## 4 Experimental Results and Discussion
### Comparison of Defect Sizes  
We measured seven defects, as shown in Figure 17, including their maximum length, width, and depth. To obtain reference values for these measurements, we utilized two tools. The first was the built-in measurement function of the laser scanner (LJ-X Observer), which provided highly accurate measurements for larger defects. The second tool was a NREEOHY electron microscope, used to magnify smaller defects that were difficult to discern with the naked eye. Figure 19 displays an image magnified 20 times, where, with the provided scale, defects could be directly measured using the built-in function. However, this tool was limited in that it could not measure the depth of the defects.  
It is important to note that the methods used to measure length and width, as shown in the figure, are intended solely for estimating the approximate area of the defect region. For more precise calculations, when area is not a factor, length and width should be computed based on the projections along the first and second principal components derived from PCA. This approach was also used when obtaining the reference values.  
Table 4 presents a detailed comparison of defect sizes, where \"R\" represents the reference values and \"M\" represents the measured values using the proposed method. The error standard for each measurement was calculated as follows:  
\[Error=\frac{|R-M|}{R} \tag{27}\]  
The data presented in the table demonstrate that our method consistently maintained relatively low and stable errors when measuring defect length, width, and depth, with average errors of 3.43%, 4.50%, and 5.12%, respectively. Notably, the average error in depth measurement was higher than that for length and width. This was due to the generally shallow nature of these defects, and the fact that defect depth was measured by examining residuals, which were influenced by the fitted curve. Although our fitted  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Defect**} & \multicolumn{2}{c}{**Length (mm)**} & \multicolumn{3}{c}{**Width (mm)**} & \multirow{2}{*}{**Error (\%)**} & \multirow{2}{*}{**Error (\%)**} & \multirow{2}{*}{**Error (\%)**} & \multirow{2}{*}{**Error (\%)**} & \multirow{2}{*}{**Error (\%)**} \\ \cline{2-2} \cline{5-8}  & **R** & & & & & & & **R** & & **M** \\ \hline
1 & 16.160 & 15.62 & 3.34 & 3.470 & 3.351 & 3.43 & 0.483 & 0.472 & 2.28 \\
2 & 0.527 & 0.549 & 4.17 & 0.235 & 0.247 & 5.11 & 0.031 & 0.042 & 3.55 \\
3 & 0.360 & 0.372 & 3.33 & 0.115 & 0.183 & 5.91 & 0.082 & 0.077 & 6.10 \\
4 & 0.615 & 0.638 & 3.74 & 0.291 & 0.304 & 4.47 & 0.052 & 0.055 & 5.77 \\
5 & 9.682 & 9.419 & 2.72 & 0.478 & 0.459 & 3.97 & 0.038 & 0.041 & 7.89 \\
6 & 7.080 & 7.254 & 2.46 & 0.365 & 0.377 & 3.29 & - & 0.013 & - \\
7 & 14.845 & 14.215 & 4.24 & 0.430 & 0.407 & 5.35 & - & 0.011 & - \\ Average Error & - & - & 3.43 & - & - & 4.50 & - & - & 5.12 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of measurements of selected defects with reference values.  
Figure 19: Measurement of tiny defects under electron microscope: tiny defects (red dashed lines); defect length and width (red arrows).  
reference contours closely matched the true shape of the part, some discrepancies remained. As shown for defects 6 and 7, their depths were negligible, and no depth information could be captured with high-precision instruments during reference value collection. It is also worth noting that the defect sizes in the table are smaller than those detectable by current image processing-based deep learning methods, further confirming the effectiveness of our approach in detecting tiny defects. This suggests its potential application in fields requiring high precision.