## Materials and Methods
### Principle of Wavelet Threshold Denoising
#### 2.2.1 Wavelet Threshold Denoising  
The definitions of continuous wavelet transform and inverse transform are as follows:  
\[W_{T}(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}x(t)\cdot\psi\bigg{(} \frac{t-b}{a}\bigg{)}dt, \tag{7}\]  
\[x(t)=\frac{1}{C_{\varphi}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W_{ T}(a,b)\psi_{a,b}(t)\overset{2}{a^{-2}}dadb, \tag{8}\]  
where \(C_{\varphi}=\int\frac{\left|\psi(w)\right|^{2}}{\left|w\right|}dw<+\infty\), \(x(t)\) stands for the image to be processed, \(\psi(\)t\()\) represents the wavelet function, \(a\) is expansion coefficient, \(b\) is translation value. In fact, the continuous wavelet transform theory is mainly used in theoretical research. In practical application, the scale parameter \(a\) and translation parameter \(b\) need to be discretized. Normally, \(a=a_{0}^{j}\), \(a=kb_{0}a_{0}^{j}\), and \(a_{0}>1\), \(b_{0}>0\), \(j\) and \(k\) are rounded. The definition of discrete wavelet is as follows: \begin{table}
\begin{tabular}{c c} \hline \hline
**Type** & **Parameter** \\ \hline System & Windows10 (64 bit) \\ Central processing unit (CPU) & AMD Ryzen 5 3500 u 2.10 GHz \\ Random access memory (RAM) & 16 GB \\ Read only memory (ROM) & SSD (512 GB) \\ Matlab version & 2019b \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The configuration and parameters of processing platform. \[\varphi_{j,k}=\frac{1}{\sqrt{a_{0}^{j}}}\varphi\Bigg{(}\frac{t-kb_{0}a_{0}^{j}}{a_ {0}^{j}}\Bigg{)}=a_{0}^{-j/2}\varphi\Big{(}a_{0}^{-j}t-kb_{0}\Big{)} \tag{9}\]  
The corresponding discrete wavelet transform is as follows:  
\[W_{T}(j,k)=a_{0}^{-j/2}\int_{-\infty}^{+\infty}T(t)\varphi_{jk}(t)dt=a_{0}^{-j/ 2}\int_{-\infty}^{+\infty}T(t)\varphi\Big{(}a_{0}^{-j}t-kb_{0}\Big{)}dt \tag{10}\]  
In practical application, the commonly used discrete method is to discretize \(a\) and \(b\), so \(a_{0}=2\), \(b_{0}=1\). The discrete wavelet is as follows:  
\[\varphi_{jk}(t)=2^{-j/2}\varphi\Big{(}2^{-j/2}t-k\Big{)} \tag{11}\]  
Since the image is a two-dimensional spatial signal, it is necessary to extend one-dimensional wavelet transform into two-dimensional wavelet transform. The wavelet threshold image denoising process is as follows:  
1. Wavelet transform. For noisy images, we select a suitable wavelet function, and use the wavelet function to perform multi-layer decomposition of noisy images through wavelet transform to obtain a set of wavelet coefficients. 2. Non-linear threshold. According to the characteristics of the noise in the noisy image, an appropriate noise threshold is selected through a threshold function, and the wavelet coefficients of the noise part in the high-frequency decomposition value are reset to zero. Commonly used threshold functions are hard threshold function, soft threshold function and semi-soft threshold function. The hard threshold function is given as follows: \[\overline{W_{j,k}}=\left\{ \begin{array}{ll}W_{j,k}&,\Big{|}W_{j,k}\geq\tau \Big{|}\\ 0&,\Big{|}W_{j,k}<\tau\Big{|}\end{array} \right..\] (12) The soft threshold function is as follows: \[\overline{W_{j,k}}= \begin{cases}\text{sgn}\Big{(}W_{j,k}\Big{)}\Big{(}\big{|} W_{j,k}\big{|}-\tau\Big{)}&,\Big{|}W_{j,k}\geq\tau\Big{|}\\ 0&,\Big{|}W_{j,k}<\tau\Big{|}\end{cases} .\] (13) The semi-soft threshold function is as follows: \[\overline{W_{j,k}}= \begin{cases}\text{sgn}\Big{(}W_{j,k}\Big{)}\Big{|}W_{j,k} \Big{|}-\tau+\frac{2\tau}{1+e^{W_{j,k}/\tau}}&,\Big{|}W_{j,k}\geq\tau\Big{|}\\ 0&,\Big{|}W_{j,k}<\tau\Big{|}\end{cases} .\] (14) where \(\overline{W_{j,k}}\) is wavelet shrinkage coefficient; \(W_{j,k}\) is wavelet coefficient; \(\text{sgn}()\) stands for the symbolic function and \(\tau\) is the wavelet threshold. 3. Inverse wavelet transform to reconstruct the image. The processed wavelet estimation coefficients are subjected to inverse wavelet transform to obtain a noise-removed image.