## 2 The Principle of the Sine-Cosine Transform Wavelet Threshold Denoising Method
Combined with the characteristics of speckle Gaussian white noise, the threshold form of the combination of fixed threshold form and noise level is selected for denoising. \[\left\{ \begin{array}{l}T_{hr}=\sigma\sqrt{2\ln\mathrm{N}}\\ \sigma=\frac{\mathrm{median}(|k_{N})}{0.6\sigma 45}\end{array} \right. \tag{5}\]  
where \(T_{hr}\) is the denoising threshold, N is the decomposition level, \(\sigma\) is the noise intensity, and \(k_{N}\) is the original wavelet coefficients in level N.  
The wavelet coefficient should be processed for denoising according to the threshold function. The threshold function mainly includes the hard threshold function and soft threshold function. A hard threshold denoising function can easily cause local jitter and remove useful signal information. There is no sudden change in the wavelet coefficients when using the soft threshold denoising function. Therefore, the soft threshold processing function is adopted for the wrapped phase patterns. When the absolute value of the wavelet coefficient is more than the given threshold, the wavelet coefficient subtracts the threshold for the soft threshold processing function. The wavelet coefficient is set to zero when it is less than the threshold. \[\eta_{H}(k,T_{hr})=\left\{ \begin{array}{l}k-T_{hr},k\geq T_{hr}\\ 0,|k|<T_{hr}\\ k+T_{hr},k\leq-T_{hr}\end{array} \right. \tag{6}\]  
where \(k\) is the original wavelet coefficients, and \(\eta_{H}(k,T_{hr})\) is wavelet coefficients processed by the threshold. The soft threshold global denoising of wavelet coefficients can be expressed as  
\[\left\{ \begin{array}{l}Spw^{\prime}=\eta_{H}(cSpw,T_{hr})\\ Cpw^{\prime}=\eta_{H}(cCpw,T_{hr})\end{array} \right. \tag{7}\]  
where \(Spw^{\prime},Cpw^{\prime}\) are the denoised wavelet coefficients of the sine and cosine patterns, respectively. (4) Wavelet reconstruction of sine and cosine patterns after threshold processing. The 2-D inverse discrete wavelet transform (IDWT) algorithm for patterns is shown in Figure 3. The 2-D IDWT algorithm reconstructs the approximation coefficients \(c\hat{f}_{j+1}\), \(cD^{(h)}_{j+1}\), \(cD^{(v)}_{j+1}\) and \(cD^{(d)}_{j+1}\) into \(c\hat{f}_{j}\) at scale \(j\). Figure 3: 2-D inverse discrete wavelet transform (IDWT) algorithm. The wavelet reconstruction transform can be expressed as  
\[\left\{ \begin{array}{l}Spw^{*}=\text{Re}(Spw^{\prime})\\ Cpw^{*}=\text{Re}(Cpw^{\prime})\end{array} \right. \tag{8}\]  
where \"Re\" means wavelet reconstruction, and \(Spw^{*}\), \(Cpw^{*}\) are the reconstructed sine and cosine patterns, respectively. Calculate the inverse tangent of sine and cosine after denoising. \[pw=\arctan\left(\frac{Spw^{*}}{Cpw^{*}}\right) \tag{9}\]  
where \(pw\) is the denoised shearography-interferometric phase fringe patterns. Calculating speckle suppression index to determine the optimal decomposition level. The speckle suppression index (SSI) reflects the ability of noise suppression [24]. And the smaller the speckle suppression index is, the better the denoising effect is. \[I_{\text{SSI}}(\text{N})=\frac{\sqrt{\sigma_{I}}}{\mu_{I}}/\frac{\sqrt{\sigma_{ F}}}{\mu_{F}} \tag{10}\]  
where \(\sigma_{I},\mu_{I}\) are the variance and mean of a denoised pattern, and \(\sigma_{F},\mu_{F}\) are the variance and mean of a noisy pattern. Calculate the speckle suppression index when the decomposition level is \(3,\cdots,N\), \((N\subseteq N^{*})\). The optimal decomposition level can be regarded as  
\[\text{N}_{0}=\text{argmin}\{\text{I}_{\text{SSI}}(\text{N})\} \tag{11}\]  
where \(N_{0}\) is the optimal decomposition level, and \(\text{argmin}\{\text{I}_{\text{SSI}}(\text{N})\}\) represents the decomposition level among \(3,\cdots,N\) when the SSI is minimum.