## 3 Experiments and Result Analysis  
### Simulation Experiments  
To verify the feasibility of the above method, three sets of four-step phase-shifting patterns with grating pitches of 16, 20, and 26 pixels are designed. Different Gaussian noises are added to the phase-shifting patterns, and the phase principal values of three sets of patterns are obtained according to the four-step phase-shifting method. The phase principal values are unwrapped by the three-frequency heterodyne principle, the absolute phase is corrected by the procedure in this paper, and the threshold value \(H\) is set as 0.459. As a comparison, two sets of phase-shifting patterns with grating pitches of 34 and 36 pixels are designed [19; 20; 21], while adding the same Gaussian noise. The phase principal values of the two sets of patterns are obtained according to the four-step phase-shifting method, and the phase principal values are unwrapped by the traditional method (the dual-frequency heterodyne procedure) and the method of [15], respectively. The numbers of calculated jumping points are shown in Table 1. The absolute phases obtained by different methods are shown in the Figures 2-4. As shown in Table 1, compared with the traditional method, the correction rate of absolute phase jumping points can reach more than 96.8%, and in the face of Gaussian noise with different mean values and variances, only a few jumping points are not corrected. Compared with the method in the literature [15], the number of jumping points corrected by the method in this paper is also bigger. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Gaussian Noise**} & \multicolumn{3}{c}{**Jumping Point**} \\ \hline
**Mean** & **Variance** & **Traditional** & **Method of** & **Method of** & **Method of** & **Correction** \\
**Value** & **Variance** & **Method** & **Literature**[15] & **this Paper** & **Rate/\%** \\ \hline
0 & 0 & 40,339 & 1668 & 1305 & 96.8 \\
0.01 & 0.002 & 112,654 & 2345 & 2249 & 98.0 \\
0.03 & 0.004 & 142,195 & 3705 & 2762 & 98.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Jumping error points. Figure 3: Phase unwrapping by the method in literature [15] under different noises. Phase expansion by the method of [15] when no Gaussian noise is added (**a**). The phase expansion is performed by the method of [15] when adding the first set of Gaussian noise (**b**). The phase expansion is performed by the method of [15] when adding the second set of Gaussian noise (**c**). Figure 2: Phase unwrapping by the traditional method under different noises. Phase expansion by conventional method when Gaussian noise is not added (**a**). Traditional method of phase expansion when adding the first set of Gaussian noise (**b**). Traditional method of phase expansion when adding the second set of Gaussian noise (**c**). Figures 2-4 show the absolute phase diagrams after the wrapped phase expansion by the traditional method of phase solving, the method of [15] and the improved method of this paper, respectively. Figure (a), Figure (b), and Figure (c) represent the absolute phase diagrams without Gaussian noise, with Gaussian noise of mean 0.01 and variance 0.002 (the first set of Gaussian noise), and with Gaussian noise of mean 0.03 and variance 0.004 (the second set of Gaussian noise), respectively, in the phase shift pattern using the current three sets of wrapped phase patterns solved by the method. It is evident in Figure 2 that the phase unwrapped by the traditional dual-frequency heterodyne method undergoes many jumps; especially after the addition of noise, the number of jumping points increases dramatically, which introduces a lot of errors and the obtained absolute phase is not accurate. Meanwhile, it will also cause a great impact on the 3D reconstruction. As can be observed from Figure 3, the method in [15] improves the threshold value setting for the jumping errors in the unwrapping phase.