## 3 Deep Learning  
In this study, we tried to introduce deep learning to phase extraction. Deep learning is the method of learning by multi-layering neural networks and has been successful in many fields, especially in the field of image processing [1]. This study uses supervised learning. Supervised learning is one of the machine learning methods and is a method for obtaining an optimal functional relation \(y=f(x)\) by learning a model using a large number of input data, \(x\), and output data, \(y\). In this study, the input, \(x\), and the output, \(y\), correspond to the interferograms and the phase-shift images, respectively. It should be noticed that the relations of the interferogram and the phase-shift is inverted in the physical process as an inverse problem; that is, the interferogram is the function of the phase-shift as shown in Equation (3). Since the phase-shift images have real values for each pixel, the model to be acquired is a regression model. Various models including fully convolutional networks [23] have been considered for network models in which both the input and output are an image. U-net [24] is a network originally designed for segmentation of cell images. The feature of U-net is that there are paths connecting the upstream and the downstream, and by the presence of these paths, it is possible to integrate and to learn the local features of the object and the general position information. The U-net is also used for the regression problems as a classification problem [25]. Since the extraction of the phase-shift images of the interferograms requires both local features and global position information, we adopted the same network model as U-net in this study. However, the detailed structure of each layer is different as shown in Figure 3. Specifically, the size of filters of each convolutional layer is \(3\times 3\), the stride of filter applications is 1 and the spatial padding width for input arrays is 2. The size of the window of each pooling layer is \(2\times 2\). Also, the size of filters of each up-convolution layer is \(4\times 4\), the stride of filter applications is 1 and the spatial padding width for input arrays is 2. However, those in the final convolutional layer are \(1\times 1\), 1 and 0, respectively. It is difficult to use the measured data as the training dataset. This is because the phase-shift distribution from all measured interferograms, including those with unfavorable conditions cannot be extracted by the Fourier transform method, and in addition, it is impossible to prepare a large amount of measured data. Therefore, the pairs of the phase-shift images generated by simulation and the interferograms evaluated by them are prepared as training data. Since the problem is the inverse problem, the output of the deep learning system, \(y\), which is the phase-shift image, \(\phi(\mathbf{r})\), is defined first; then the input of the learning, \(x\), which is the interferogram, \(i(\mathbf{r})\), is evaluated by the simulation of interference according to Equation (3). As a physical model, we considered a model that a probe beam with non-uniform amplitude is incident to an object which is a hot gas around candle flames. The amplitude, \(i_{0}(\mathbf{r})\), is defined by a 2-D Gaussian function and the phase-shift image, \(\phi(\mathbf{r})\), is also expressed by a linear combination of the 2-D Gaussian functions. \[i_{0}(\mathbf{r}) =\,\mathrm{gauss}(\mathbf{r}\,;\,\mathbf{r}_{i_{0}},0,a_{i_{0}},b_{i_{0}}), \tag{7}\] \[\phi(\mathbf{r}) =\sum_{n}\phi_{n}\,\mathrm{gauss}(\mathbf{r}\,;\,\mathbf{r}_{n},\theta_{ n},a_{n},b_{n}), \tag{8}\]  
where \(\,\mathrm{gauss}()\) is the 2-D Gaussian function of which contour is an ellipse:  
\[\mathrm{gauss}(\mathbf{r}\,;\,\mathbf{r}^{\prime},\theta,a,b) =e^{-(\mathbf{r}-\mathbf{r}^{\prime})^{\top}}\mathbf{R}^{\mathrm{T}}( \theta)\mathbf{W}^{-2}(a,b)\mathbf{R}(\theta)(\mathbf{r}-\mathbf{r}^{\prime}) \tag{9}\] \[\mathbf{R}(\theta) = \begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix} ,\quad\mathbf{W}^{-2}(a,b)=