & 4.10\% & 3.11\% & 7.21\% \\  & (305, 386) & 501.29 & 6.69 & 2.45\% & 1.26\% & 3.71\% \\  & (315, 386) & 505.29 & 10.56 & 3.59\% & 2.46\% & 6.05\% \\ \hline \multirow{2}{*}{\(\varphi\)28} & (318, 368) & 468.17 & 28.87 & 7.94\% & 8.50\% & 16.44\% \\  & (318, 378) & 482.45 & 8.46 & 2.08\% & 2.19\% & 4.27\% \\ \cline{1-1}  & (318, 388) & 467.11 & 18.60 & 5.09\% & 5.93\% & 11.02\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Measurement error statistics after center adjustment. Figure 12: Correction process and results. (**a**) \(\varphi\)12 & image center (295, 386) (**b**) \(\varphi\)12 & image center (315, 386) (**c**) \(\varphi\)28 & image center (318, 368) (**d**) \(\varphi\)28 & image center (318, 388). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(r_{\text{max}}\)} & \multirow{2}{*}{**Pixel Distance**} & \multirow{2}{*}{**Mean/Pixel**} & \multirow{2}{*}{**Standard**} & \multirow{2}{*}{**Unit Pixel**} & \multirow{2}{*}{**Positive**} & \multirow{2}{*}{**Negative**} & \multirow{2}{*}{**Deviation**} \\  & & & & **Deviation** & & **Deviation** & & **Range** \\ \hline \multirow{2}{*}{253} & 469.22; 493.71; 479.55; 472.97; 483.12 & \multirow{2}{*}{479.72} & \multirow{2}{*}{8.52} & \multirow{2}{*}{0.0237 \(\times\) 0.0237} & \multirow{2}{*}{2.92\%} & \multirow{2}{*}{2.19\%} & \multirow{2}{*}{5.11\%} \\  & 494.86; 513.59; 496.19; 502.56; 499.14 & \multirow{2}{*}{501.29} & \multirow{2}{*}{6.69} & \multirow{2}{*}{0.0233 \(\times\) 0.0233} & \multirow{2}{*}{2.45\%} & \multirow{2}{*}{1.26\%} & \multirow{2}{*}{3.71\%} \\ \hline \multirow{2}{*}{263} & 520.20; 513.95; 505.94; 517.42;508.10 & \multirow{2}{*}{513.12} & \multirow{2}{*}{5.41} & \multirow{2}{*}{0.0228 \(\times\) 0.0228} & \multirow{2}{*}{0.84\%} & \multirow{2}{*}{1.40\%} & \multirow{2}{*}{2.24\%} \\  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Pixel Position and Measurement Error Statistics. \(R\) and the maximum image radius \(r_{\rm max}\) of the phase plane image, where \(i=R/r_{\rm max}\). \(j\) is determined by the true height of the calibration object and number of pixels in the vertical direction before axial correction, with the vertical pixel number being determined by an axial distance growth function \(\delta=G(r)\). To study the influence of the unit pixel circumferential arc length \(i\) on the correction accuracy, local images of the same size are intercepted and corrected based on different maximum radii \(r_{\rm max}\) using \(\varphi\)12 mm image as an example. The results are shown in Figure 13, where (a) \(r_{\rm max}\)1 = 253 pixels, (b) \(r_{\rm max}\)2 = 258 pixels, (c) \(r_{\rm max}\)3 = 263 pixels, and all image size are 125 \(\times\) 125 pixels. The pixel distance of the pentagon's side length in the corrected image is measured, and the measurement error is calculated as shown in Table 5.  
It can be observed from Table 5 that as the outer diameter \(r_{\rm max}\) of the endoscopic image increases while keeping the aperture fixed, the spatial size represented by each unit pixel decreases, resulting in improved measurement accuracy. The range of deviation in the measurement results also reduces from 5.11% to 2.24%.