Phase Extraction from Single Interferogram Including Closed-Fringe Using Deep Learning
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applied sciences Article Phase Extraction from Single Interferogram Including Closed-Fringe Using Deep Learning Daichi Kando 1,* , Satoshi Tomioka 2,*, Naoki Miyamoto 2 and Ryosuke Ueda 3 1 Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan 2 Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan 3 Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba 305-8573, Japan * Correspondence: [email protected] (D.K.); [email protected] (S.T.) Received: 16 June 2019; Accepted: 23 August 2019; Published: 28 August 2019 Abstract: In an optical measurement system using an interferometer, a phase extracting technique from interferogram is the key issue. When the object is varying in time, the Fourier-transform method is commonly used since this method can extract a phase image from a single interferogram. However, there is a limitation, that an interferogram including closed-fringes cannot be applied. The closed-fringes appear when intervals of the background fringes are long. In some experimental setups, which need to change the alignments of optical components such as a 3-D optical tomographic system, the interval of the fringes cannot be controlled. To extract the phase from the interferogram including the closed-fringes we propose the use of deep learning. A large amount of the pairs of the interferograms and phase-shift images are prepared, and the trained network, the input for which is an interferogram and the output a corresponding phase-shift image, is obtained using supervised learning. From comparisons of the extracted phase, we can demonstrate that the accuracy of the trained network is superior to that of the Fourier-transform method. Furthermore, the trained network can be applicable to the interferogram including the closed-fringes, which is impossible with the Fourier transform method. Keywords: deep learning; convolutional network; U-net; phase extraction; fringe analysis; closed-fringe; interferometer 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 Appl. Sci. 2019, 9, 3529; doi:10.3390/app9173529 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3529 2 of 13 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 [8–10] 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 [15–20], which detail processes will be shown in Section2. 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. To apply the Fourier-transform method, there is a limitation regarding the intervals of the background fringes. If the interval is too long, the fringe pattern caused by the object includes a closed-fringe pattern and the filtering process cannot be applied except in a special case [21]. In the case where the background fringe cannot be controlled, some interferograms may have closed-fringes and a complete set of the phase-shift images for CT cannot be obtained. As a result, the quality of the 3-D reconstructed distribution is reduced. In this study, we will demonstrate the applicability of deep learning to extracting the phase-shift image from the fringe pattern even when the closed-fringe is included through a comparison to the Fourier-transform method. The outline of this paper is as follows. In Section2, the Fourier-transform method to extract the phase-shift image is reviewed. In Section3, we show the phase extraction by the deep learning. The comparison of the numerical results and discussions is presented in Section4. Finally, the summary is presented in Section5. 2. Fourier-Transform Method The Fourier-transform method is commonly used to obtain a phase-shift image from a single interferogram including background fringes. In this section, taking the phase extraction from the interferogram shown in Figure1a as an example, the procedure of the Fourier-transform method is shown and the advantages and disadvantages are shown. In the interferometer, a single optical wave is split into two waves that pass along different optical paths; one is called a reference wave which does not pass the object and the other called an object wave passing through the object with an additional phase-shift, f. When their wave vectors are different but Appl. Sci. 2019, 9, 3529 3 of 13 their amplitudes are provided by the same function, electric fields of both waves are expressed with the complex unit j as r r E (r) = E0(r) exp [−j k · r] , (1) o r E (r) = E0(r) exp [−j ((k + dk) · r + f(r))] , (2) where superscripts ‘r’ and ‘o’ denotes the quantities related to the reference wave and those to the object wave, respectively, and the difference of the wave vector is denoted by dk. These waves are superposed on a screen and the interferogram, i(r), is observed as a power of the superposed electric field: r o 2 2 i(r) = jE (r) + E (r)j = 2jE0(r)j (1 + cos(f(r) + dk · r)) ∗ = 2i0(r) + if(r) exp (+j dk · r) + if(r) exp (−j dk · r) (3) 2 i0(r) = jE0(r)j , if(r) = i0(r) exp (j f(r)), where the symbol ‘*’ denotes a complex conjugate. To obtain the phase-shift f(r) from this equation, the first and the last term on the right-hand side in Equation (3) can be removed by using a 2-D Fourier transform. Applying the Fourier transform to Equation (3) and using a convolutional theorem, we obtain the following equation. ∗ I(k) = F fi(r)g = 2I0(k) + If(k − dk) + If(−k − dk), (4) I0(k) = F fi0(r)g , If(k) = F if(r) , where F fg represents a 2-D Fourier transform operator. Figure1b shows the Fourier transform of i(r) shown in Figure1a. We can find three peaks in the Fourier transformed signal, I(k). These peaks correspond to three terms in the right-hand side of Equation (4); the peak appearing at the center in the frequency space, k, is a DC component the distribution of which is related to the profile of the waves i0(r), and the other two peaks spreading around k = ±dk include the information of f(r) related to the object.