## 3 Numerical Simulation Analysis
### Noise Immunity of Algorithm  
Gaussian noise is introduced to the ideal slope and curvature using the same simulation conditions as described in Section 3.1. This is carried out to compare the relative reconstruction errors of each algorithm when considering the presence of noise. The level of noise is quantified using the signal-to-noise ratio (\(SNR\)), which is defined as the ratio of the root mean square (RMS) of the ideal slope or curvature to the RMS of the added noise as follows:  
\[SNR=\sqrt{\frac{{{\
u_{0}}^{2}}}{{n^{2}}}}, \tag{17}\]  
where is the ideal slope curvature information, and \(n\) is the introduced Gaussian noise that satisfies the analogue noise with mean 0 and variance 1 and satisfies the Gaussian distribution. The reconstructed information is as follows:  
\[\
u={{\
u_{0}}+{n_{0}}/{SNR}}. \tag{18}\]  
Wavefront reconstruction simulations are performed at signal-to-noise ratios of 10 and 30. The simulation results are repeated 20 times for each algorithm and then averaged to reduce the effect of randomness.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**LP17** & \multicolumn{2}{c}{**Computational Time (s)**} & \\ \cline{2-4}
**Algorithm** & **100 \(\times\) 100** & **200 \(\times\) 200** & **300 \(\times\) 300** \\ \hline FA & 0.02869 & 0.10363 & 0.24451 \\ BA & 0.05532 & 0.29948 & 0.79579 \\ PA & 0.04897 & 0.14419 & 0.39198 \\ ZA & 0.04248 & 0.19756 & 0.58311 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Reconstruction time for different reconstruction algorithms.  
Figure 6: Plot of speed of slope-curvature hybrid reconstruction algorithms.  
A comparison of the wavefront recovery reconstruction results for a signal-to-noise ratio of 10 considering noise is shown in Figure 7. As can be seen from the simulation results in Figure 7, in the presence of measurement noise, the present method introduces more integral loop points compared to existing algorithms in that there is a smaller relative reconstruction error for higher-order aberrations, as well as a stronger noise immunity. Compared to the PA and BA, the present algorithm introduces more reconstruction information, resulting in better noise immunity, and, compared to the ZA, the noise immunity remains basically the same due to the same amount of information being introduced.  
Figure 8 shows the comparison between the wavefront recovery reconstruction results of the method described in this study and the conventional method when the signal-to-noise ratio is 30, considering noise. From the simulation results of Figure 8, we can see that, as the signal-to-noise ratio increases from 10 to 30, the relative reconstruction errors of the Barwick algorithm and the Pathak algorithm do not change significantly, while the relative reconstruction errors of this algorithm are significantly reduced and have a strong noise immunity. For the low-order polynomials, the reconstruction surface type is relatively simple, and the reconstruction noise immunity advantage of the FA over the BA and PA is not obvious, and several methods have good reconstruction accuracy. As the number of polynomial terms increases, the reconstructed surface pattern becomes complex, and the noise immunity advantage of the FA begins to show. Compared with the ZA, it also has some accuracy advantages in some face shapes.  
Figure 7: Comparison of the reconstruction error of each reconstruction algorithm for a signal-to-noise ratio (SNR) of 10: (**a**) PA and FA, (**b**) BA and FA, (**c**) ZA and FA.