## 3 Case Study
### Results and Discussions  
Several indicators can be utilized to describe the accuracy of binary classification, including pixel accuracy (PA), true positive rate (TPR), true negative rate (TNR), positive predictive value (PPV), negative predictive value (NPV), etc. In this study, TPR and TNR are used to evaluate model performance, which are defined as [26]  
\[\text{TPR}=\frac{\text{TP}}{\text{TP}+\text{FN}} \tag{3}\]  
\[\text{TNR}=\frac{\text{TN}}{\text{FP}+\text{TN}} \tag{4}\]  
where TP, FP, FN, and TN represent the patch number of true positives, false positives, false negatives, and true negatives, respectively.  
\begin{table}
\begin{tabular}{c c} \hline
**Epoch** & **Learning Rate** \\ \hline
1–20 & 0.01 \\
21–40 & 0.001 \\
41–60 & 0.0001 \\
61–80 & 0.00001 \\
81–100 & 0.000001 \\ \hline \end{tabular}
\end{table}
Table 2: Learning rate.  
Figure 9: A sample of the training data for the CNN-GAT model.  
The eight raw images used for testing are not involved in the establishment of the databank, so the detection results over these images can show the performance of the proposed CNN-GAT model. A sliding window of 224 \(\times\) 224 pixels scans with a stride of 112 pixels is used over the test images. The patches inside the window and their neighbors are input into the trained CNN-GAT model to execute the classification. An example test result is exhibited in Figure 11. The patches classified as \"crack\" constitute the crack area and show the position of the cracks and verify the detectability of the proposed approach. The confusion matrix of the test results is illustrated in Figure 12. It can be figured out that both the concerned indicators are over 90%, which confirms the ability of the CNN-GAT model to detect cracks.  
Figure 11: An example test result of the CNN-GAT model: (**a**) original test image, (**b**) illustration of cracks, (**c**) patches classified as “crack” by the CNN-GAT model (shown in white), and (**d**) crack detection result.  
Figure 10: The training process of the CNN-GAT model.  
The pre-trained CNN model in Section 3.2 is also applied to the test images, and the test results are compared with those of the CNN-GAT model. Since the architecture of the pre-trained CNN model is the same as that of the front CNN part in the proposed CNN-GAT model, such a comparison can clarify the advantage of adding the GAT after the CNN. An example test result of the CNN is shown in Figure 13, in which the crack zone is not identified completely and a relatively large number of patches without cracks are classified as \"crack\" by mistake. Obviously, the performance of the CNN-GAT model surpasses that of the CNN through the comparison. The confusion matrix of the test results with the pre-trained CNN is illustrated in Figure 14. It is observed that the TPR of the CNN model is only 0.7821, while the TPR of the CNN-GAT model is as high as 0.9106, which implies that the proposed method is much more sensitive to cracks. Meanwhile, the TNR of the proposed CNN-GAT is almost as high as that of the CNN, which indicates that the increase in sensitivity to cracks will not cause an increase in sensitivity to other interfering factors. This can be attributed to the fact that the established graph structure in the proposed framework takes into account the spatial continuity of cracks, so the CNN-GAT model can concentrate on the features related to cracks instead of other irrelevant features. Through the quantitative comparison of TPR and TNR, it can be inferred that the introduction of GAT could enhance the crack identification ability of the CNN-based model, which also encourages researchers to add extra graph neural networks to other models for improvement.  
Figure 12: Confusion matrix of the test results by the CNN-GAT model.  
Figure 13: An example test result of (**a**) the CNN-GAT model and (**b**) the pre-trained CNN model.