# Optical Element Surface Defect Size Recognition Based on Decision Regression Tree
## 4 Results and Discussion
### Revision of Simulation Light Intensity Distribution  
The simulation light intensity distribution in Figure 5 are ideal results without calculating the diffraction effect caused by the aperture of the actual MSDI system. Hence, we measure the PSF of the MSDI system by the method mentioned in Section 3.1 and use PSF to revise the distribution. The measurement result of PSF is shown in Figure 6. Figure 6a is the ideal gray distribution of the step source, and its expression is given in Equation (5). In Figure 6a, there is a discontinuity point, where the gray value jumps from 0 to 255. To measure the PSF, the gray distribution of the actual step source is acquired by the MSDI system, as shown in Figure 6b. Slightly different from the ideal result in Figure 6a, the actual gray distribution is continuous and doesn't have a discontinuity point. After taking the derivative of the actual gray distribution in Figure 6b, the PSF in a discrete form is shown in Figure 6c. Since the typical expression of the PSF is Gaussian type [23], we use the Gaussian fitting function to fit the discrete points, and the fitting expression result can be seen in Figure 6c. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(\lambda\) (\(\upmu\)m) & w (\(\upmu\)m) & d (\(\upmu\)m) & \(n_{\text{Air}}\) & \(n_{\text{SiO}_{2}}\) & \(\beta\) \\ \hline
**Setting value** & 0.4–0.75 & 0.5–5 & \(<\)1 & 1 & \(n(\lambda)\) & 9.7\(\times\) \\ \hline \hline \end{tabular} \(\lambda\): incident light wavelength; \(w\): defect width; \(d\): defect depth; \(n_{\text{Air}}\): refractive index of the air; \(n_{\text{SiO}_{2}}\): refractive index of the SiO\({}_{2}\); \(\beta\): magnification of the objective lens.  
\end{table} 
Table 1: Parameters settings of the simulation model. Convolving the distribution in Figure 5 with the PSF, we get a revised distribution, as shown in Figure 7. The revised distribution is unimodal in Figure 7a-c and is bimodal in Figure 7d-f. Compared with Figure 5a-f, the span of the revised distribution in Figure 7a-f broadens. The bimodal distribution in Figure 5b,c becomes unimodal in Figure 7b,c, with the effect of the PSF. Figure 7d is the revised distribution of scratch, where \(w=3\)\(\upmu\)m, and we can see that the bimodal distribution begins to appear, which means the actual diffraction limit resolution of our MSDI system is around 3 \(\upmu\)m. With the increase in the scratch width, the distance between two peaks in bimodal distribution increases, as shown in Figure 7d-f. In order to validate the validity of the revision result by PSF, the images of five scratches are acquired by our MSDI system, and its width is measured by SEM. We utilize the simulation model to get the light intensity distribution of the scratch in the far-field whose width is equal to the value SEM measured. Figure 8 is the comparison between the revised simulation light intensity distribution and the actual gray distribution of the scratch at the same scratch width. Figure 8a corresponds to the scratch width of 1.17 \(\upmu\)m; both the two kinds of distributions are unimodal, and the revised simulation light intensity distribution feature parameter \(v_{\text{sim}}\) is slightly larger than the experiment gray distribution feature parameter \(v_{\text{exp}}\). For the scratch width of 2.23 \(\upmu\)m, \(v_{\text{sim}}\) is almost the same with \(v_{\text{exp}}\), as can be seen in Figure 8b. Figure 8c corresponds to the scratch width of 3.24 \(\upmu\)m, and the difference between \(v_{\text{sim}}\) and \(v_{\text{exp}}\) is about 1.9 pixels. In Figure 8d,e, both two kinds of distributions are bimodal, and their feature parameter difference is within 1 pixel. The results in Figure 8a-e indicate that the simulation light intensity distribution revised by PSF is almost cons  
Figure 6: (**a**) Gray distribution of ideal step source. Inset: Ideal image of step source; (**b**) Gray distribution of actual step source. Inset: Actual image of step source; (**c**) Gray distribution of point spread function (PSF).