We propose an algorithm that can improve the quality of the reconstructed image from the single hologram recorded by the optical system of the parallel four-step phase-shifting digital holography. The proposed algorithm applies the image-reconstruction algorithm of parallel two-step phase-shifting digital holography to the hologram so as to reduce errors in the reconstructed image and eliminate ghosts. We numerically and experimentally confirmed that the proposed algorithm decreased 25% in terms of root mean square error in amplitude, and eliminated the ghosts, respectively.
©2010 Optical Society of America
Digital holography (DH) [1–5] is a technique that records holograms by use of an image sensor and reconstructs the image of objects by computer. This technique is capable of three-dimensional (3-D) image capturing and has been actively researched in many fields such as particle measurement [6,7], object recognition [8–10], phase and object imaging [11–13], phase-shift extraction [14,15], and encryption and information security [16–18]. In particular, in-line DH has been frequently employed because the pixel pitch of an image sensor is too large to record fine interference fringes generated by off-axis holography. In in-line DH, the reference wave is perpendicularly incident to an image sensor, so that the quality of the reconstructed image degrades. This is because the 0th-order diffraction image and the conjugate image are superimposed on the desired image of objects. Phase-shifting digital holography (PSDH)  is one of the powerful techniques which can obtain only the desired image. PSDH sequentially records several holograms by use of reference waves with several phase shifts to retrieve the original complex amplitude of the object wave. However, it is useless for the instantaneous measurement of moving objects due to the sequential recording of several holograms. Then, we proposed parallel PSDH [20–24] which was capable of the instantaneous measurement. This technique records an interference fringe image, in which several holograms required for phase-shifting interferometry are spatially multiplexed, with a single-shot exposure. Several parallel PSDHs have been reported [20–26]. However, PSDH has such the problem that the sampling interval of interference fringes is large because of the spatial multiplexing. Then, aliasing arises and the quality of the reconstructed images degrades due to aliasing. Although most of parallel PSDHs uses four phase shifts [20,21,25–27], the quality of the reconstructed image is degraded according to the number of the multiplexing. It is one of the most important issues for the parallel PSDHs to improve the quality of the reconstructed image. Then, we proposed a parallel PSDH which uses only two phase shifts to improve the image quality [28,29]. This parallel PSDH is based on the phase-shifting interferometry by use of two phase shits . Although the technique of Ref. 30 sequentially records two holograms, the parallel PSDH we proposed records the information of the two holograms with a single-shot exposure. However, no image sensor required for parallel two-step PSDH  has been developed yet. On the other hand, some image sensors that can be used for parallel four-step PSDH have already been commercially available [31,32]. Moreover, only the results of the preliminary experiment using sequentially recorded holograms were shown and parallel PSDHs have not been experimentally demonstrated yet in Refs [28,29].
In this paper, we propose an algorithm for improving the quality of the reconstructed image in parallel four-step PSDH and experimentally demonstrate a parallel PSDH using the proposed algorithm. This algorithm applies the image-reconstruction algorithm of parallel two-step PSDH to the hologram recorded by the image sensor, which is commercially available, required for parallel four-step PSDH.
Figure 1(a) shows an example of the optical implementation of parallel four-step PSDH. This implementation is based on the one shown in Ref. 25. The wave emitted from the light source passes through half-wave plate and is collimated. After that, the wave is split into two waves by the first polarizing beam splitter (PBS). The polarization directions of the two waves are orthogonal to each other. One wave illuminates the object and the wave diffracted or diffused from the object is called the object wave. The other is called the reference wave. The second PBS aligns the object wave and the reference wave. A quarter-wave plate (QWP) is oriented at 45° to the polarization direction of the reference wave. Then, the QWP transform the object wave and the reference wave into circular polarization waves. Because the polarization directions of the two waves are orthogonal to each other before entering the QWP, the two circular polarization waves rotate counter each other. The interference fringe image formed by the reference wave and the object wave is recorded by an image sensor on which a phase-shifting array device is attached. The polarization array shown in Fig. 1(b) is used for the phase-shifting array device. Therefore, a hologram containing the information of the four holograms, which is required for four-step PSDH, can be recorded with a single-shot exposure as shown in Fig. 1(c).
Figure 2 shows the schematic flow of parallel four-step PSDH. Holograms of each phase shift are extracted from the recorded hologram. The values of the vacant pixels are interpolated by the use of those of the adjacent pixels. Then, the four holograms I(0), I(-π/2), I(-π), I(−3π/2) required for four-step phase-shifting interferometry are generated by computer. By applying the phase-shifting method  to the four holograms, the complex amplitude distribution on the image sensor plane is obtained. Then, the 0th-order diffraction image and the conjugate image are eliminated and only the image of the object can be reconstructed by computer.
We propose an algorithm that can improve the quality of the images reconstructed from the single hologram recorded by the optical system of parallel four-step PSDH. In the reconstruction, we apply the image-reconstruction algorithm of parallel two-step PSDH  to the four holograms recorded by the image sensor with the phase-shifting array device required for parallel four-step PSDH.
Figure 3 shows the schematic flow of the proposed algorithm. The four holograms I(0), I(-π/2), I(-π), I(−3π/2) are generated from the single hologram recorded by the image sensor with the phase-shifting array device required for parallel four-step PSDH as shown in Fig. 1. We calculate the odd rows of the complex-amplitude distributions of object on the image sensor plane u0(x, y) by applying the calculation algorithm of parallel two-step PSDH to the data of pixels in the odd rows of I(0) and I(-π/2). The even rows of the complex-amplitude distributions of object on the image sensor plane uπ(x, y) is similarly calculated by using the data of pixels in the even rows in I(-π) and I(−3π/2). And then, we obtain the complex amplitude distribution u(x, y) by combining the data of pixels in the odd rows of u0(x, y) with those in the even rows of uπ(x, y). Finally, the image of object can be reconstructed by Fresnel transform  of u(x, y).
To confirm the validity of the proposed algorithm, we conducted a numerical simulation. Figures 4(a) and 4(b) show the objects. Each image consists of 256 × 256 pixels. The pixel size of an image sensor and the wavelength of a light source were assumed to be 5 μm × 5 μm and 532 nm, respectively. It was assumed that the distance between the object and the image sensor was 26 cm. Figures 4(c) and 4(d) are the images reconstructed by the proposed algorithm. The images reconstructed by parallel four-step PSDH are shown in Figs. 4(e) and 4(f), for comparison. In addition, the magnified images of the reconstructed images by the proposed algorithm and by the parallel four-step one are shown in Figs. 4(g) and 4(h), respectively. While the details of the object were blurry in Fig. 4(h), the proposed algorithm enabled to clearly reconstruct the details of the object as shown in Fig. 4(g). In parallel PSDH, the error by the interpolation of holograms causes the degradation of the image quality. The larger the number of the interpolated pixels is, the lower the quality of the image reconstruction is. The number of the interpolated pixels in the proposed algorithm is less than that in the parallel four-step one in the calculation of image reconstruction. Therefore, the proposed algorithm can improve the quality of the reconstructed image owing to the reduction of the number of the interpolated pixels. To quantitatively evaluate the numerical results, we calculated root mean square errors (RMSEs) between the original images and the reconstructed images obtained by each algorithm. Table 1 shows the RMSEs. The pixel values in the amplitude and phase distributions were normalized in range from 0 to 255 and from 0 to 2π, respectively. The closer RMSEs are to 0, the less error in reconstructed images is. The RMSEs of the proposed algorithm is superior to and equal to those of the parallel four-step algorithm in the amplitude distributions and the phase distributions, respectively. The proposed algorithm decreased 25% in terms of RMSE in amplitude. Thus, we confirm the validity of the proposed algorithm.
To demonstrate the proposed algorithm experimentally, we recorded holograms using the image sensor on which the phase-shifting array device used for parallel four-step PSDH was attached. We used a CCD camera with 1167 (H) × 874 (V) pixels. The configuration of the phase-shifting array device was the same as that shown in Fig. 1(b). A Nd:YVO4 laser operated at 532 nm was used as an optical source. Figure 5(a) shows the object made of a sheet of a transparent film on which the letter “R” was printed. The size of the letter was 13 mm (H) × 15 mm (V). The distance between the object and the image sensor was 26 cm. Figures 5(b) and 5(c) show the reconstructed image by the proposed algorithm and the parallel four-step algorithm, respectively. These images were reconstructed from only the central parts of 512 × 512 pixels of recorded holograms. Ghosts appeared at the side of the desired image in the parallel four-step algorithm. On the other hand, the proposed algorithm succeeded in eliminating the 0th-order diffraction image and the conjugate image from the desired image and clearly reconstructed the image of object. Although clarification of the cause of the ghost suppression by the proposed algorithm is under way, we consider that the proposed algorithm improved the image quality by decreasing the number of the interpolated pixels that affected the reconstructed image.
The proposed algorithm can more precisely record the holograms and more clearly reconstruct images than the parallel four-step one. Thus, we have experimentally verified that the proposed algorithm was capable of improving the quality of images reconstructed from a hologram recorded by parallel four-step PSDH.
In conclusion, we proposed an algorithm that can improve the image quality reconstructed by parallel four-step PSDH. The proposed algorithm applies the reconstruction algorithm of parallel two-step PSDH to the hologram recorded by the optical system of parallel four-step PSDH. The effectiveness of the proposed algorithm was numerically confirmed by evaluating the results using RMSEs. Furthermore, we experimentally demonstrated that the proposed algorithm can more clearly reconstruct the images of object. The proposed algorithm contributes to high-accuracy 3-D measurement for dynamically moving objects such as flow, particles, micro-electro-mechanical systems, living cells and so on.
This study was partially supported by Industrial Technology Research Grant Program from New Energy and Industrial Technology Development Organization (NEDO) of Japan.
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