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Abstract
We propose a holographic stereogram synthesis method which uses holograms that are optically captured by self-interference incoherent digital holography (SIDH) based on a geometric phase lens. SIDH is a promising solution for hologram acquisition under low-coherence lighting condition. A mechanical scanning system is constructed to acquire multiple perspective holograms. Numerical simulations and experimental analyses conducted using high-resolution diffractive optical element demonstrate that the proposed method can produce a wide-viewing hologram which can realize realistic 3D scenarios with depth cues such as accommodation and motion parallax. The future objectives include the implementation of a multiple-camera system for holographic videos.
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1. Introduction
A holographic display reconstructs optical field and provides complete depth cues. Various approaches have been proposed over several decades to efficiently modulate the optical field and produce high-quality three-dimensional (3D) images [1–4]. However, a high space-bandwidth product of a spatial light modulator (SLM) must be developed along with an efficient holographic data generation approach to present a realistic 3D visual experience using a holographic display.
The existing hologram generation methods are primarily based on computer-generated holograms (CGHs). The CGHs synthesis algorithm aims to determine the solution to propagation equation in a specific 3D formation, which is an ill-posed problem. CGHs can be categorized into direct and iterative methods. The direct method calculates the optical field from the physical condition using a propagation equation [5]. Various sources of 3D information have been employed, including point [6–8], layer [9–11], polygon [12–14], and light field [15–21]. However, the ideally calculated result does not sufficiently match the actual optical configuration, which leads to degraded image quality. Additionally, the computation time increases with the increase complexity of 3D information. An iterative method such as the Gerchberg-Saxton (GS) algorithm propagates a complex wave back and forth between the image and the SLM planes [22–24]. This approach is simple and straightforward, but the iteration requires optimization and incurs a higher computational cost.
Data-driven approaches are one of the latest solutions in modern computer science [24–27]. The development in deep neural networks (DNNs) and general-purpose graphics processing units (GPGPUs) has made deep learning as a powerful solution for conventional computer vision problems. Various studies have proposed the application of deep learning in the field of general holographic tasks such as holographic generation and reconstruction. However, machine learning requires numerous datasets for high-precision learning and high computing power to train the models.
However, generating the CGH in a real-world scenario requires additional 3D information. The speed of generation of 3D information from virtual scenarios has rapidly increased owing to the development of computer graphics. However, supplementary devices are required to extract 3D information from the real scenarios. Even if precise 3D information is obtained by using an accurate depth sensing device, complicated calculations are required during the hologram synthesis process.
This paper proposes a method for synthesizing wide-angle holographic stereograms using optically captured holograms. Holographic stereograms are ray-based approaches which convert light fields into holograms by adding phase and depth factors [19–22]. They comprise holographic elements (hogels) which consider the wavefield as a summation of a spatio-angular distribution of light rays. A ray-based approach enables the approximation of the optical field, which helps perform effective calculations. Additionally, a holographic stereogram is used as the basic principle of holographic printing [28]. Self-interference incoherent digital holography (SIDH) or Fresnel incoherent correlation holography (FINCH) is a promising incoherent hologram recording method [29–36], which splits the light emitted from an object into two waves and modulates each light wave differently. The two light waves are emitted from the same point, they have mutual coherence which enables interference on the image sensor even under low-coherence lighting conditions. SIDH or FINCH can be implemented using SLM [30], birefringence lenses [31], or metalens [32]. This study proposed the implementation of SIDH using a Pancharatnam-berry phase lens or a geometric phase (GP) lens, which modulate light based on the polarization state [33–35]. Because the proposed holographic camera has a compact configuration and acquires a phase-shifted hologram with a single exposure, we constructed a rotational scanning holographic camera system. Each directional holographic scenario was multiplied by the corresponding plane carrier wave. To enhance the visibility of synthetic holographic stereograms, holographic post-processing which compensates a wavefront by using optimization, was applied to each hologram. Hence, the proposed method can provide precise depth cues with a smooth parallax and occlusion. The digital reconstruction of a 25k resolution hologram and the optical reconstruction of 25k and 100k high-resolution holograms demonstrate the effectiveness of the angular synthesis approach and realistic 3D scenarios.
2. Proposed holographic stereogram synthesis method
2.1 Holographic stereogram synthesis
The stereogram produces 3D depth cues by composing images of different perspectives, and the holographic stereogram consists of the perspective view by hologram element or ‘hogel’. In this study, the hogels were replaced by angular-multiplexed holograms. Figure 1 presents a schematic of the concept. The finally synthesized hologram is called ‘master-hologram,’ and each hologram obtained through rotation is called ‘sub-hologram.’ Each perspective reproduced by the master-hologram comprises a sub-hologram multiplied by the plane carrier wave corresponding to the capturing direction. The plane carrier wave can be expressed as a k-vector.
The proposed holographic stereogram synthesis approach consists of three steps. Firstly, the sub-hologram is acquired. Figure 2 illustrates the sub-hologram capturing scheme and the flow chart of the holographic stereogram synthesis procedure. Figure 2(a) presents the optical configuration of the GP lens based SIDH. In the sub-hologram acquisition, the GP lens splits the incident wave into two waves, based on the circular polarization state. According to the principle of self-interference, these two waves are coherent because they originate from the same point. Additionally, the parallel phase shift method is used to generate the complex hologram since the GP lens simultaneously induces phase shift with the wavefront modulation [33–35]. The light passes through the polarizer again after passing through the GP lens, to ensure that both the waves have the same polarization; the phase difference is determined by the relative angle difference between the two polarizers. If the angle variance is four steps with 45° per step, four phase-shifted images with 90° per step are obtained and recombined to the complex hologram without bias and twin image noise. The polarized image sensor consists of the micro-polarizer array on the pixel array, which can acquire the intensity value of the polarization component of 0°, 45°, 90°, and 135°, and each extracted intensity hologram is recombined into the complex hologram.
Fig. 2. (a) Schematic of the optical configuration of geometric phase based SIDH system. Solid and dashed lines illustrate the optical path of the marginal rays emitted from the rotation center, (b) the flow chart of the proposed method.
Secondly, free space propagation is performed to the position of the master-hologram corresponding to the rotation center. The sub-hologram must match the geometric configuration of the capturing scheme. Therefore, the angular spectrum propagator is used to prevent the sampling problem [37]. Since the sampling period of the angular spectrum propagation remains constant during the propagation, the propagation result exhibit same magnification with the raw hologram. After the backpropagation, to match the resolution of the master-hologram, the sub-hologram is resized to the resolution of the master-hologram using the bicubic interpolation method.
Finally, the plane carrier wave corresponding to the capturing direction is multiplied and then added to form the master-hologram. The resolution and sampling period of the carrier wave are matched with that of the master-hologram. The field of the master-hologram, which is synthesized from the sub-hologram of $M \times N$ is given as follows:
The angular sampling distance of each sub-hologram is determined by its maximum spatial frequency, which is obtained based on the specifications of the image sensor and the Nyquist theorem. The largest spatial frequency is ${f_{max\; }} = 1/2\varDelta x$, which can determine the synthetic interval of the sub-hologram as follows:
2.2 Sub-hologram preprocessing
The proposed holographic stereogram carries holographic information on the plane carrier waves. Therefore, the image quality of the sub-hologram determines the image quality of the master-hologram. The proposed SIDH system is focused on the field-of-view (FOV), depth resolution, and sacrifice beam condition matching of the conventional FINCH, which presents high resolution, to capture a macroscopic object. The FOV of the SIDH system is determined by the ratio of the distances between the GP lens and object and the image sensor and GP lens. However, during the process of adjusting the distance condition to satisfy the FOV, the image quality degrades due to the illumination conditions, spectral bandwidth of the light source, and defects in the GP lens. The illumination condition is directly related to the clarity of the holographic fringe pattern; the narrower the spectral bandwidth of the incoherent light source, the higher the resolution of the self-interference hologram that is generated. Defects in the GP lens considerably affect the phase modulation accuracy. The GP lens is fabricated through the photo-alignment of liquid crystal cells. If the liquid crystal alignment is distorted by an external factor during the process of fabricating the GP lens, the phase modulation efficiency decreases, and it remains as an additional error term of a specific polarization component. This is not removed as a bias or twin image noise in the phase shift hologram, but remains as a residual bias and degrades the quality of the reconstructed image.
Among the various approaches used to calibrate holograms [38–40], the black box approach is applied in this study to numerically suppress the noise of the holographic signal. The aforementioned optical aberration factors ultimately construct one aberration term on the sub-hologram. The consequential aberration field is defined and the aberration term modeled as a quadratic surface is determined using a numerical optimization technique.
Figure 3 illustrates the wavefront compensation procedure. While obtaining a phase-shift hologram using polarization, the aberration terms generated by the various factors introduce errors in the polarization component, as shown in Figs. 3(a) and 3(d). Therefore, the numerical aberration compensation process aims to approximate the aberration field as the real and imaginary parts of the hologram. The compensated complex amplitude, ${U_{com}}$., can be expressed as
where ${U_{ori}}$ is the original complex amplitude, and ${U_{abe}}$. is the aberration complex amplitude. Figures 3(b) and 3(e) present the modeled aberration surface. According to the sparse optimization technique described in Ref. [40], ${U_{abe}}$ is considered for the surface model in the second-order asFig. 3. Wavefront optimization procedure applied at real and imaginary parts of raw hologram. (a–c) Real part compensation procedure, (d–f) imaginary part compensation procedure.
Figure 4 presents the results of applying wavefront compensation to SIDH. The removal of background noise in the digital reconstruction of the ensated hologram is demonstrated through the comparison with the raw hologram. Particularly, it is eliminated that the window-like noise at the peripheral region which is similar to the ‘ringing’ noise of the coherent image. The focus of the reconstruction is maintained, indicating that the proposed wavefront correction preserves the phase information. This is because the average of the complex data was adjusted to a value close to zero through wavefront compensation. The preserved holographic information leads to a precise accommodation cue during reconstruction. However, there is fluctuation in the phase-angle part after the compensation, as shown in Fig. 4(h). There are two resultant aberration surfaces because the compensation is utilized to the real and imaginary parts of the hologram. Therefore, these two surfaces act independently to compensate the wavefront; the continuity for an arbitrary local phase appears to be lost. Nevertheless, the reconstruction result is not significantly affected by this phase fluctuation.
Fig. 4. Wavefront compensation result comparison and digital reconstruction. (a–d) Raw hologram, (e–h) Compensated hologram (i) digital reconstruction of raw hologram, (j) digital reconstruction of compensated hologram.
The brightness of the peripheral region in the digital reconstruction of the compensated hologram was reduced, as shown in Fig. 4(j). The brightness of the image was determined by the aperture limitation. In SIDH and FINCH, the overall hologram is composed as a point spread hologram from a point source. This approach has two main limitations: the optical path length difference and the Nyquist frequency. The optical path length difference is based on the focal length of the GP lens and optical configuration of the objective illumination and must be smaller than the coherence length. The Nyquist frequency is related to the specifications of the image sensor; an aliasing problem occurs if the highest spatial frequency of the point spread hologram is smaller than the Nyquist frequency. Compared with the Nyquist frequency, the optical path difference is the dominant factor of the aperture limitation, which defines the minimum size of the point spread hologram. The aperture limitation owing to the size of the image sensor results in the cropping of the periphery of the hologram, which makes the periphery of the reconstructed image appear dark.
3. Experiment
3.1 Sub-hologram generation
The proposed method was verified by performing a numerical simulation and a proof-of-concept (PoC) demonstration using a diffractive optical element (DOE). A multi-angle rotation system was set up and integrated with a holographic camera based on our previous research [33–35]. The GP lens based SIDH can be installed in a motorized scanning system because it is robust against vibrations. The implemented motorized stage has a range of motion of 90° and 45° in horizontal and vertical directions, respectively. Figure 5 illustrates the capture conditions. The holograms of the emissive and the reflective objects placed 50 mm apart from the rotation center, were captured. The emissive target consisted of a LED surface light source and patterned aperture of letters ‘ETRI’ and ‘KHU’, and the reflective target was a white statue. The distance between the rotation center and GP lens was 200 mm for the emissive target and 250 mm for the reflective target. A polarization image sensor was used to implement the single-shot SIDH system, with a resolution of 1024 × 1024 pixels and a pixel pitch of 6.9 $\mathrm{\mu m}$. In our experiment, a 550 nm bandpass filter with a bandwidth of ±50 nm and a GP lens with focal length 265 mm were used. Furthermore, the distance from the GP lens to the image sensor was 8 mm, which determines the field-of-view (FOV) of the holographic camera, which was set as 23.6°.
Fig. 5. (a) Concept of the sub-hologram capture system. The experimental setup of (b) emissive target, (c) reflective target.
3.2 Numerical simulation
The digital reconstruction was verified by synthesizing the master-hologram with a resolution of 25,000 × 25,000 and the sampling period of 1 $\mathrm{\mu m}$, with the sub-hologram of the emissive target. It is easier to confirm the holographic signal than the reflective target as the brightness of the incident light determines the quality of the holographic fringe pattern. According to Eq. (3), the angular sampling distance was 1.29°, and the number of sub-holograms was 25 × 25.
Figure 6 presents the spatial frequency domain and digital reconstruction of the master-hologram. The signal distribution was arranged periodically in the frequency domain, as shown in Fig. 6(a). The periodic arrangement of holographic signals in the Fourier domain was obtained through the multiplication of the plane carrier wave of each k-vector. The view corresponding to the angle was reconstructed by selecting a specific area, as shown in Figs. 6(b)–6(g). Each reconstructed view image can provide accommodation according to the propagation equation and a smooth parallax owing to the viewing directions.
Fig. 6. Numerical simulation result of synthesized hologram. (a) Spatial frequency domain, digital reconstruction results when focusing on the (b–d) backward letter ‘KHU’ and (e-g) forward letter ‘ETRI’, viewing from (b, e) -10°, (c, f) center, (d, g) +10°.
3.3 Optical experiment
The holographic stereogram made of silica DOE was fabricated using lithography to demonstrate optical reconstruction. Off-axis processing, phase encoding, and binarization techniques were applied to the complex holographic stereogram. Off-axis processing on the angular domain was employed at 20° to avoid the illumination bias. The conjugate signal induced by binary modulation was deleted in the off-axis processing. Subsequently, phase angle encoding and binarization based on zero were performed by applying the inverse Fourier transform.
Holographic stereograms with two resolutions, i.e., 25k and 100k were produced. The 25k hologram synthesis conditions were identical to those in the simulation in terms of the resolution and pixel pitch. The synthesis condition of the 100k high-resolution hologram included a sampling period of 0.5 $\mathrm{\mu m}$, synthesis interval of 0.64°, and perspective of 99 × 75. Figure 7 depicts the fabricated holographic stereogram of a 25k hologram without illumination. The dimensions of the holographic stereogram were 25 mm × 25 mm. The hologram was reconstructed by using tilted LED illumination at an incident angle of 12°. The wavelength was 550 nm and the field of view was 31.9°.
Figure 8 presents each result of the overall synthesis procedure and optical reconstruction comparison. Figure 8(a) depicts the phase-angle part of the compensated sub-hologram which has the resolution of $({{N_x},\; {N_y}} )= ({1024,\; 1024} ).$ Fig. 8(b) depicts the result of applying backpropagation to the rotation center and resizing to the resolution of $({N_x^{\prime},\; N_y^{\prime}} )= ({25,000,\; 25,000} ).$ The bicubic algorithm was used as the interpolation method. Figure 8(c) depicts the plane carrier wave, and Fig. 8(d) is a single perspective sub-hologram which multiplied with the resized hologram data and carrier wave. The master-hologram shown in Fig. 8(e) was obtained by adding all the sub-holograms obtained by repeating this process at all the measured angles. The phase-angle part of the hologram was extracted and binarized to reconstruct the synthesized holographic stereogram, as shown in Fig. 8(f).
Fig. 8. Step-by-step phase-angle part hologram visualization of the proposed method. (a) Raw hologram with resolution of $({{N_x},\; {N_y}} )$, and sampling period of $({\varDelta x,\; \varDelta y} )$, (b) back propagation and resizing result, (c) arbitrary plane carrier wave, (d) plane carrier multiplied result, (e) the result of summation of sub-hologram, (f) binarization of phase-angle part of master-hologram, (g) optical reconstruction without wavefront compensation, (h) backward focused optical reconstruction with wavefront compensation, (i) frontward focused optical reconstruction with wavefront compensation.
The effectiveness of the wavefront compensation was demonstrated through a comparison between the compensation adapted proof of concept and the uncompensated one, as shown in Figs. 8(g), 8(h), and 8(i). While the optical reconstruction of the uncompensated data can record the holographic signal, it is disrupted by noise and cannot be observed properly, as shown in Fig. 8(g). Conversely, in the case of the corrected data, clear images and focus cues can be observed, as shown in Figs. 8(h) and 8(i).
Figure 9 presents the optical reconstruction result of the 25k holographic stereogram of the emissive target. The same angle as that of the digital reconstruction result shown in Fig. 6, was observed. The synthesized conditions were a viewing zone of 25 × 25 and an angular distance of 1.29°. The theoretical diffraction angle was 15.9°. However, the observation was conducted from -10° to +10° because the diffraction efficiency decreases as the angle of observation increases, which is the characteristic of DOE. Variation of focus and motion parallax are observed in the results of Fig. 9. The dark regions at the periphery of the letters are induced by overlapping images of different angles.
Fig. 9. Optical reconstruction of the 25k holographic stereogram. (a-c) Backward focus, (d-g) Frontward focus, viewing from (a, d) -10°, (b, f) center, (c, g) 10°.
Figure 10 and visualizations 1 and 2 depict the fabricated 100k high-resolution holographic stereogram and the resultant optical reconstruction. The hologram of the reflective target was noisier than the emissive target due to the scattering effect on the surface. The same wavefront compensation was applied to the sub-holograms, and the master-hologram of the 100k resolution was synthesized. The dimensions of the produced holographic stereogram were 50 mm × 50 mm, and the viewing angle of the holographic stereogram was approximately 66.7° from -33° to 33°. However, the diffractive efficiency was relatively low compared with the close view of the central axis. Figures 10(b)–10(g) present the holographic stereogram observed at a viewing angle of approximately 25° where the diffraction efficiency was relatively preserved. The optical reconstruction results presented a natural 3D image even when viewed with the naked eye, as shown in Figs. 10(h) and 10(i). In the case of the periphery, a dim image was observed owing to the diffractive efficiency of the DOE, but the depth cue and occlusion were also confirmed.
Fig. 10. Experimental result of the 100k holographic stereogram result. (a) Fabricated 100k hologram without illumination. (b–d) Backward focus, (e–g) frontward focus, viewing from (b, e) -25°, (c, f) center, (d, g) 25°, (h–i) close view of (c) and (f) (see Visualization 1 and Visualization 2).
The proposed method can be used to develop a high-resolution hologram synthesis algorithm which uses holographic cameras. A higher resolution is required to provide a more natural 3D visual experience using holographic displays. However, the current CGHs generation techniques which are based on the FFT algorithm require more computing power as the resolution of the hologram increases to more than 100k. The proposed holographic stereogram approach is a potential solution for high-resolution holograms, and a hologram can be generated without requiring additional 3D information such as a depth sensor. However, the numerical approximation-based solution is used in this case because the noise suppression is required. The characteristics of the optical element must be improved along with the optical structure of the holographic camera to acquire high-quality holographic stereograms.
4. Conclusion
In summary, this study presented a self-interference incoherent digital hologram-based holographic stereogram synthesis method. The method was used to synthesize a high space-bandwidth hologram which excludes additional 3D information for separate CGH generation. Wavefront correction through an optimization technique was applied to SIDH to address the arbitrary noise in the hologram, thereby improving the image quality of the holographic stereogram. The effectiveness of the proposed method was demonstrated through numerical simulation and optical reconstruction, and accommodation and motion parallax were produced. The future objectives include the implementation of a multiple-camera system for holographic videos. Therefore, careful assessment of the holographic camera is required, and the physical limitations of the image sensor must be overcome. We believe that this attempt inspires various fields of the incoherent holographic camera.
Funding
Institute for Information and Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT)(2019-0-00001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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