## 3 Simulation and Comparative Analysis  
Shearography-interferometric phase fringe patterns of 600 \(\times\) 600 pixels with different noise and fringe densities are provided. Assume that the out-of-plane displacement \(W(x,y)\) satisfies the following:  
\[W(x,y)=w\cdot\exp\left(-\frac{x^{2}+y^{2}}{2}\right)\ldots w\subseteq[10,40 ]\&w\subseteq N^{*} \tag{12}\]  
where \(x,y\in[-3,3]\) are the normalized distance units. Set the shearing direction along \(x\). Then, the relationship between the unwrapping phase \(\varphi(x,y)\) and the out-of-plane displacement \(W(x,y)\) satisfies the following:  
\[\varphi(x,y)=\frac{4\pi}{\lambda}\frac{\delta W(x,y)}{\delta x}\Delta x \tag{13}\]  
where \(\lambda\) is the wavelength of light in the system, \(\Delta x\) is the amount of shear in the \(x\) direction, and \(\delta W(x,y)/\delta x\) is \(W(x,y)\) the difference with respect to \(x\). Calculate the wrapping phase of \(\varphi^{\prime}(x,y)\). \[\varphi^{\prime}(x,y)=mod[\varphi(x,y),2\pi] \tag{14}\]  
We then generate four phase-shifted fringe patterns according to the following formula [25]:  
\[I_{i}(x,y)=1+\cos\left[\varphi^{\prime}(x,y)+\delta_{i}\right]+\text{ noise }(x,y) \tag{15}\]  
where \(i=1\), \(2\), \(3\), \(4\) is the phase-shifting step, \(\delta_{i}=(i-1)\pi/2\) is the phase-shift amounts, and \(\text{noise}(x,y)\) is the Gaussian white noise for each pattern. Then, we can use the standard four-step phase shift method to restore the phase. The noise of the power intensity I = 0.2, 0.5, 0.7, 1 dBw, in which dBw represents the absolute noise intensity, is added to the noiseless shearography-interferometric phase fringe patterns obtained by simulation with w = 10, 20, 30. Then, the sine-cosine transform wavelet threshold denoising method with decomposition levels of 3, 4, 5, and 6 is used to denoise the pattern. According to the four-step phase shift method, the wrapped patterns of Gaussian white noise with different noise intensities and fringe densities are processed with different wavelet decomposition levels, as shown in Figures 4-6. Figure 4: The wrapped patterns of Gaussian white noise with different noise intensities and fringe densities are processed by varying levels of wavelet decomposition with the parameter w in Formula (12) is 10. Figure 5: The wrapped patterns of Gaussian white noise with different noise intensities and fringe densities are processed by varying levels of wavelet decomposition with the parameter w in Formula (12) is 20. The speckle suppression indices of the above fringe patterns are calculated and compared, as shown in Figure 7.  
The three-group shearography-interferometric phase fringe patterns under different noises are processed by the sine-cosine transform wavelet threshold denoising method with varying decomposition levels. As shown in Figure 7a-c, the distribution of SSI with low fringe density is concentrated, while it is just the opposite with high fringe density. The SSI analysis results show that the decomposition level has little effect on the filtering effect of low-density fringes. On the contrary, the decomposition level will significantly affect the filtering effect for high-density fringes, so it is particularly important to determine the decomposition level. The SSI with five-level decomposition is always the smallest regardless of fringe density and noise intensity. The denoising effect of five-level decomposition in the sine-cosine transform wavelet threshold denoising method is the best for shearography-interferometric phase fringe patterns. The four kinds of noisy wrapped patterns with different fringe densities and noise intensity are processed by the sine-cosine transform wavelet threshold denoising method. The axial phases of the noiseless and denoised speckle pattern are extracted, as shown in Figure 8. As we can see, the noise is well removed, and the changing trend is consistent. The denoised phase is preserved well, with the details maintained.