## 1 Introduction  
In the last two decades, deep learning gone through a remarkable evolution, especially in image processing; for example, the classification of objects included in the image [1], noise reduction [2], pan-sharpening [3], caption generation [4] and image style transfer [5]. In this paper, we propose the method using deep learning to extract phase-shift images from a fringe patterns which is obtained from an interferometer. The extraction of the phase-shift is necessary to evaluate a refractive index of a matter of interest. According to classical electromagnetism, the refractive indices of matters can be represented by the Drude-Lorentz model which is a spring model with dumping caused by electric dipoles and free electrons. In this model, the refractive index depends on a density of the dipoles and the free electrons. In the case of air, the density of molecules with the electric dipoles depends on temperature of the air. These relations show that a three-dimensional (3-D) temperature distribution of gas around hot matter, such as a flame, can be indirectly evaluated through a 3-D measurement of a distribution of the refractive index. As the other medium than gas, 3-D distributions of electron density in plasma and concentration of solute in solution for liquid-state media could be evaluated. To determine the 3-D quantity, the technique of computed tomography (CT) is commonly used in medical diagnostics. In the medical CT, 2-D projection images of probe beams with different incident directions passing through a human body are measured; then, the 3-D distribution of an attenuation coefficient is reconstructed from these images. The key to the reconstruction is that the projection images include integrals of the attenuation coefficient distributed in the body along the directions of the probe beam. Upon replacing the projection from the absorption with phase-shift which is an integral of the refractive index along a path of optical beams, the 3-D refractive index can be reconstructed by the same way to the medical CT. To obtain the phase-shift caused by an object of interest, interferometer is commonly used. The authors developed a 3-D gas temperature measurement system [6,7] which uses mechanical stages to change the directions of the incident beams of the interferometer. The first observed image by the interferometer--called an interferogram or a fringe pattern--is represented by a sinusoidal function in 2-D space of the phase-shift. To obtain the phase-shift image from the interferogram, there are two commonly used methods; the phase-shift method [8, 9, 10] and the Fourier-transform method [8, 11, 12]. The mathematical operation in the phase-shift method is simple; however, it requires the three or four interferograms for the same object with different reference phase-shift which applied one of two optical paths in the interferometer. Since the incident direction varies in the 3-D measurement system which means the object is always rotating, it is difficult to apply the phase-shift method. In contrast, the Fourier-transform method can be applied for a single interferogram. However, it requires several steps of mathematical operations: a 2-D Fourier transform, filtering in 2-D spectral domain [13, 14], an inverse Fourier transform, removing background carrier pattern, and phase unwrapping [15, 16, 17, 18, 19, 20], which detail processes will be shown in Section 2. The mechanical system to control the incident direction induces vibrations of optical elements used in the system. To reduce the effect of the vibration exposure time of the camera to obtain the interferogram must be shortened. However, it causes a new problem; the obtained interferograms have much noise, which increases error to the phase-shift image at the phase unwrapping process. There is another problem in the system. The background fringe pattern cannot be controlled with high precision because the precision of the mechanical stages is dozens of times larger than the wavelength.