November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS
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Phase-shift binary digital holography Ryoichi Horisaki1,* and Tatsuki Tahara2 1
Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan 2 Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan *Corresponding author: [email protected]‑u.ac.jp Received August 6, 2014; revised October 3, 2014; accepted October 10, 2014; posted October 14, 2014 (Doc. ID 220543); published November 3, 2014 We propose phase-shift digital holography (DH) with a one-bit image sensor. In this method, the propagating complex field from an object is binarized by a one-bit sensor using a phase-shifter. The complex field on the hologram plane is then calculated with the one-bit image data. The object field is recovered via Fresnel back-propagation of the calculated hologram and filtering to suppress some artifacts caused by the binarization. The concept was demonstrated in preliminary experiments by using a synthetically binarized hologram with single-shot and multi-shot phase-shift DH. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (110.1758) Computational imaging; (100.3020) Image reconstructionrestoration. http://dx.doi.org/10.1364/OL.39.006375
Holography is a method for producing and observing the complex field of an object [1,2]. Current research topics in holography are roughly categorized into computergenerated holography (CGH) and digital holography (DH). CGH is a holographic display technique in which a hologram for generating a target complex field is digitally calculated on a computer, and the object’s complex field is reproduced by a diffractive optical element such as a spatial light modulator (SLM), which physically realizes the calculated hologram [3,4]. To alleviate some limitations of currently available SLMs, various CGH techniques have been proposed, for example, quantized CGH, which assumes an amplitude or phase mask with few quantization levels, including binary masks having just two levels [2]. DH is a holographic imaging technique in which the complex field is captured by an image sensor, and the object is digitally reconstructed by a computer [5]. DH has been used to realize three-dimensional imaging, phase imaging, and so on [2]. Quantization in DH has also been considered for reducing the data size of holograms for storing and transferring them electronically [6,7]. Binary sensors with two quantization levels have become an interesting research field. By using binary sensors with spatial and temporal oversampling, it is possible to computationally adjust the tradeoff between the spatial and temporal resolutions after image capturing [8]. DH using a binary sensor with oversampling for reducing speckle noise has also been demonstrated with a binary object [9]. This method employs multi-shot imaging with a moving diffuser to change the speckle pattern in each shot. A method has been proposed for achieving single-shot acquisition of a multi-bit image with a binary image sensor based on compressive sensing, which enables observation of a large object dataset by using a measurement dataset smaller than that used in conventional approaches [10,11]. This single-shot binary sensing method is a promising approach for decreasing production costs and drastically increasing the frame rate of sensors. Image degradation caused by computational binarization, not the binary image sensor itself, in phase-shift DH 0146-9592/14/226375-04$15.00/0
(PSDH) has been reported [6,7]; however, this degradation has not yet been modeled or suppressed. Because of this serious image degradation, binary DH has been considered to be impractical. In this Letter, we model the image degradation approximately but practically, and show how to suppress it. This enables us to maintain a practical level of image quality in PSDH, even when using a binary image sensor. We describe our phase-shift binary digital holography (PSBDH) method here. We also report results of preliminary experiments conducted to demonstrate PSBDH based on parallel PSDH (PPSDH) and conventional PSDH with scattering objects [12,13]. Our concept significantly increases the potential of DH for high-speed sensing with low-cost hardware. In conventional PSDH, the propagating complex field from an object is captured multiple times with a different phase shift introduced in each capturing step by using a phase-shifter [14]. The amplitude Ax and the phase Φx of the complex field Ux Ax expiΦx are recovered by the following procedure, where four phaseshifts with pitch π∕2 and a plane-wave reference beam are assumed (the y axis is omitted for simplicity): Ax
q Ix; 0 − Ix; π2 Ix; π∕2 − Ix; 3π∕22 ; (1)
Φx tan−1
Ix; π∕2 − Ix; 3π∕2 ; Ix; 0 − Ix; π
(2)
where Ix; θ is an intensity image with phase shift θ. References [6,7] have analyzed quantization, including binarization, to reduce the data size of the calculated complex field U from four phase-shifted images I captured by a full-bit image sensor. In contrast, here we consider the error in the complex field U from four phaseshifted images I captured by an one-bit image sensor. As shown in Eqs. (1) and (2), the autocorrelation terms jUj2 in the phase-shifted images I are canceled out even with binarization. However, possible values of the © 2014 Optical Society of America
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We model these artifacts with the phase-only hologram, as mentioned above, and use the model to suppress the artifacts in PSBDH. The phase-only hologram is written as expiΦx
n X Ux aUx 1 − jU 0 xjk jUxj k0
aUx
n X k0
Fig. 1. Simulation of the PSBDH. (a) The amplitude of a scattering object. The amplitudes of the Fresnel back-propagation results of (b) PSBDH and (c) the phase-only hologram in PSDH. (d) The amplitude after filtering (b).
complex field U B in p PSBDH are limited to three ampli tude levels (0, 1, 2) and eight phase levels (−π, −3π∕4, −π∕2, −π∕4, 0, π∕4, π∕2, 3π∕4). Thus, the error in the PSBDH is assumed to come from the quantization of the amplitude and phase. The simulation results of PSBDH, assuming a scattering object, are shown in Fig. 1. The object shown in Fig. 1(a) consists of two squares with random phases and different amplitudes. In the simulation, the pixel count N 2 of the object and the image sensor was 512 × 512 pixels, the distance z between them was 50 mm, the pitch Δ of the detectors on the sensor was 5.0 μm, and the wavelength λ of the illumination was 532 nm. The mean value of the four phase-shifted images I was set as the threshold value for the binarization. This process may be implemented by an image sensor with on-chip histogram equalization [15]. The Fresnel back-propagation result of PSBDH is shown in Fig. 1(b) [1]. The scattering square object was surrounded by a halo, which has been discussed in Ref. [7], and some ghosts appeared at the upper left and lower right. The peak signal-to-noise ratio (PSNR) between Figs. 1(a) and 1(b) was 28.2 dB. Similar artifacts also appeared in the back-propagation result of the phase-only hologram in the conventional PSDH with a full-bit image sensor, as shown in Fig. 1(c), where the number of amplitude levels was one, and the phase was not quantized. Thus, these artifacts were mainly caused by the amplitude quantization and not the phase quantization, as suggested in Ref. [7]. The artifacts caused by the amplitude quantization with a few levels can be approximated by amplitude equalization (quantization with a single level) of the phase-only hologram in PSDH.
n1 Ck1 −jU
0
xjk ;
(3)
(4)
where jU 0 j is a normalized jUj, where jU 0 j ajUj < 1, and a> 0 is a constant introduced to derive the rightmost part of Eq. (3), by performing Taylor-series expansion, and is also used in the following discussion. The binomial theorem is used to derive Eq. (4), where • C• is the binomial coefficient. As shown in Eq. (4), the amplitude quantization results in the multiplication of U and a weighted sum of jU 0 j raised to the k-th power. This means that the Fresnel back-propagation result can be expressed as the convolution of the object and the weighted summation of the multi-autocorrelation of the object, where the object is convolved with itself multiple times. Scattering objects are assumed in this Letter. Thus, in the object plane, the higher k-th terms approach small delta function because jU 0 j < 1. Therefore, the effect on the back-propagation results from the lower k-th terms is stronger than that from the higher k-th terms. The results in Figs. 1(b) and 1(c) support this discussion. The figures may be interpreted as the convolution between the scattering object and its autocorrelation. The above discussion indicates that the object in PSBDH can be retrieved by deconvolution with the multi-autocorrelation of the object. The object and its multi-autocorrelation are unknown. However, the object is available in the back-propagation result in Fig. 1(b), although artifacts are present because the object is scattered and its multi-autocorrelation must have a delta peak at the origin. Therefore, for the deconvolution, we use the back-propagation result instead of the object itself. The deconvolution process is as follows: 1. The complex field U B in PSBDH is backpropagated with the Fresnel kernel. The resultant field is defined as uB . 2. The multi-autocorrelation of the field uB is calculated by using the central part of Eq. (3) as 1 ; (5) h F−1 jFuB j where F· is the Fourier transform and is used for the forward-propagation here. 3. The Wiener deconvolved amplitude is calculated with the multi-autocorrelation result h: Fjhj ˆ F−1 FjuB j juj ; (6) jFjhjj2 c where c is a constant for the regularization, and is the
November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS
ˆ are kernel for the conjugation. Negative values in juj replaced with zero. Figure 1(d) shows the deconvolution result for the back-propagation in PSBDH, shown in Fig. 1(b). The artifacts in Fig. 1(b) were suppressed successfully. The PSNR between Figs. 1(a) and 1(d) was 34.4 dB. The improvement in the PSNR achieved by the deconvolution was 6.4 dB. The phase information of the object is compromised in this process. To recover a high-quality image in PSBDH, each point on the object should contribute to the whole plane of the hologram to reduce information loss due to binarization. Thus, the image sensor must be kept at a certain distance zB from the object in PSBDH. We assume that the spatial resolution Δ and the one-dimensional pixel count N of the image sensor are the same as those of the object, which is illuminated by a light source with wavelength λ. The numerical aperture (NA) is calculated by NA 0.5NΔ∕z, and the resolvable size Δ on the object is Δ 0.5λ∕NA when the sampling theorem is satisfied [16]. Thus, each pixel of the object spreads over the whole image sensor when the following condition is satisfied: z ≥ zB
NΔ2 : λ
(7)
The critical distance zB in the first simulation was 24 mm, which satisfied the condition in Eq. (7). The simulation result in the case where this condition is not satisfied is shown in Fig. 2. The object distance was 5 mm, and the object’s complex field and the other parameters were the same as those in the first simulation. Artifacts appearing in the Fresnel back-propagation result in Fig. 2(a) had a different shape compared with those in the first simulation in Fig. 1(b). The artifacts were reduced by the deconvolution; however, the intensities of the square objects became closer, as shown in Fig. 2(b). This means that the dynamic range in the back-propagation result was decreased by the binarization. First, we demonstrated PSBDH with synthetic experimental data for a two-dimensional scattering object captured by PPSDH. PPSDH is spatially resolved PSDH, which employs an array composed of sub-arrays of 2 × 2 different pixel-wise micropolarizers on the image sensor and polarization of the object and reference waves
Fig. 2. Simulation with the object in Fig. 1(a) at a near distance, showing the amplitudes of (a) the Fresnel back-propagation result and (b) its filtering result.
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[12,13]. It allows single-shot acquisition of the four phase-shifted images I. In PPSDH, the phase-shifted images are extracted from the single captured image, and then they are interpolated. PPS binary DH (PPSBDH), where the PPSDH is implemented by using a one-bit image sensor, may achieve fast threedimensional imaging with low-cost hardware. In the experiment, the PPSDH was implemented with an eight-bit image sensor, where the pixel count N 2 was 512 × 512 pixels and the pixel pitch Δ was 4.65 μm. The wavelength λ of the illumination was 532 nm. In this case, the critical distance zB in Eq. (7) was calculated to be 21 mm. In the experiment, the object was a die, and the object distance z was 250 mm, which satisfied the condition in Eq. (7). The experimentally captured image in PPSDH was computationally binarized. The threshold value was the mean value of the captured image, and the complex field on the hologram was calculated with Eqs. (1) and (2). If the denominator of Eq. (2) became zero, the phase Φ was calculated as π∕2, 0, or −π∕2 when the numerator was positive, zero, or negative, respectively. The experimental result is shown in Fig. 3. The back-propagation result of PPSBDH without the deconvolution, shown in Fig. 3(b), had a halo around the die and a lower contrast compared with that of the conventional PPSDH shown in Fig. 3(a). The halo was successfully suppressed by the deconvolution, and the contrast was improved, as shown in Fig. 3(c). However, inherent speckle noise was enhanced by the deconvolution. This speckle noise could be reduced by adopting one of the denoising methods reported in the literature on DH [17]. Next, we conducted a preliminary experiment to demonstrate PSBDH with a three-dimensional object by using
Fig. 3. Preliminary experimental results of PPSBDH with a two-dimensional object. The amplitudes of the Fresnel backpropagation result (a) without and (b) with binarization. (c) The amplitude after filtering (b). The scale bar is 10 mm.
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scattering objects and proposed a filtering method for suppressing them. We determined a condition for the object distance in PSBDH for achieving better performance. The preliminary results showed the promise of PSBDH. Our proposed concept can realize three-dimensional imaging at high-speed with a low-cost image sensor. In this Letter, we assumed a scattering object at a far distance and approximated the errors in PSBDH as a phase-only hologram using conventional PSDH. The phase information was also compromised in the error reduction. We should investigate a more accurate error model, including weakly scattering objects and near objects, as well as a reconstruction algorithm for achieving higher quality and functional imaging. The design of the image sensor is also important because of the thresholding required in the binarization in PSBDH. On-chip histogram equalization is one possible solution for this issue [15]. In future work, we will build a PPSBDH system for high-performance sensing. The authors thank Prof. Y. Awatsuji at Kyoto Institute of Technology for providing the hologram data used here. Fig. 4. Preliminary experimental results of PSBDH with a three-dimensional object. The amplitudes of the Fresnel back-propagation result (a) without and (b) with binarization. (c) The amplitude after filtering (b). The left and right results in each subfigure were obtained when focusing on near and far targets, respectively. The scale bar is 10 mm.
conventional PSDH. The results are shown in Fig. 4. The four phase-shifted images were sequentially captured by an eight-bit image sensor experimentally, where the pixel count N 2 was 1000 × 2000 pixels, and the pixel pitch Δ was 3.45 μm. The four captured images were computationally binarized, where the threshold value was the mean value of them, to emulate the binary image sensor. The wavelength λ of the illumination was 633 nm. The critical distance zB in Eq. (7) was 38 mm, and N was set to 2000. The object consisted of two toys. The left and right ones were located at 350 mm and 430 mm, respectively, from the image sensor. The back-propagation results without and with binarization are shown in Figs. 4(a) and 4(b), respectively. Each toy was focused well even with binarization, but binarization reduced the contrast due to the halo. The deconvolution for each back-propagation result with binarization in Fig. 4(b) improved the contrast successfully, as shown in Fig. 4(c). We proposed PSBDH and conducted preliminary experiments using experimental data obtained with conventional PSDH and PPSDH. In PSBDH, the phaseshifted images are captured by a one-bit image sensor. This binarization causes some artifacts in the Fresnel back-propagation result. We modeled the artifacts from
References 1. J. W. Goodman, Introduction to Fourier Optics (McGrawHill, 1996). 2. G. Nehmetallah and P. P. Banerjee, Adv. Opt. Photon. 4, 472 (2012). 3. B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966). 4. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967). 5. J. Goodman and R. Lawrence, Appl. Phys. Lett. 11, 77 (1967). 6. O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, Appl. Opt. 41, 6187 (2002). 7. I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, Appl. Opt. 45, 975 (2006). 8. F. Yang, Y. M. Lu, L. Sbaiz, and M. Vetterli, Image Process. IEEE Trans. 21, 1421 (2012). 9. N. Pandey and B. Hennelly, Appl. Opt. 50, B58 (2011). 10. P. T. Boufounos and R. G. Baraniuk, in Proceedings of 42nd Annual Conference on Information Sciences and Systems (IEEE, 2008), pp. 16–21. 11. A. Bourquard, F. Aguet, and M. Unser, Opt. Express 18, 4876 (2010). 12. Y. Awatsuji, M. Sasada, and T. Kubota, Appl. Phys. Lett. 85, 1069 (2004). 13. T. Tahara, Y. Ito, P. Xia, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, Opt. Lett. 38, 2463 (2013). 14. I. Yamaguchi and T. Zhang, Opt. Lett. 22, 1268 (1997). 15. C. Shoushun and A. Bermak, IEEE Trans. Very Large Scale Integr. Syst. 15, 346 (2007). 16. U. Schnars and W. P. O. Jüptner, Meas. Sci. Technol. 13, R85 (2002). 17. A. Uzan, Y. Rivenson, and A. Stern, Appl. Opt. 52, A195 (2013).
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Phase-shift binary digital holography Ryoichi Horisaki1,* and Tatsuki Tahara2 1
Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan 2 Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan *Corresponding author: [email protected]‑u.ac.jp Received August 6, 2014; revised October 3, 2014; accepted October 10, 2014; posted October 14, 2014 (Doc. ID 220543); published November 3, 2014 We propose phase-shift digital holography (DH) with a one-bit image sensor. In this method, the propagating complex field from an object is binarized by a one-bit sensor using a phase-shifter. The complex field on the hologram plane is then calculated with the one-bit image data. The object field is recovered via Fresnel back-propagation of the calculated hologram and filtering to suppress some artifacts caused by the binarization. The concept was demonstrated in preliminary experiments by using a synthetically binarized hologram with single-shot and multi-shot phase-shift DH. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (110.1758) Computational imaging; (100.3020) Image reconstructionrestoration. http://dx.doi.org/10.1364/OL.39.006375
Holography is a method for producing and observing the complex field of an object [1,2]. Current research topics in holography are roughly categorized into computergenerated holography (CGH) and digital holography (DH). CGH is a holographic display technique in which a hologram for generating a target complex field is digitally calculated on a computer, and the object’s complex field is reproduced by a diffractive optical element such as a spatial light modulator (SLM), which physically realizes the calculated hologram [3,4]. To alleviate some limitations of currently available SLMs, various CGH techniques have been proposed, for example, quantized CGH, which assumes an amplitude or phase mask with few quantization levels, including binary masks having just two levels [2]. DH is a holographic imaging technique in which the complex field is captured by an image sensor, and the object is digitally reconstructed by a computer [5]. DH has been used to realize three-dimensional imaging, phase imaging, and so on [2]. Quantization in DH has also been considered for reducing the data size of holograms for storing and transferring them electronically [6,7]. Binary sensors with two quantization levels have become an interesting research field. By using binary sensors with spatial and temporal oversampling, it is possible to computationally adjust the tradeoff between the spatial and temporal resolutions after image capturing [8]. DH using a binary sensor with oversampling for reducing speckle noise has also been demonstrated with a binary object [9]. This method employs multi-shot imaging with a moving diffuser to change the speckle pattern in each shot. A method has been proposed for achieving single-shot acquisition of a multi-bit image with a binary image sensor based on compressive sensing, which enables observation of a large object dataset by using a measurement dataset smaller than that used in conventional approaches [10,11]. This single-shot binary sensing method is a promising approach for decreasing production costs and drastically increasing the frame rate of sensors. Image degradation caused by computational binarization, not the binary image sensor itself, in phase-shift DH 0146-9592/14/226375-04$15.00/0
(PSDH) has been reported [6,7]; however, this degradation has not yet been modeled or suppressed. Because of this serious image degradation, binary DH has been considered to be impractical. In this Letter, we model the image degradation approximately but practically, and show how to suppress it. This enables us to maintain a practical level of image quality in PSDH, even when using a binary image sensor. We describe our phase-shift binary digital holography (PSBDH) method here. We also report results of preliminary experiments conducted to demonstrate PSBDH based on parallel PSDH (PPSDH) and conventional PSDH with scattering objects [12,13]. Our concept significantly increases the potential of DH for high-speed sensing with low-cost hardware. In conventional PSDH, the propagating complex field from an object is captured multiple times with a different phase shift introduced in each capturing step by using a phase-shifter [14]. The amplitude Ax and the phase Φx of the complex field Ux Ax expiΦx are recovered by the following procedure, where four phaseshifts with pitch π∕2 and a plane-wave reference beam are assumed (the y axis is omitted for simplicity): Ax
q Ix; 0 − Ix; π2 Ix; π∕2 − Ix; 3π∕22 ; (1)
Φx tan−1
Ix; π∕2 − Ix; 3π∕2 ; Ix; 0 − Ix; π
(2)
where Ix; θ is an intensity image with phase shift θ. References [6,7] have analyzed quantization, including binarization, to reduce the data size of the calculated complex field U from four phase-shifted images I captured by a full-bit image sensor. In contrast, here we consider the error in the complex field U from four phaseshifted images I captured by an one-bit image sensor. As shown in Eqs. (1) and (2), the autocorrelation terms jUj2 in the phase-shifted images I are canceled out even with binarization. However, possible values of the © 2014 Optical Society of America
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We model these artifacts with the phase-only hologram, as mentioned above, and use the model to suppress the artifacts in PSBDH. The phase-only hologram is written as expiΦx
n X Ux aUx 1 − jU 0 xjk jUxj k0
aUx
n X k0
Fig. 1. Simulation of the PSBDH. (a) The amplitude of a scattering object. The amplitudes of the Fresnel back-propagation results of (b) PSBDH and (c) the phase-only hologram in PSDH. (d) The amplitude after filtering (b).
complex field U B in p PSBDH are limited to three ampli tude levels (0, 1, 2) and eight phase levels (−π, −3π∕4, −π∕2, −π∕4, 0, π∕4, π∕2, 3π∕4). Thus, the error in the PSBDH is assumed to come from the quantization of the amplitude and phase. The simulation results of PSBDH, assuming a scattering object, are shown in Fig. 1. The object shown in Fig. 1(a) consists of two squares with random phases and different amplitudes. In the simulation, the pixel count N 2 of the object and the image sensor was 512 × 512 pixels, the distance z between them was 50 mm, the pitch Δ of the detectors on the sensor was 5.0 μm, and the wavelength λ of the illumination was 532 nm. The mean value of the four phase-shifted images I was set as the threshold value for the binarization. This process may be implemented by an image sensor with on-chip histogram equalization [15]. The Fresnel back-propagation result of PSBDH is shown in Fig. 1(b) [1]. The scattering square object was surrounded by a halo, which has been discussed in Ref. [7], and some ghosts appeared at the upper left and lower right. The peak signal-to-noise ratio (PSNR) between Figs. 1(a) and 1(b) was 28.2 dB. Similar artifacts also appeared in the back-propagation result of the phase-only hologram in the conventional PSDH with a full-bit image sensor, as shown in Fig. 1(c), where the number of amplitude levels was one, and the phase was not quantized. Thus, these artifacts were mainly caused by the amplitude quantization and not the phase quantization, as suggested in Ref. [7]. The artifacts caused by the amplitude quantization with a few levels can be approximated by amplitude equalization (quantization with a single level) of the phase-only hologram in PSDH.
n1 Ck1 −jU
0
xjk ;
(3)
(4)
where jU 0 j is a normalized jUj, where jU 0 j ajUj < 1, and a> 0 is a constant introduced to derive the rightmost part of Eq. (3), by performing Taylor-series expansion, and is also used in the following discussion. The binomial theorem is used to derive Eq. (4), where • C• is the binomial coefficient. As shown in Eq. (4), the amplitude quantization results in the multiplication of U and a weighted sum of jU 0 j raised to the k-th power. This means that the Fresnel back-propagation result can be expressed as the convolution of the object and the weighted summation of the multi-autocorrelation of the object, where the object is convolved with itself multiple times. Scattering objects are assumed in this Letter. Thus, in the object plane, the higher k-th terms approach small delta function because jU 0 j < 1. Therefore, the effect on the back-propagation results from the lower k-th terms is stronger than that from the higher k-th terms. The results in Figs. 1(b) and 1(c) support this discussion. The figures may be interpreted as the convolution between the scattering object and its autocorrelation. The above discussion indicates that the object in PSBDH can be retrieved by deconvolution with the multi-autocorrelation of the object. The object and its multi-autocorrelation are unknown. However, the object is available in the back-propagation result in Fig. 1(b), although artifacts are present because the object is scattered and its multi-autocorrelation must have a delta peak at the origin. Therefore, for the deconvolution, we use the back-propagation result instead of the object itself. The deconvolution process is as follows: 1. The complex field U B in PSBDH is backpropagated with the Fresnel kernel. The resultant field is defined as uB . 2. The multi-autocorrelation of the field uB is calculated by using the central part of Eq. (3) as 1 ; (5) h F−1 jFuB j where F· is the Fourier transform and is used for the forward-propagation here. 3. The Wiener deconvolved amplitude is calculated with the multi-autocorrelation result h: Fjhj ˆ F−1 FjuB j juj ; (6) jFjhjj2 c where c is a constant for the regularization, and is the
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ˆ are kernel for the conjugation. Negative values in juj replaced with zero. Figure 1(d) shows the deconvolution result for the back-propagation in PSBDH, shown in Fig. 1(b). The artifacts in Fig. 1(b) were suppressed successfully. The PSNR between Figs. 1(a) and 1(d) was 34.4 dB. The improvement in the PSNR achieved by the deconvolution was 6.4 dB. The phase information of the object is compromised in this process. To recover a high-quality image in PSBDH, each point on the object should contribute to the whole plane of the hologram to reduce information loss due to binarization. Thus, the image sensor must be kept at a certain distance zB from the object in PSBDH. We assume that the spatial resolution Δ and the one-dimensional pixel count N of the image sensor are the same as those of the object, which is illuminated by a light source with wavelength λ. The numerical aperture (NA) is calculated by NA 0.5NΔ∕z, and the resolvable size Δ on the object is Δ 0.5λ∕NA when the sampling theorem is satisfied [16]. Thus, each pixel of the object spreads over the whole image sensor when the following condition is satisfied: z ≥ zB
NΔ2 : λ
(7)
The critical distance zB in the first simulation was 24 mm, which satisfied the condition in Eq. (7). The simulation result in the case where this condition is not satisfied is shown in Fig. 2. The object distance was 5 mm, and the object’s complex field and the other parameters were the same as those in the first simulation. Artifacts appearing in the Fresnel back-propagation result in Fig. 2(a) had a different shape compared with those in the first simulation in Fig. 1(b). The artifacts were reduced by the deconvolution; however, the intensities of the square objects became closer, as shown in Fig. 2(b). This means that the dynamic range in the back-propagation result was decreased by the binarization. First, we demonstrated PSBDH with synthetic experimental data for a two-dimensional scattering object captured by PPSDH. PPSDH is spatially resolved PSDH, which employs an array composed of sub-arrays of 2 × 2 different pixel-wise micropolarizers on the image sensor and polarization of the object and reference waves
Fig. 2. Simulation with the object in Fig. 1(a) at a near distance, showing the amplitudes of (a) the Fresnel back-propagation result and (b) its filtering result.
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[12,13]. It allows single-shot acquisition of the four phase-shifted images I. In PPSDH, the phase-shifted images are extracted from the single captured image, and then they are interpolated. PPS binary DH (PPSBDH), where the PPSDH is implemented by using a one-bit image sensor, may achieve fast threedimensional imaging with low-cost hardware. In the experiment, the PPSDH was implemented with an eight-bit image sensor, where the pixel count N 2 was 512 × 512 pixels and the pixel pitch Δ was 4.65 μm. The wavelength λ of the illumination was 532 nm. In this case, the critical distance zB in Eq. (7) was calculated to be 21 mm. In the experiment, the object was a die, and the object distance z was 250 mm, which satisfied the condition in Eq. (7). The experimentally captured image in PPSDH was computationally binarized. The threshold value was the mean value of the captured image, and the complex field on the hologram was calculated with Eqs. (1) and (2). If the denominator of Eq. (2) became zero, the phase Φ was calculated as π∕2, 0, or −π∕2 when the numerator was positive, zero, or negative, respectively. The experimental result is shown in Fig. 3. The back-propagation result of PPSBDH without the deconvolution, shown in Fig. 3(b), had a halo around the die and a lower contrast compared with that of the conventional PPSDH shown in Fig. 3(a). The halo was successfully suppressed by the deconvolution, and the contrast was improved, as shown in Fig. 3(c). However, inherent speckle noise was enhanced by the deconvolution. This speckle noise could be reduced by adopting one of the denoising methods reported in the literature on DH [17]. Next, we conducted a preliminary experiment to demonstrate PSBDH with a three-dimensional object by using
Fig. 3. Preliminary experimental results of PPSBDH with a two-dimensional object. The amplitudes of the Fresnel backpropagation result (a) without and (b) with binarization. (c) The amplitude after filtering (b). The scale bar is 10 mm.
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scattering objects and proposed a filtering method for suppressing them. We determined a condition for the object distance in PSBDH for achieving better performance. The preliminary results showed the promise of PSBDH. Our proposed concept can realize three-dimensional imaging at high-speed with a low-cost image sensor. In this Letter, we assumed a scattering object at a far distance and approximated the errors in PSBDH as a phase-only hologram using conventional PSDH. The phase information was also compromised in the error reduction. We should investigate a more accurate error model, including weakly scattering objects and near objects, as well as a reconstruction algorithm for achieving higher quality and functional imaging. The design of the image sensor is also important because of the thresholding required in the binarization in PSBDH. On-chip histogram equalization is one possible solution for this issue [15]. In future work, we will build a PPSBDH system for high-performance sensing. The authors thank Prof. Y. Awatsuji at Kyoto Institute of Technology for providing the hologram data used here. Fig. 4. Preliminary experimental results of PSBDH with a three-dimensional object. The amplitudes of the Fresnel back-propagation result (a) without and (b) with binarization. (c) The amplitude after filtering (b). The left and right results in each subfigure were obtained when focusing on near and far targets, respectively. The scale bar is 10 mm.
conventional PSDH. The results are shown in Fig. 4. The four phase-shifted images were sequentially captured by an eight-bit image sensor experimentally, where the pixel count N 2 was 1000 × 2000 pixels, and the pixel pitch Δ was 3.45 μm. The four captured images were computationally binarized, where the threshold value was the mean value of them, to emulate the binary image sensor. The wavelength λ of the illumination was 633 nm. The critical distance zB in Eq. (7) was 38 mm, and N was set to 2000. The object consisted of two toys. The left and right ones were located at 350 mm and 430 mm, respectively, from the image sensor. The back-propagation results without and with binarization are shown in Figs. 4(a) and 4(b), respectively. Each toy was focused well even with binarization, but binarization reduced the contrast due to the halo. The deconvolution for each back-propagation result with binarization in Fig. 4(b) improved the contrast successfully, as shown in Fig. 4(c). We proposed PSBDH and conducted preliminary experiments using experimental data obtained with conventional PSDH and PPSDH. In PSBDH, the phaseshifted images are captured by a one-bit image sensor. This binarization causes some artifacts in the Fresnel back-propagation result. We modeled the artifacts from
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