## 4 Results and Discussion
### Extracted Phase-Shift from Simulated Interferogram
The reason for the difference is caused by the size of area with small phase value, of which pixels are depicted by dark pixels in \(\phi_{\text{GT}}\). The area with small phase in (a) is smaller than (c). In the Fourier-transform method  
Figure 7: Examples of extracted phase-shift and relative error. Four different sample data are shown as (**a–d**). The images in the first column, \(i(\mathbf{r})\), are interferograms which are input to the phase extraction methods. The images in the second column, \(\phi_{\text{GT}}\) is the grand true phase which is used to prepare \(i(\mathbf{r})\) as a forward simulation. The remained two columns are of the phase-shift images estimated by the deep learning, \(\phi_{\text{DL}}\), and that by the rule-based algorithm, \(\phi_{\text{RB}}\). the carrier frequency, \(\delta k\), is important, if it has error, \(\Delta\delta k\), the error of the phase, \(\Delta\phi\), spread to whole image as a planar (linear in 2-D space) function with a constant gradient as  
\[\Delta\phi(\mathbf{r})=\Delta\delta\mathbf{k}\cdot\mathbf{r}, \tag{15}\]  
As mentioned in Section 2, \(\delta k\) is determined from the peak position in \(k\)-space. When the non-zero phase area spread to whole image, an intensity of the peak caused by \(\delta k\) becomes smaller and \(\delta k\) may not be detected. From \(\phi_{\text{RB}}\) in (a), a leftward phase gradient is observed on both sides of the image. Similar to (a), \(\phi_{\text{RB}}\) in (b) a rightward smaller gradient is also observed. These errors can be compensated after the phase extraction. However, the case of the closed-fringe cannot be compensated the error. As discussed in this section, the rule-based algorithm has many factors to induce the phase error. In contrast, once the trained network with appropriate parameters is acquired, the phase is extracted accurately. Deep learning is especially applicable to the interferogram including the closed-fringe, which cannot be solved by the Fourier-transform method. These results show that the phase extraction from the simulated interferograms using deep learning was successful and better than the rule-based algorithms. As some comparisons, the results of U-nets with different parameters and an FCN are shown. The Figure 8 shows the histograms similar to the Figure 4, and they represent the extraction accuracy from each network. The U-net of (a) is the same as that of the Figure 3. The U-nets from (b) to (d) are the ones with the following parameters changed from (a). The parameter \(c\) is the number of output channels in the first convolutional layer, \(g\) is the number of groups that make up U-net when two consecutive convolutional layers are regarded as one group, and \(k\) is the kernel size of each convolutional layer. The parameters of (a) are \(c=32\), \(g=9\), and \(k=5\), whereas (b) is \(g=7\), (c) is \(c=16\), and (d) is \(k=3\). In addition to these, (e) FCN was prepared, which has almost the same learning time as (c) U-net. From the results of Figure 8, it can be said that U-net is suitable for the phase extraction problem. Further, it can be expected that the extraction accuracy will be further improved by increasing the parameters c, g and k but the learning cost will be increased accordingly. Since the average of relative RMSE when using (a) is about 0.027 and the accuracy is sufficient, the U-net used in this paper is optimal for the phase extraction problem. Figure 8: Distributions of the relative root mean squared error, by the various U-nets and an FCN for 10,000 test data. The parameter \(c\) is the number of output channels in the first convolutional layer, \(g\) is the number of groups that make up U-net when two consecutive convolutional layers are regarded as one group, and \(k\) is the kernel size of each convolutional layer.