## 4 Results and Discussion
### Estimation of Defect Depth and Width  
In this subsection, we explain the application of the support vector regression model mentioned in Section 3.3 for estimating the depth and width of the defect, and we analyze the error. To evaluate the performance of a support vector regression model, leave-one-out cross-validation is [23] applicable in situations where the sample set is small. For each of the \(n=113\) samples, one sample was taken as the test set each time with the other samples used as the training set. Thus, 113 trainings and tests have been performed. The absolute error of each test is \(\big{|}l_{i}-\hat{l}_{i}\big{|}\), where \(l_{i}\) and \(\hat{l}_{i}\) are respectively the parameter (depth or width) ground truth and the estimated value of the \(i\)th test sample. The mean absolute error (MAE)\(=\frac{1}{n}\sum_{i=1}^{n}\big{|}l_{i}-\hat{l}_{i}\big{|}\) and the standard deviation of absolute error (SAE)\(=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\big{(}\big{|}l_{i}-\hat{l}_{i}\big{|}-MAE \big{)}^{2}}\) are considered for performance comparison. The mean relative error (MRE) is \(\frac{1}{n}\sum_{i=1}^{n}\big{|}l_{i}-\hat{l}_{i}\big{|}/l_{i}\). We also consider the standard deviation of relative error (SRE) to show the  
Figure 10: The mean absolute error (MAE) and the standard deviation of absolute error (SAE) performance comparison for defect depth and width estimation. (**a**) Depth estimation MAE vs. dimension number, (**b**) depth estimation SAE vs. dimension number, (**c**) width estimation MAE vs. dimension number, (**d**) width estimation SAE vs. dimension number. precision of the estimation; that is, \(\sqrt{\frac{1}{n}\sum_{i=1}^{n}(r_{i}-MRE)^{2}}\), where \(r_{i}=\big{|}l_{i}-l_{i}\big{|}/l_{i}\) is the relative error of the \(i\)th sample. The input features of the regression model are the points in the low-dimensional manifold space \(\mathcal{Y}=\big{\{}y_{i}:y_{i}\in\mathcal{R}^{d}\big{\}}_{i=1}^{n}\) resulting from reduction of the sample space \(\mathcal{X}=\big{\{}x_{i}:x_{i}\in\mathcal{R}^{D}\big{\}}_{i=1}^{n}\) by using PCA, Isomap, LLE, or LE (see Figure 5 for the flowchart). The choice of the insensitivity parameter \(\varepsilon\) in the SVR model greatly influences performance. If \(\varepsilon\) is too small, it can lead to too many support vectors, and the model can overfit. If \(\varepsilon\) is too high, too few support vectors can lead to underfitting. According to the result in Figure 11, \(\varepsilon=1\) is a suitable choice for all four dimensionality reduction methods. They all obtain lower MAE when the first 20 dimensional features are input; therefore, we fixed this parameter in later experiments. Considering the depth and width of the defect respectively, as shown in Figure 10, we compared the MAE and SAE of the SVR model with different dimensionality reduction methods and different numbers of features. These comparisons almost uniformly reflect the following rules. For both defect depth (Figure 10a) and defect width (Figure 10c), all MAEs of the four combined methods (i.e., PCA + SVR, Isomap + SVR, LLE + SVR, and LE + SVR) show a downward trend with the addition of 2 to 20 features. This is because the more features that are used, the richer the discriminant information is. Conversely, after 60 features, excessive redundancy of information hinders the regression performance. Additionally, the MAEs of the four combined methods increase slowly. Therefore, it is ideal to select 20 to 60 dimensional features. As shown in Figure 10b,d, by contrast, the SAEs of the four methods present a gradual downward trend. When the feature dimensions exceed 20, the SAEs of the four methods are relatively small and stable. Isomap + SVR has a comparative advantage over the other three methods when estimating defect depth. Figure 10a shows that the MAE of the defect depth can be as small as 1.5 \(\upmu\)m. When estimating the defect width, Isomap + SVR is slightly more effective than LLE + SVR and LE + SVR.