## 4 Fault Identification
### Infrared Feature Extraction  
We obtained 120 rod sample segmentations from the obtained samples through infrared image segmentation, which were regrouped in 40 sample sets. Each set includes three different samples and reflects three different conductive states. In each set, brightness differences of the three sample segmentations exist, and uneven gray distributions and gray gradient distributions under the abnormal heating condition are observed. These different items can be described as follows:  
1. [label=(0)]
2. Gray mean (_mean_)  
The gray mean reflects the average temperature of the copper rod. The higher the gray mean is, the higher the temperature of the copper rod:  
\[mean=\frac{1}{M\times N}\sum_{i}^{M}\sum_{j}^{N}I(i,j) \tag{3}\]  
where \(M\) and \(N\) are the number of rows and columns of the copper rod sample gray image \(I\). 3. Gray standard deviation (_std_)  
The gray standard deviation reflects the discrete degree of the surface temperature distribution of the copper rod. The larger the value of the gray standard deviation, the higher the dispersion degree of the temperature:  
\[std=\sqrt{\frac{1}{M\times N}\sum_{i}^{M}\sum_{j}^{N}\left[I(i,j)-mean\right] ^{2}} \tag{4}\]
4. Mean gradient (_G_)  
The mean gradient reflects the gradient variation of the copper rod surface temperature. From the conductive end to the isolated end, the temperature in the copper rod gradually decreases and approaches the environmental temperature. Gray gradient variation can be quantified in rod sample segmentations and this gradient variation is obvious under the abnormal heating condition but not clear under the other conditions. Figure 13 shows the gradient variation in different directions. Figure 12: Result of the copper rod segmentation. Actually, conventional Hu moments include seven invariant moments, but after comparing all rod sample segmentations, it is found that only the first three Hu moments (\(\Phi_{1},\Phi_{2},\Phi_{3}\)) show obvious differences, which are used for the image feature in this paper. Hu moments always have a large range and can be negative values. The parameter \(H_{i}\) is defined, which is the natural logarithm value of the absolute values of Hu moments:  
\[H_{i}=\ln[\Phi_{i}]\quad i=1,2,3 \tag{11}\]  
(5) Gray difference between the target copper rod and the isolated end of the adjacent copper rod (\(D\))  
To reduce the influence of the environmental temperature on the identification results, the gray difference can be used to identify differences in the infrared feature between the target copper rod and the isolated end of the adjacent copper rod because the mean gray in the area of the isolated end represents the current environmental temperature. \(D\) is defined as follows:  
\[ \begin{array}{l}D=mean1-mean2\\ mean1=\frac{1}{M_{1}\times N_{1}}\sum\limits_{i=1}^{M_{1}}\sum\limits_{j=1}^{N_{1 }}I_{1}(i,j)\\ mean2=\frac{1}{40\times N_{2}}\sum\limits_{i=M_{2}-40}^{M_{2}}\sum\limits_{j=1}^{N _{2}}I_{2}(i,j)\end{array}  \tag{12}\]  
where \(mean1\) is the gray mean of the target copper rod area; \(M_{1}\) and \(N_{1}\) are the number of rows and columns; \(mean2\) is the gray mean of the adjacent copper rod area, which is used for comparison; \(N_{1}\) is the number of columns; and the number of rows is limited to 40. Figure 14 shows an illustration of the calculation method for the infrared feature \(D\). Table 1 shows the results of infrared feature extraction and the fault labels for part of the rod sample segmentations. The fault labels for the abnormal heating state, normal operation state, and short circuit state are set to 0, 1, and 2, respectively.