## 2 Defect Optical Model and Saturated Imaging System  
The aim of this study is to detect internal transparent microdefects in polarizers. The gray level of the convex or concave defects has a consistent relationship with the image distance [15]. One of our previous work [13] indicates that the transparent microdefects of polarizers can be approximated as a microscale planoconvex lens model, as shown in Figure 1a. Using structured-light illumination [7], we can effectively improve the imaging contrast at the position of microdefects and increase the detection accuracy.  
The optical simulation software TracePro is used to model and simulate the dents and bumps of a polarizer, as shown in Figure 1b. Further simulations demonstrated that the saturated imaging brightness varies with the defect depth. It is therefore feasible to infer the depth of the defects from information such as the intensity of the images. An example defect is shown in Figure 2.  
Based on the aforementioned transparent defect model, we established a corresponding backlit experimental environment to obtain high-quality grayscale images of target defects. The platform is composed primarily of a backlight, a polarizer (as a lens), and a camera (Figure 3). In our previous study [15], the saturated imaging technique was innovatively applied to the detection of transparent microdefects, which could be accurately detected in black stripes by adjusting the light source intensity, exposure time, and camera gain.  
Figure 1: Defect model and TracePro simulation model. (**a**) Microscale planoconvex lens model, and (**b**) TracePro simulation model.  
Figure 2: TracePro simulation example of different defect depths resulting in different irradiances. (**a**) 2 \(\upmu\)m, (**b**) 4 \(\upmu\)m, (**c**) 6 \(\upmu\)m, (**d**) 8 \(\upmu\)m, (**e**) 10 \(\upmu\)m, (**f**) 12 \(\upmu\)m, (**g**) 14 \(\upmu\)m, and (**h**) 16 \(\upmu\)m. (**i**) Light intensity versus defect depth.  
We used the KEYENCE VK-X250K confocal microscope to measure the shape of the defects. The measurement time for a defect area of 1.1 mm \(\times\) 1.1 mm is nearly 60 s. Therefore, this type of measuring instrument cannot meet efficiency requirements for complete inspection of products.  
Using saturated imaging and confocal microscopy measurement, we obtained sample data. Some examples and corresponding information are shown in Table 1. Grey level is plotted against defect depth in Figure 4. The experimental results are consistent with the TracePro simulation, indicating that it is appropriate to infer depth or other attributes of defect shapes from the intensity of the image without changing other conditions. In Section 3 we discuss the proposed estimation methods.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Depth (\(\mu\)m)**} & \multicolumn{2}{c}{**Defect Gray Level**} & \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Depth (\(\mu\)m)**} & \multicolumn{2}{c}{**Defect Gray Level**} \\ \hline
1 & 8.011 & 93 & 6 & 7.265 & 87 \\
2 & 6.783 & 75 & 7 & 13.884 & 123 \\
3 & 5.640 & 65 & 8 & 2.726 & 27 \\
4 & 14.631 & 139 & 9 & 11.242 & 112 \\
5 & 18.117 & 141 & 10 & 4.286 & 51 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Ten random samples out of 113.  
Figure 4: Saturated imaging grey level varies with defect depth.  
Figure 3: Saturated imaging system platform. (**a**) System setup, (**b**) black and white stripes of structured light, (**c**) example image of a defective area.