We propose an image-reconstruction algorithm of parallel phase-shifting digital holography (PPSDH) which is a technique of single-shot phase-shifting interferometry. In the conventional algorithms in PPSDH, the residual 0th-order diffraction wave and the conjugate images cannot be removed completely and a part of space-bandwidth information is discarded. The proposed algorithm can remove these residual images by modifying the calculation of phase-shifting interferometry and by using Fourier transform technique, respectively. Then, several types of complex amplitudes are derived from a recorded hologram according to the directions in which the neighboring pixels used for carrying out the spatial phase-shifting interferometry are aligned. Several distributions are Fourier-transformed and wide space-bandwidth information of the object wave is obtained by selecting the spectrum among the Fourier-transformed images in each region of the spatial frequency domain and synthesizing a Fourier-transformed image from the spectrum.
©2012 Optical Society of America
Digital holography is one of interferometry and is a technique for recording a hologram using an image sensor and reconstructing the wavefront of objects [1–4]. The technique is capable of both three-dimensional (3D) and phase measurements with a single-shot exposure. Also the technique has been actively researched in many fields: microscopy [5–7], particle and flow measurement [8–10], object recognition [11–13], quantitative phase imaging [14–16], and so on. There are two configurations in the embodiment of digital holography: in-line and off-axis configurations. Since the pixel size of an image sensor is too large to record fine interference fringes so far, both resolution and field of view are severely limited in off-axis configuration because most of space-bandwidth of the image sensor is discarded . In in-line configuration, there is a problem that the unwanted images, which are the 0th-order diffraction wave and the conjugate image, get superposed on the object image. Although phase-shifting digital holography was proposed  to remove the unwanted image, it is useless for the instantaneous measurement of objects by the technique because of the requirement of sequentially recording multiple phase-shifted holograms.
To achieve both wide-area and single-shot 3D measurement without the unwanted images, parallel phase-shifting digital holography (PPSDH) was proposed [18–30]. In this technique, a single image sensor records multiple phase-shifted holograms with a single-shot exposure by using space-division multiplexing of holograms. Up to now, application to microscopy , displacement measurement , and high-speed motion-picture recording of quantitative phase distribution  have been presented by making use of single-shot 3D imaging ability of PPSDH. Several types of parallel phase-shifting digital holography have been reported in terms of the number of phase shifts and the parallel two-step technique can achieve the highest-accuracy measurement for wide 3D area among them .
However, there are three problems in PPSDH: (1) Residual 0th-order wave remains because the intensity of the 0th-order wave is not always constant at a concerned pixel and the neighboring pixels, (2) the conjugate image cannot be completely removed when phase-shift error occurs severely [30,31], and (3) fringe resolution decreases in the direction of interpolation [20,22] or the pixels used for phase-shifting interferometry  and then the performance of both the resolution and the field of view of PPSDH are degraded because of sacrificing a part of space-bandwidth of the object wave. To solve these all problems, this time we propose an image-reconstruction algorithm for reconstructing wide space-bandwidth information in PPSDH. We modify Meng’s two-step phase-shifting interferometry  to remove residual 0th-order wave in PPSDH and apply Fourier transform technique  to PPSDH to remove the residual conjugate image. Several types of complex amplitudes are derived from a recorded hologram according to the directions in which the neighboring pixels used for carrying out the spatial phase-shifting interferometry are aligned. Then, the most accurate spectrum is selected from the Fourier-transformed (FT) images of the complex amplitudes in the spatial frequency domain. After that, wide space-bandwidth information of object wave is obtained by synthesizing a FT image from the spectrum. The effectiveness of the proposed technique was experimentally verified.
2. Parallel phase-shifting digital holography (PPSDH)
Figure 1 shows the basic concept of PPSDH using two phase shifts. An image sensor records two phase-shifted holograms with a single-shot exposure by utilizing space-division multiplexing technique. 3-D image is numerically reconstructed by computer from the recorded hologram.
In PPSDH, complex amplitude at a pixel is obtained by using the value of a concerned pixel and, the value of its neighboring pixel or the values of its neighboring pixels. Up to now, two types of the image-reconstruction algorithms have been proposed for PPSDH [20,32]. This time we call the algorithm reported in Ref. 20 interpolation method and that reported in Ref. 32 neighboring-pixel-use method. However, fringe resolution decreases in the conventional algorithms. Figure 2 shows the schematics of the algorithms and direction dependencies of the error in these algorithms. Figure 2(c) describes that, if the calculation of phase-shifting interferometry is conducted by using neighboring-pixels in the vertical direction for fine horizontal-striped fringes, phase-shift error occurs severely. Fine fringes are generated by the large incident angle of the object wave in in-line configuration . This means that the phase of the object wave is drastically different from those of neighboring pixels. In spatial phase-shifting interferometry, it is assumed that the phase of the object wave does not vary drastically and it has been reported that phase-shift error occurs when the assumption is not satisfied . On the other hand, if the calculation is conducted by using neighboring-pixels in the horizontal direction for fine horizontal-striped fringes, phase-shift error does not occur. If the phase-shift error occurs, the conjugate image remains and the information of object wave decreases by the calculation error of phase-shifting interferometry. This leads to the sacrifice of the space-bandwidth used for recording object wave because phase-shift error occurs in the high spatial frequency region. In the interpolation method, it has been reported that the interpolation error occurs for fine fringes and then phase-shift error and the decrease of fringe resolution occur . As a result, space-bandwidth is sacrificed and then not only the resolution decreases but also measured area narrows in Fresnel region because of the relationship between the space-bandwidth and the object size in Fresnel hologram .
3. Proposed algorithm
Figure 3 shows the flow of the proposed image-reconstruction algorithm. In parallel two-step technique using Meng’s technique, intensity distribution of the reference wave Ir(x,y) is recorded before and/or after capturing a spatially phase-shifted hologram I(x,y) . Firstly, we subtract Ir(x,y) from the hologram I(x,y) to relieve the problem of the non-uniformity of 0th-order wave between neighboring pixels.Equation (1) is especially effective when Ir(x,y) is non-uniform because Ir(x,y) should be constant between neighboring pixels in PPSDH. Then, the modified two-step phase-shifting interferometry for PPSDH is calculated. The expressions are given byEq. (2) as follows.Eqs. (4)-(6), respectively. U1(x,y) and U2(x,y) have much spatial information in the vertical and horizontal directions because both phase-shift error and the decrease of fringe resolution does not occur in these directions. Although the residual conjugate image remains after the calculation of the modified phase-shifting interferometry, the residual conjugate image can be removed by applying Fourier transform technique. This is because the residual conjugate image does not overlap on the object image . Thus, each residual conjugate image of each complex amplitude distribution is removed in spatial frequency domain. After that, wide space-bandwidth information of object wave Uo(x,y) is generated from the selective extraction of the spatial frequency distributions of U1(x,y) and U2(x,y). FT image of U1(x,y) has more information than that of U2(x,y) regarding the vertical direction in spatial frequency region, and vice versa. Mathematical expression is as follows.Eq. (7), 3D image of objects is reconstructed by calculating an inverse Fourier transform and diffraction integral.
4. Preliminary experiment
We conducted a preliminary experiment to verify the effectiveness of the proposed algorithm experimentally and quantitatively. We constructed a sequential phase-shifting digital holography system shown in Fig. 4 and recorded two phase-shifted holograms, and Ir(x,y). α was set to be π/2 by using a mirror with Piezo actuator. The image reconstructed by the sequential phase-shifting digital holography was obtained as a standard of quantitative evaluation. After that, to compare the reconstructed image quantitatively by the proposed algorithm with the standard, the hologram I(x,y) to be obtained by PPSDH using two phase shifts was equivalently and numerically generated from the two phase-shifted holograms. Object images reconstructed by the conventional [22,32] and the proposed algorithms were obtained by using the generated hologram and Ir(x,y). For comparing PPSDH using two phase shifts with spatial-carrier phase-shifting technique [30,31], the hologram to be obtained by spatial-carrier technique was numerically generated from the complex amplitude by sequential technique and the image was reconstructed. Spatial carrier of the reference wave was introduced to the vertical direction. α was set to be π/2 not to occur aliasing because space-bandwidth of object wave was wide. The residual conjugate image was removed by Fourier transform and spatial-frequency filtering technique. A He-Ne laser operated at 632.8nm was used as the light source. A SONY CCD camera (the number of pixels 2448 (V) × 2050 (H), pixel size 3.45μm × 3.45μm, resolution 12 bits) was used as the image sensor.
Figure 5 shows the images reconstructed by the sequential phase-shifting technique, spatial-carrier phase-shifting technique, PPSDH using the conventional and the proposed algorithms, respectively. The distance between the image sensor and a miniature model of a duck was 300mm. The focused image on the duck was shown in Fig. 5. As shown in Fig. 5(b), whole object image was not reconstructed by spatial-carrier phase-shifting technique because the whole size of the object was too large to be captured without the superposition of the unwanted images by off-axis configuration. This time, the space-bandwidths used forrecording object wave in the horizontal and vertical directions were more than a half of that of an image sensor. However, spatial-carrier phase-shifting technique has no ability of capturing this wide space-bandwidth because the bandwidth of spatial-carrier technique in one of each direction is less than a half of that of an image sensor . As a result, a part of the object image was not reconstructed. In contrast, whole object image was reconstructed by parallel two-step technique. However, unwanted images appeared in Figs. 5(c)-5(e), which were reconstructed by the conventional algorithms. This time, neighboring-pixel-use method using horizontal pixels was ineffective, as shown in Fig. 5(c), in particular. This is because the measured area in the horizontal direction was wider than that in the vertical direction and fine horizontal-striped fringes were formed. In contrast, clear object image was reconstructed by the proposed algorithm, as shown in Fig. 5(f). Figures 5(g)-5(j) show the subtracted images between the conventional algorithms and the proposed one. Figures 5(g) and 5(i) show the undesired components and Figs. 5(h) and 5(j) show the recovered information by the proposed algorithm. Figures 5(g) and 5(i) show that the residual 0th-diffraction and conjugate images appeared in the ellipsoid areas surrounded by green and red lines, respectively. These residual images were removed comparatively by the modified phase-shifting interferometry and Fourier transform technique in the proposed algorithm. Figures 5(h) and 5(j) show that object images in the ellipsoid areas surrounded by a blue line are brightly reconstructed without direction dependency, which was brought by the procedure of Eq. (7) in the proposed algorithm. On the other hand, Figs. 5(c) and 5(d), which are reconstructed by the neighboring-pixel-use method using the pixels in horizontal and vertical directions, show severe direction dependency. This is because the proposed algorithm can avoid the phase-shift error in neighboring-pixel-use method and the blur of fine fringes caused by the interpolation method, and space-bandwidth information is not sacrificed in comparison to the conventional algorithms. Figures 5(k)-5(m) show FT images of complex amplitudes. It was clarified that the spectrum of the residual images were observed and high space-frequency information was lost by the conventional algorithms while wide space-bandwidth information of the object wave was obtained without degradation by the proposed algorithm. We calculate normalized root-mean-square errors (NRMSEs) and correlation coefficients (CC) between the amplitude images by the sequential phase-shifting technique and those by spatial phase-shifting techniques using the algorithms. The amplitude distribution obtained by sequential phase-shifting technique is normalized from 0 to 255. Table 1 shows the calculation results. Both the NRMSE and CC of the image reconstructed by the proposed algorithm was the best among them. From the viewpoint of the values of the NRMSE and CC, the effectiveness of the proposed algorithm was experimentally and quantitatively verified.
We have proposed an algorithm for both removing the residual images appeared by the conventional algorithm of PPSDH and reconstructing wide space-bandwidth information of object wave. The effectiveness of the algorithm was verified experimentally and quantitatively. It can be considered that not only image quality but also the resolution and the field of view are improved in PPSDH, thanks to the proposed algorithm. This algorithm will contribute to high-quality 3D motion-picture measurement based on PPSDH and its applications, such as microscopy, quantitative phase imaging, and shape measurement.
This study was partially supported by Grant-in-Aid for JSPS Fellows from Japan Society for the Promotion of Science (JSPS), and by the Funding Program for Next Generation World- Leading Researchers GR064 of JSPS.
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