## 4 Results and Discussion
### Estimation of Defect Depth and Width
It retains a substantial advantage over PCA + SVR, and the MAE could be as small as 12 \(\upmu\)m. The relative error is discussed in the following paragraphs. Figure 12 illustrates the curves of cumulative accuracy score versus error level for estimating defect depth and width. Cumulative accuracy score is defined as follows:  
\[CS=\frac{n_{err\leq l}}{n}, \tag{1}\]  
Figure 11: MAE of defect depth estimation for the first 20 dimensional features under different epsilon values. where \(n\) is the total number of test samples and \(n_{err\leq l}\) indicates the number of test samples whose absolute error \(err\) is less than the error level \(el\). The curve reveals the cumulative ratio of the estimated result below a certain error level. Accordingly, the closer the curve is to the upper left corner of the axis, the more accurate is the prediction. Figure 12a,b compares the performance of dimensionality reduction to 5, 20, 40, 70, and 110 dimensions when using the Isomap + SVR method. The perfo  
Figure 12: Comparison curve of cumulative accuracy score versus error level. (**a**) Depth estimation using isometric mapping (ISOMAP) + support vector regression (SVR) with different subspace dimensions, (**b**) width estimation using ISOMAP + SVR with different subspace dimensions, (**c**) depth estimation using SVR with 20 subspace dimensions, (**d**) width estimation using SVR with 20 subspace dimensions, (**e**) depth estimation using quadratic regression (QR) with 20 subspace dimensions, (**f**) width estimation using QR with 20 subspace dimensions. from 5 to 20 dimensions. Starting from 70 dimensions, however, it gradually degrades. The results indicate that the performance of 20 and that of 40 dimensions are similar. Thus, the range from 20 to 40 is preferable, which is in agreement with the MAE versus dimension curve in Figure 10. Figure 12a,b respectively shows that the depth estimation error is less than 1 \(\upmu\)m (approximately 5.4% relative to the maximum depth) in 85% of the samples when the dimensions are reduced to 40, and that the width estimation error is less than 30 \(\upmu\)m (approximately 5.5% relative to the maximum width) in 80% of the samples. Figure 12c,d displays the performance of four dimensionality reduction + SVR methods and SVR alone when the data are reduced to 20 dimensions. If dimensionality reduction is not executed, the number of support vectors obtained through SVR training is approximately 80. The number of support vectors is approximately 60 when we reduce the sample dimensions to 20. The performance of SVR alone without dimensionality reduction is poorer than with dimensionality reduction. This is because more support vectors probably lead to overfitting. For both depth estimation and width estimation, Isomap achieves the optimal results among the four dimensionality reduction methods. PCA is a linear dimensionality reduction method that cannot represent the internal structure of data. Isomap is a typical nonlinear dimensionality reduction method, and it could more effectively learn the intrinsic variables, such as defect depth and width, in a set of saturated images. Figure 12e,f compares the performance of the four dimensionality reduction methods plus quadratic regression (QR) in the case of data dimensionality reduction to 20 dimensions. All combination methods in Figure 12c,d are clearly more effective than those in Figure 12e,f. This is a result of the small sample size. The QR model is extremely sensitive to outliers and noise, which leads to serious overfitting of the model and poor generalization ability. When considering dimensionality reduction to only 20 dimensions, the MREs of defect depth and width are calculated separately. The results are shown in the Table 2. All relative errors have a small variance value, indicating that the relative error and absolute error distributions are concentrated. This is consistent with the curves near the upper left corner of the axis shown in Figure 12a-d.