## 4 Results and Discussion
### Extracted Phase-Shift from Simulated Interferogram  
We evaluated the errors of two methods; one is the trained network and the other is the rule-based method which consists of the procedures of the Fourier-transform method and the phase unwrapping shown in Section 2. Figure 5 presents the distribution of error. The error for each image is evaluated by a relative root mean squared error defined as  
\[\hat{e}=\sqrt{\left\langle\left(\phi(r)-\phi_{\text{GT}}(r)\right)^{2}\right \rangle_{\text{pixels}}\Big{/}\left\langle\left(\phi_{\text{GT}}(r)\right)^{2 }\right\rangle_{\text{pixels}}}. \tag{14}\]  
Figure 4: Evolution of the evaluation function \(\hat{E}\). The horizontal axis indicates the epoch number, and the vertical axis indicates the evaluation function \(\hat{E}\). Training data represents the data used for the training and test data represent the data not used for the training. The most frequent error of the deep learning method is around 2% and most of the samples are distributed less than 10% error. In contrast, in case of the rule-based method, the peak is found almost 20%; furthermore, the decrease of the error is slow. In the figure, the samples with \(\hat{e}>1.0\) (100%) are omitted and they are excluded in the evaluation of the error average, which will be shown below, the number of those samples reaches almost 2000 among 10,000 test data. The average of \(\hat{e}\) using the trained network is 0.027, meanwhile that using the rule-based algorithms is 0.455. The error of the rule-based algorithm is almost twenty times larger than that of the deep learning, despite excluding the samples with enormous errors by the rule-based algorithm. From this comparison it is evident that phase extraction from interferograms by using the deep learning is superior to that using the rule-based algorithms. The correlation of the extraction error between the two methods is shown in Figure 6. We can understand that there are no correlations between the rule-based algorithm and the deep learning. It means that the reasons of the error are different, and the interferogram which causes large error to the phase are different. Figure 5: Distribution of the root mean squared error, \(\hat{E}\), by the deep learning and the rule-based algorithms for 10,000 test data. The data which have the relative root mean squared error of more than 1.0 are not shown. Figure 6: Correlation of errors between the deep learning and the rule-based method. Both the axes represent the relative error defined in Equation (14). Figure 7 presents comparisons between the extracted phases by the deep learning and that by the Fourier-transform methods for typical interferograms;  
1. [label=()]
2. the estimation errors, \(\hat{E}\), by both the deep learning and the Fourier-transform method from the interferogram, \(i(\mathbf{r})\), belong to the maximum of the histogram shown in Figure 5; which are called a mode,
3. \(\hat{E}\) of the Fourier-transform method is smaller than the mode, \(\hat{E}\) by both the deep learning and the rule-based algorithm are the mode of the histogram shown in Figure 5,
4. and (d): the interferograms include closed-fringes. The interferograms, \(i(\mathbf{r})\), shown in the figure are some of the test data, which were not used for training in the deep learning method. In the case of results from the deep learning, \(\phi_{\text{DL}}\), we cannot find the significant difference of \(i(\mathbf{r})\) from the images, since their errors are sufficiently small. For the case of the rule-based algorithm, we can find that the interferograms, \(i(\mathbf{r})\), shown in (c) and (d) includes closed-fringe, and their results, \(\phi_{\text{RB}}\), include large error as predicted in Section 2. Between these two, the error of (d) is larger than (c), and the number of closed-fringes has the same relation. Therefore, the closed-fringe strongly reduces the accuracy. Both \(i(\mathbf{r})\) in (a) and (b) do not include the closed-fringes; however, there is a difference between the error of them.