## 2 Improved EWT
### Component Selecting Based on MI and Kurtosis  
After re-dividing the spectrum through the boundary-optimization method and decomposing \(f(t)\), it is necessary to select suitable components containing the defect signal for signal reconstruction. MI reflects the relationship between each component and \(f(t)\). From Figure 4, it can be observed that \(MI\) between \(f_{1}(t)\) and \(f_{n}(t)\), and \(MI\) between \(f_{3}(t)\) and \(f_{n}(t)\) should be both less than \(M_{m}\), which is more likely to include defect signals. However, as shown in Figure 5, \(f_{3}(t)\) is distributed in the high-frequency region, and from the frequency-domain perspective, the possibility of \(f_{3}(t)\) containing a defect signal is extremely low. Therefore, if the components selected are only based on MI, components without a defect may be selected, resulting in the reconstructed signal still containing high interference. Therefore, in addition to MI, kurtosis is also used to select the suitable components.  
Kurtosis is a dimensionless parameter that characterizes a signal waveform's peak degree. For an arbitrary input signal \(x\), define the kurtosis as:  
\[K=\frac{E(x-\mu_{0})^{4}}{\sigma^{4}} \tag{10}\]  
\(\mu_{0}\) is the mean of \(x\). \(\sigma\) is the standard deviation of \(x\). \(E(\cdot)\) represents compute the mathematical expectation.  
\(K\) is very sensitive to the mutagenic component in a signal. If a signal approximately follows normal distribution, then \(K\) is about 3, such as Gaussian white noise. If a signal has an impact or protrusion, then \(K\) is greater than 3. Therefore, when using kurtosis to analyze the defect MFL signal, the \(K\) of the signal has the following characteristics: if no defect is detected, the signal only contains noise, and its amplitude is close to normal distribution,  
Figure 5: New components after boundary optimization.  
then \(K\) is about 3. If some defects are detected, the signal has obvious protrusions, and does not follow normal distribution, then \(K\) is greater than 3. The steeper the peak value of the signal, the greater the \(K\). A large number of detection data have been studied, and this pattern has been proven to be correct.  
The process of components selecting is as shown in Figure 6:  
1. Calculate \(MI\) between components \(f_{n}(t)\) and \(f(t)\), and calculate \(K\) of \(f_{n}(t)\).
2. Compare each \(MI\) with the mean \(MI_{m}\) and \(K\) of each \(f_{n}(t)\) with 3. The components whose \(MI\) is not greater than \(MI_{m}\) and \(K\) is greater than 3 are retained. Other components are deleted.
3. Reconstruct the undeleted components according to Equation (8) to obtain the reconstructed signal.  
The \(MI\) and \(K\) of each component in Figure 5 are shown in Table 1. Firstly, by comparing each \(MI\) with \(MI_{m}\), \(f_{1}(t)\) and \(f_{3}(t)\) are selected. And then, observing \(K\) of the two, only the \(K\) of \(f_{1}(t)\) is greater than 3. Therefore, only \(f_{1}(t)\) is retained, and other components are deleted. The reconstructed signal is \(f_{1}(t)\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \(MI\) & \(K\) \\ \hline \(f_{1}(t)\) & 0.8001 & 5.4834 \\ \(f_{2}(t)\) & 0.8235 & 2.8071 \\ \(f_{3}(t)\) & 0.8162 & 2.7437 \\ \(f_{4}(t)\) & 0.8432 & 1.9591 \\ mean & 0.8208 & \(-\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: \(MI\) and \(K\) of each component.  
Figure 6: Component-selection process.