# Improved Least Squares Phase Unwrapping Method Based on Chebyshev Filter
## 3 Results
### The Simulated Data Results  
The simulated wrapped phase was generated using the PEAKS function in MATLAB with a peak to valley of 10, with an adding Gaussian noise of 0 mean and 1 standard deviation. The simulated data have a horizontal and vertical size of 512 \(\times\) 512, as shown in Figure 2. The color bar on the right side of the figures represents the phase value, the unit is rad. Figure 1: Flow chart of the proposed method. In Figure 2a, the highest phase is 83.84 rad and the lowest phase is \(-\)68.23 rad. The wrapped phase map is shown in Figure 2b. Since the wrapped phase is distributed between \(-\pi\) and \(\pi\), noise with a standard deviation of 1 is added, which satisfies the high-noise situation. When the Y direction is at the 250 th pixel, the phase gradient map of Figure 2b is shown in Figure 3, which shows that the simulated phase has a steep phase gradient. Therefore, the simulated phase used has a large noise and steep phase gradient, which has a positive effect on verifying the method proposed in this paper. Then, LS, PCUA, RPUA, and the proposed method are used for phase unwrapping. The unwrapping error maps are shown in Figure 4. Table 1 presents the time and RMSE values for each of the four methods. As shown in Figure 4a, the LS phase unwrapping method has a lot of unwrapped areas. As shown in Table 1, the advantage of the LS phase unwrapping method is that it can perform phase unwrapping fastest. But it has the largest RMSE and the worst unwrapping effect. As shown in Figure 4b,c, PCUA and RPUA have greatly improved the LS phase unwrapping method, and have good phase unwrapping results, but there are still a few unwrapped areas. As shown in Table 1, PCUA and RPUA have smaller RMSE compared to the LS phase unwrapping method but require more unwrapping time, which has room for improvement. As shown in Figure 4d, the proposed method has the least unwrapped areas, and as shown in Table 1, the proposed method has the smallest RMSE, which is reduced by 94.53% compared with LS, 31.50% compared with PCUA and 15.11% compared with RPUA. The proposed method has a faster unwrapping speed compared with PCUA and RPUA, which is improved by 55.29%  
Figure 3: The phase gradient of Figure 2b when the Y direction is at the 250 th pixel. Figure 2: The simulated original phase and wrapped phase. (**a**) The simulated original phase. (**b**) The wrapped phase. and 10.91%. Therefore, this simulation experiment shows that the proposed method has a better effect than LS, PCUA, and RPUA. Furthermore, to test the robustness of the proposed method, phase unwrapping tests were performed by increasing the phase gradient while keeping the noise constant. Based on the above simulated experimental data, the same MATLAB PEAKS function was used to generate experimental data. While the noise standard deviation remained unchanged at 1, the peak to the valley gradually increased from 10 to 15. The RMSE of the unwrapping results with different phase gradients is shown in Figure 5. Among them, the RMSE of LS phase unwrapping is the largest and gradually increases as the phase gradient increases. The unwrapping effect is the worst and is not suitable for steep phase unwrapping. PCUA, RPUA, and the proposed method can all achieve lower RMSE than LS and can perform stable phase unwrapping under a high phase gradient compared to LS. However, when the peaks to the valley is 13.5, the RMSE of the unwrapping results of PCUA and RPUA \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Times (s)** & **RMSE (rad)** \\ \hline LS & 0.0468 & 7.2621 \\ PCUA & 5.8232 & 0.5797 \\ RPUA & 2.9227 & 0.4678 \\ Proposed & 2.6037 & 0.3971 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The time and RMSE values for each of the four methods. Figure 4: The error maps result from simulated phase unwrapping. (**a**) LS phase unwrapping error map. (**b**) PCUA phase unwrapping error map. (**c**) RPUA phase unwrapping error map. (**d**) The proposed method phase unwrapping error map. will mutate and be unstable.