## 2 Improved EWT
### A Boundary-Optimization Method Based on MI  
Spectrum division is the basis for signal reconstruction and determines the quantity and quality of the decomposed signals. Due to the adaptive decomposition process of EWT, the signal boundary may be redundant if the signal is complex, such as the MFL detection signal, with vibration noise and other interferences. The redundant boundary leads to the obtention of unnecessary components and disrupts the integrity of the spectrum. In order to reduce the boundary redundancy in EWT, a boundary optimization method based on MI is proposed.  
MI considers the mutual independence between variables from the perspective of probability distribution. Assuming that two random variables are \(X\) and \(Y\), MI can quantitatively represent the degree of interdependence between the two, where  
\[MI(X,Y)=H(Y)-H(Y|X) \tag{9}\]  
\(MI(X,Y)\) is the MI of \(X\) and \(Y\), \(H(Y)\) is the entropy of \(Y\), and \(H(Y|X)\) is the entropy of \(Y\) under the condition of \(X\). The stronger the correlation between \(X\) and \(Y\), the smaller the \(H(Y|X)\), and the greater the \(MI(X,Y)\).  
After dividing the spectrum using the scale space method to obtain some initial components, the correlation between each component and the original signal based on MI is calculated. The initial components are screened via correlation, and the spectrum is redivided to eliminate redundant components.  
The boundary-optimization process is shown in Figure 2. Let \(i\) represent any one of the initial components, and component \(i-1\) and \(i+1\) are the adjacent components of component \(i\). Suppose \(MI_{i}\) is the MI between component \(i\) and the original signal \(f(t)\), \(MI_{m}\) is the mean of all the \(MI_{i}\). Calculate each \(MI_{i}\) and compare it with \(MI_{m}\). If \(MI_{i}>MI_{m}\) and \(MI_{i+1}>MI_{m}\), or \(MI_{i}\leq MI_{m}\) and \(MI_{i+1}\leq MI_{m}\), then merge component \(i\) and \(i+1\). If \(MI_{i}>MI_{m}\) and \(MI_{i-1}>MI_{m}\), or \(MI_{i}\leq MI_{m}\) and \(MI_{i-1}\leq MI_{m}\), then merge component \(i\) and \(i-1\). Otherwise, the components remain independent. After merging, some new boundaries can be obtained to re-divide the spectrum.  
Figure 2: Boundary-optimization process.  
Here is an example to introduce this method. The signal consists of three components: a defect MFL signal in the x-direction, as show in a red rectangle in Figure 3, a random interference, and vibration noise.  
The signal was adaptively divided into 23 initial components using the scale space method. Some components were redundant. \(MI_{i}\) were calculated and are shown in Figure 4. \(MI_{m}\) is marked with a solid red line, and the new boundaries obtained via the boundary-optimization method are marked with dashed red lines. After re-dividing the spectrum, we constructed a filter bank according to Equations (2) and (3) to obtain four modal components, as shown in Figure 5. Record the new components as \(f_{n}(t)\).  
According to the principle of MI and the composition characteristics of the signal, the larger the \(MI_{i}\), the more noise ingredients unrelated to defect information are contained in the component \(i\). From the bar graph of \(MI_{i}\), it can be seen that the first component is located in the low-frequency part and has the smallest \(MI_{i}\), indicating that it has the smallest correlation with the tested signal. Meanwhile, combined with Figure 3, a significant peak  
Figure 4: \(MI\) between the initial components and the tested signal.  
Figure 3: Tested signal-construction process.  
appears at the defect location, as shown in a red rectangle in Figure 5, indicating that this component contains a defect signal.  
The defect component, vibration noise component, and random interference component in the signal are successfully separated, and the number of components is reduced from the initial 23 to 4. Obviously, this method has no redundant components and decomposes signals accurately.