## 4 Results and Discussion
### Extracted Phase-Shift from Interferogram Obtained by an Experiment  
In order to demonstrate the validity of the deep learning method in the phase extraction estimation from the interferograms, we applied the algorithms to interferograms which are obtained by an experiment to measure a 3D refractive index distribution in which the object is three candle flames [7]. Figure 9 shows triplets of an interferogram, \(i(\mathbf{r})\), and the extracted phase-shifts by two methods; one is the deep learning, \(\phi_{\mathrm{DL}}(\mathbf{r})\), and the other is the rule-based method, \(\phi_{\mathrm{RB}}(\mathbf{r})\), where the rule-based method is a set of sequential procedures of the Fourier-transform method and the phase unwrapping process of which detail is shown in Section 2. The extracted phase obtained by the rule-based method can be the true solution, if condition of the interferogram is good. However, the interferograms shown in Figure 9 have high density of the fringes and include much noise. In this condition, the results include much error. To evaluate the accuracy we roughly estimated the phase-shift at a sampling point by counting the number of fringes across a reference line which is parallel to the background fringe. The reference lines are drawn with red color in \(i(\mathbf{r})\) in Figure 9 and the estimated number of fringes are shown as \(N\) below \(i(\mathbf{r})\). Since the interval between fringes in the interferogram is \(2\pi\), the number of fringes crossing the reference line is equal to the phase-shift divided by \(2\pi\). By counting the number of contour lines from background to the point of interest, \(\phi_{\mathrm{DL}}(\mathbf{r})\) and \(\phi_{\mathrm{RB}}(\mathbf{r})\) in (a)-(d) are almost equal to \(N\). However, the distorted phase distributions are found \(\phi_{\mathrm{RB}}(\mathbf{r})\) in (b) and (c) where the fringes are not found in \(i(\mathbf{r})\). In the deep learning method, we can assume that there no phase-shift at both sides by the learning data. In contrast, the rule-based cannot generally impose this condition. In (d), \(\phi_{\mathrm{RB}}(\mathbf{r})\) show unnatural distribution caused by the dense fringes. From these results, the phase-shift estimated by the deep learning seems to good performance similar to the previous subsection. However, the deep learning method sometimes returns incorrect estimation which are shown in (e) and (f). In both cases the phase-shifts are underestimated in \(\phi_{\mathrm{DL}}(\mathbf{r})\). In \(i(\mathbf{r})\) in (e) and (f) includes dense and unclear fringes.  
Figure 9: Examples of extracted phase-shift from experimental interferograms. Six different sample data are shown as **(a–f)**. The images in the first row, \(i(\mathbf{r})\), are interferograms which are input to the phase extraction methods. The red line segment in each \(i(\mathbf{r})\) indicates a reference line to count the number of fringes from background area manually. The counted number of fringes along the line segment, \(N\), is shown below \(i(\mathbf{r})\). The second and third rows are the phase-shift images estimated by the deep learning, \(\phi_{\mathrm{DL}}\), and that by the rule-based algorithm, \(\phi_{\mathrm{RB}}\), respectively. In each figure, the curved lines depict contour lines of the phase-shift with intervals \(2\pi\). The size of images is \(128\times 128\) pixels.