## 5 Results Analysis
### Quantitative Analysis  
(1) Feature Extraction to Thermal Response  
Prior to extracting the characteristic, it is necessary to establish reference positions and reference maximum heating frames. The reference position is the crevice where the rail surface is defectively grooved, as demonstrated in Figure 9a. As mentioned in Section 5.1, the heating process results in a large negative skewness of the defect location compared to the surrounding area, helping to locate the reference position. Figure 9b illustrates the skewness observed along the lines of the reference positions of six defects (using the excitation frequency of 25 kHz as an example). The reference maximum heating frames can be selected through the temperature curves. This experiment provides a detailed examination of the relationship between temperature information extracted from sweeping frequency heating infrared images and the depth and layer number of defects. The experiments are controlled for the same heating power at different excitation frequencies, and TSR values are used to characterize and distinguish different types of defects. During the heating and cooling phases of the ECPTT, the temperature changes are close to a logarithmic curve [23]. TSR quantifies the information about the temperature change into a value by taking a logarithm, reinforcing the linear relationship between the different thermal signals. Therefore, the value of TSR is more capable of distinguishing the characteristics of different thermal images. Table 5 shows the values of TSR in the defect region at different frequencies. The mean, the standard deviation, and the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Excitation** & **Defect 1** & **Defect 2** & **Defect 3** & **Defect 4** & **Defect 5** & **Defect 6** \\ \hline
9 kHz & (a) & (b) & (c) & (d) & (e) & (f) \\
16 kHz & (g) & (h) & (i) & (j) & (k) & (k) \\
25 kHz & (m) & (n) & (o) & (p) & (q) & (r) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Thermal image of center trench plane coil. coefficient of variation in TSR are 1.1906, 0.0792, and 0.0665, respectively. The coefficient of variation measures the dispersion of the data; it can be seen that the dispersion of TSR values are pronounced enough to distinguish different types of defects. Consequently, the TSR value is used to analyze the differences among various types of defects. (2) Feature Analysis for Distinguishing Multi-layer Defects  
(a) The Depth of Defects  
The temperature curves of characteristic points for single-layer and double-layer defects with different depths are shown in Figure 10. From the TSR noticed in Figure 11, it can be seen that when the excitation frequency increases, the temperature curves of shallow defects have a significant increase, while the temperature curve rise of deeper defects is significantly smaller. The result indicates that changing frequency excitation is beneficial for distincting defects with different depths. (b) The Layer of Defects  
The temperature curves of multi-layer defects in three excitation frequencies at characteristic points are shown in Figure 11. In this experiment, defects 1, 2, and 3 are single-layer, double-layer, and triple-layer defects in the same depth, respectively. As the excitation frequency increases, the temperature curve of single-layer defects varies smoothly, while the temperature curve of multi-layer defects changes abruptly at a certain frequency value \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Defect Number**} & \multicolumn{3}{c}{**Excitation Frequency**} \\ \cline{2-4}  & **9 kHz** & **16 kHz** & **25 kHz** \\ \hline
1 & 1.17 & 1.18 & 1.27 \\ \hline
2 & 1.10 & 1.24 & 1.30 \\ \hline
3 & 1.04 & 1.10 & 1.38 \\ \hline
4 & 1.19 & 1.20 & 1.26 \\ \hline
5 & 1.15 & 1.15 & 1.20 \\ \hline
6 & 1.15 & 1.17 & 1.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Thevalues of TSR at different frequencies. Figure 9: Position and skewness of the feature area. (double-layer defect in 16 kHz and triple-layer defect in 25 kHz).