## 2 Materials and Methods
### Intrinsic Feature Extraction Based on Time-Frequency Image
#### 2.2.3 Feature extraction  
After obtaining the time-frequency image and normalized time-frequency image of acoustic emission signal, it is necessary to extract the features based on the time-frequency image. The characteristics of the time-frequency diagram include energy characteristics, time-domain and frequency-domain expansion characteristics and time-frequency intrinsic characteristics.  
Energy characteristics: The time-frequency information of the signal is obtained by analyzing the energy distribution characteristics in the time-frequency image. It includes the energy of a specific frequency band, the total energy and the statistical characteristics of the time-frequency diagram compressed along the time axis or the frequency axis, as shown in Tables 1 and 2.  
The time-frequency diagram is a two-dimensional matrix. The distribution of time-frequency energy with time can be obtained by summing the coefficients of each column:  
\[P_{t}[n]=\sum_{k}P[n,k] \tag{4}\]  
Sum the coefficients of each row separately to obtain the distribution of time-frequency energy with frequency:  
\[P_{f}[k]=\sum_{n}P[n,k] \tag{5}\]  
Both distributions are one-dimensional vectors, for which statistical features are extracted separately, as shown in Table 2.  
Time-domain and frequency-domain expansion features: Time domain features are based on the basic assumption that there are different probability distributions between normal signals and abnormal signals and that the characterization of monitoring targets is realized by extracting the statistical features of signals. The common time-domain characteristics are usually for one-dimensional signals. If the time-frequency image is regarded as a two-dimensional signal, the time-domain expansion characteristics of the time-frequency diagram can be obtained. This paper gives some extended features corresponding to common time-frequency features, as shown in Table 3. When it is necessary to distinguish between two different signals with a similar frequency spectrum, it is solved by expanding the frequency domain features to the time-frequency domain. Table 4 shows some common frequency-domain features and corresponding extended features.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Energy Characteristics** & **Expression** \\ \hline Specific frequency band energy & \(EB_{(l,f)}=\sum\limits_{k=\left[(l-1)A+1\right]}^{[iA]}P[n,k]\) \\ total energy & \(E_{(l,f)}=\sum\limits_{n=1}^{N}\sum\limits_{k=1}^{M}P[n,k]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Energy characteristics of the time–frequency diagram.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distribution over** & **Expression** & **Distribution with** & **Expression** \\
**Time** & & **Frequency** & **Expression** \\ \hline variance & \(\hat{\sigma}_{t}^{2}=var(P_{t}[n])\) & variance & \(\hat{\sigma}_{f}^{2}=var\left(P_{f}[k]\right)\) \\ skewness & \(\hat{\gamma}_{t}=ske(P_{t}[n])\) & skewness & \(\hat{\gamma}_{f}=ske\left(P_{f}[k]\right)\) \\ kurtosis & \(\hat{k}_{t}=kur(P_{t}[n])\). & kurtosis & \(\hat{k}_{f}=kur\left(P_{f}[k]\right)\) \\  & & peak frequency & \(\hat{f}_{p_{f_{max}}}=F\left(\left[P_{f_{max}}\right]\right)\) \\  & & Frequency center of gravity & \(\hat{f}_{m}=\frac{\sum_{k}F[k]P_{f}[k]}{M}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical characteristics of the compressed time–frequency matrix along the time axis or the frequency axis.  
Time-frequency intrinsic features: Some features are essentially time-frequency features, which cannot be defined in the time domain or the frequency domain, that is, the intrinsic features of a time-frequency image. In this paper, the intrinsic characteristics and their expressions based on a time-frequency diagram are given in Table 5.