# A Time-Frequency Image Quality Evaluation Method Based on Improved LIME
## 2 Materials and Methods
### Technical Route
#### 2.3.1 Time-Frequency Transform  
The acquired vibration signal is transformed by time-frequency transform to obtain a time-frequency matrix, and the time-frequency image is obtained after grayscale processing. There are many kinds of time-frequency transform methods, and it is unrealistic to evaluate all of them individually. This paper selects 5 common time-frequency transform methods for evaluation and comparison, as follows. 1. Short-time Fourier transform  
Short-time Fourier transform (STFT) is one of the earliest time-frequency transform methods. STFT, which approximates the instantaneous frequency of time-varying signals through window processing, is a common means to analyze nonstationary signals. For the signal \(x_{t}\), \(STFT\) is defined as follows:  
\[STFT(t,\omega)=\int_{-\infty}^{\infty}x(\tau)H^{*}(\tau-t)e^{-j\omega\tau}d\tau \tag{7}\]  
where \(\omega\) is the angular frequency and \(H\) is the window function. 2. Wavelet transform  
Wavelet transform adds time translation and scaling parameters to the basis function and has an adaptive microscopic ability to frequency changes, which often makes it better reflect the characteristics of time-varying signals. For the signal \(x_{t}\), its wavelet transform is defined as follows:  
\[WT(a,b)=\int_{-\infty}^{\infty}x(t)\psi\bigg{(}\frac{t-b}{a}\bigg{)}dt \tag{8}\]  
Figure 2: Quality evaluation method based on improved LIME. Winger-Ville transform is also one of the earliest time-frequency conversion methods, which was first applied to quantum mechanics and later introduced into signal analysis. It has the advantages of symmetry, time-shift invariance, and frequency modulation invariance. For the signal \(x_{t}\), its Winger-Ville transform is defined as follows:  
\[WVT(t,\omega)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{-j2\pi\omega \tau}d\tau \tag{9}\]  
4. Empirical mode decomposition  
Since 2000, based on data-adaptive processing technology, empirical mode decomposition (EMD) has occupied a pivotal position in the field of nonstationary signal processing. As a representative of the signal time-frequency transform preprocessing method, EMD performs signal decomposition according to the timescale characteristics of the data themselves, without presetting the basis function. For the signal \(x_{t}\), its EMD is as follows:  
\[x(t)=\sum_{n=1}^{N}imf_{n}(x)+res(x) \tag{10}\]  
where \(imf\) is the \(n\)-order intrinsic mode function and \(res\) is the residual. 5. Synchrosqueezing transform  
In 2011, [PERSON] et al. proposed Synchrosqueezing transform (\(SST\)) from the aspect of postprocessing of time-frequency transform [22]. By compressing the value of the frequency interval [\(\omega_{i}-\Delta\omega\), \(\omega_{i}+\Delta\omega\)] near \(\omega\) in the frequency of the time-frequency matrix toward \(\omega\), the time-frequency energy aggregation is improved. Its process can be expressed as follows:  
\[SST(\omega_{i})=\Delta\omega^{-1}\sum_{|TF_{x}-\omega_{i}|\leq\frac{\Delta \omega}{2}}TF_{x}^{-1} \tag{11}\]  
where \(TF\) is the time-frequency matrix obtained by the time-frequency transformation of signal \(x_{t}\). A one-dimensional vibration signal can be transformed into a two-dimensional time-frequency matrix by using a time-frequency transform method. However, the time-frequency matrix formed by different methods may be very different in the amplitude range, as shown in Figure 3. Even if it is the same method, the amplitude range is not the same under different preset parameters. As a result, when the same time-frequency transform method with different preset parameters or different simultaneous frequency transform methods are adopted, the time-frequency image generated by the same signal has a great difference in color performance. Therefore, in engineering applications, the time-frequency matrix is usually transformed into a grayscale image by grayscale processing as the input of the model.