Enhanced resolution of holographic stereograms by moving or diffusing a virtual pinhole array

Zi Wang; Zi Wang; Guoqiang Lv; Guoqiang Lv; Qibin Feng; Anting Wang; Hai Ming

doi:10.1364/OE.396639

1. Introduction

Holographic display is considered as the most real three-dimensional (3D) display since it recovers the optical wave field of the real object [1]. To avoid the complex optical interference in the holographic capturing process, the computer-generated hologram (CGH) has been a research hotspot in the recent years. Generally, there are two approaches in CGH, the wave-optics based approach and the ray-optics based approach.

In the wave-optics based approach, the depth information of the object is represented by a collection of many points or layers [2,3]. The hologram is calculated by independently propagating the wave field from each point or layer to the hologram plane and summing them together. This approach can provide accurate depth cues, but the view-dependent properties, such as occlusion, shading, glossiness and specular reflection, are hard to present. This will decrease the fidelity of the 3D image.

The ray-optics based approach, also know as the holographic stereogram (HS), encodes the light ray information to a hologram with abundant parallax images of the 3D object, computationally or optically [4–6]. The directional information of each light ray is encoded as a plane wave phase with a specific spatial frequency. The fast Fourier transform (FFT) is always used to accelerate the computation. Since the parallax images are captured either by camera or by physical rendering, the view-dependent properties mentioned above can be easily encoded into the hologram, making the 3D image photorealistic. Many researches have been reported on the spatial resolution enhancement of HS from several aspects such as adding depth information [7,8], reducing the FFT quantization error [9], near-field ray sampling [10], increasing sampling rate [11], decreasing the volume pixel (voxel) size [12] and light field compression [13].

One fundamental issue existing in integral imaging (II) and HS display is the spatio-angular resolution tradeoff. The product of spatial resolution and angular resolution is always a constant. Thus, simply increasing the spatial resolution comes at the cost of reduced angular resolution. The time-multiplexing and spatio-multiplexing techniques are usually used for spatial resolution enhancement by increasing the total pixel number, but with high costs and a complex system [14]. In the spatio-multiplexing technique, multiple display panels or projectors are spliced. In the time-multiplexing technique, each display panel or projector projects multiple subsets of display content onto the corresponding area sequentially in time domain.

Except for the fast calculation with FFT, another fast calculation method is to encode a pinhole-type integral imaging (PII) information into a hologram [15]. It is even more efficient than conventional FFT-based HS generation. The method works by recording the diffraction patterns produced by the pinholes. The look-up table method is introduced to highly reduce the calculation time.

In this paper, it will be demonstrated that by proper use of random phase, the PII-based hologram will present two different working modes, which are called the “pinhole mode” and the “lens mode”. These two modes have different 3D display characteristics. In the pinhole mode, the spatial resolution is mainly determined by the pinhole number, while in the lens mode, the spatial resolution is mainly determined by the elemental image (EI) resolution. Based on their different features, two different methods are proposed for the spatial resolution enhancement of both modes. By moving the virtual pinhole during hologram recording, the spatial resolution or sampling rate of the “pinhole mode” is enhanced. By adding a virtual random phase diffusor during hologram recording, the “pinhole mode” is transformed into the “lens mode” with a higher fill factor. In this case, a high-resolution EIA can be encoded into a relatively low-resolution hologram to enhance the spatial resolution of the “lens mode”. Both methods enhance the spatial resolution without increasing the total pixel number or the space-bandwidth product (SBP) of the hologram. Thus, the spatio-angular resolution tradeoff inherited from PII is relieved a lot.

2. Pinhole mode of PII-based hologram

Figure 1(a) shows the principle of recording stage of PII-based hologram. The light field is reconstructed by the pinhole type II system, and the CGH plane is used to record the object beam by simulating light propagation from pinholes to CGH. Each EI is projected onto the CGH through the corresponding virtual pinhole and magnified by a magnification ratio M = L/g. The light emitted from each pixel of EI can only pass through the corresponding pinhole, so each pinhole can be thought as a point light source which emits different pyramid-shaped light rays in different directions. The intensities of different light rays are determined by the pixel values of EI. To the center pinhole, the recorded complex amplitude U(x, y) on the CGH can be expressed as the multiplication of the spherical wave phase u(x, y) and the projected EI:

(1)$$U(x,y) = u(x,y) \cdot Proj[EI(x,y)],\textrm{ }u(x,y) = \frac{1}{R}\exp (i\frac{{2\pi }}{\lambda }R),$$

where $R = \sqrt {{{(x - {x_p})}^2} + {{(y - {y_p})}^2} + {L^2}}$ is the distance between the center pinhole (x_p, y_p) and a coordinate (x, y) on the CGH, EI(x,y) is the original EI, and Proj[·] is the projection operator on each EI. The projection operator Proj[·] can be expressed as:

(2)$$Proj[EI(x,y)] = Ro{t^{180}}[M[EI(x,y)]],$$

where Rot¹⁸⁰[·] is the rotation operator to rotate each EI by 180 degrees and M[·] is the magnification operator. Since magnification is performed, the pixel number of projected EI is increased by either interpolation or simple replication. For other pinholes, the recorded complex amplitude can be obtained by shifting the spherical wave phase u(x, y) and multiplied with the corresponding projected EI. Then, the total diffraction pattern on the CGH U_total(x, y) can be obtained by adding them together:

(3)$${U_{total}}(x,y) = \sum\limits_{m ={-} N/2}^{N/2} {\sum\limits_{n ={-} N/2}^{N/2} {u(x - mp,y - np) \cdot Proj[E{I^{m,n}}(x,y)],} }$$

where m and n are the sequence numbers of EI in the direction of x and y, p is the pinhole pitch, and N is the number of EI in one dimension. The computing speed is fast since we can precalculate and store the data of the spherical wave phase u(x, y) and reference it in the real-time calculation.

Fig. 1. (a) Principle of recording stage of PII-based hologram. (b) The observer’s eye samples one light ray from each virtual pinhole in the display stage of “pinhole mode”.

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In the display stage, the hologram will reproduce the recorded light rays, as well as the virtual pinhole array. As shown in Fig. 1(b), when the observer sees through the hologram, the observer is actually seeing the 3D image through the reproduced virtual pinhole array. It is a very similar situation to that of PII. That is, the observer’s eye samples one light ray from each virtual or real pinhole, so that the spatial resolution is determined by the pinhole number. Note, the absolute spatial resolution SR is defined as SR=1/p (lines/mm). That is why it is called the “pinhole mode”.

Next, simulation and optical experiment are performed. Figure 2(a) shows the bee model built in the 3ds Max modeling software. Figure 2(b) shows that the depth range of the bee model is about 10 mm. The camera array consisted of 80${\times}$80 virtual pinhole cameras, and the pitch and focal length were set as p=0.25 mm and g=2.5 mm. Figure 2(c) shows the captured EI array (EIA) which contains 4000${\times}$4000 pixels with 5 µm pixel pitch. The recording hologram plane was set 2.5 mm distant from the pinhole array to record complex amplitude of the pinholes. In this case the magnification ratio is 1. The short recording distance is to ensure fast computing speed. Using the MATLAB R2016b with an intel E3-1230 processor (3.4 GHz) and memory of 8 Gbytes, the hologram generation only took about 70 ms, while the conventional FFT-based HS generation took about 699 ms, which performed 80${\times}$80 FFT calculations. Assuming N${\times} $N elemental images each with N₀${\times} $N₀ pixels, the total computational amount of FFT is ${N^2}\beta N_0^2{\log _2}{N_0}$, where $\beta$ is the number of arithmetic operations (e.g., additions and multiplications) in the FFT. While in the proposed method, the computational amount is only $2{N^2}N_0^2$and is much smaller [15]. Figures 2(d) and 2(e) show the simulation reconstruction and optical reconstruction of the “pinhole mode” hologram. It is confirmed that the spatial resolution is determined by the pinhole number since each pinhole provides one voxel. The optical reconstruction is performed on a binary amplitude-only hologram which is printed on a glass substrate coated with chromium film. The amplitude-only hologram I(x, y) is encoded as:

(4)$$I(x,y) = 2Re [{U_{total}}(x,y){r^ \ast }(x,y)] + C,$$

where r (x, y) was the reference plane wave with incident angle of 3° and r* (x, y) was the conjugation of r (x, y). C was a constant real value to make I(x, y) non-negative. The hologram contained 4000${\times}$4000 pixels with 5 µm pixel pitch. A solid laser with wavelength of 532 nm was used to illuminate the hologram after being collimated by a lens. The hologram served as a rectangular window with side length of 20 mm. The photo was taken by a single lens reflex camera directly through the hologram window. It was a virtual 3D image.

Fig. 2. (a) The bee model built in the 3ds Max modeling software. (b) The depth range of the bee model. (c) The EIA of the bee model. (d) The simulation reconstruction result of pinhole mode. (e) The optical reconstruction result of pinhole mode.

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3. Resolution enhancement for pinhole mode with MALT

Since the spatial resolution is determined by the pinhole number, one can increase the spatial resolution by simply increasing the pinhole number and decreasing the pinhole pitch. However, with a given pixel number, increased pinhole number means decreased EI resolution or angular resolution. A high angular resolution is important for continuous motion parallax, and an extremely low angular resolution will in turn have a bad influence on the spatial resolution [16]. To solve this spatio-angular resolution tradeoff and increase the spatial resolution, the moving array lenslet technique (MALT) is used. The MALT is a technique to increase the sampling rate of integral imaging through fast moving the lens array along the horizontal and vertical axis [17,18]. Due to the afterimage effect, one can sample dense light rays and perceive high-quality 3D images. Here, instead of moving lens array, we move the pinhole array and record multiple light field information on the CGH plane, as shown in Fig. 3(a). Of course, multiple EIAs captured by MALT are prepared in advance. Each group of EIA is converted into the corresponding sub-hologram, and these sub-holograms have relative shift in real space. Assuming the pinhole array shifting K${\times} $K times in two dimensions, the moving step is p/K. The sub-holograms are added together in one hologram H (x, y):

(5)$$H(x,y) = \sum\limits_i^K {\sum\limits_j^K {U_{total}^{ij}(x - ip/K,y - jp/K)} } ,$$

where i and j are the moving times of the pinhole array along the horizontal and vertical axis. Equation (5) shows that the final hologram can be obtained by shifting the sub-holograms depending on the movement of the pinhole array and adding them all together.

Fig. 3. (a) Principle of recording stage of PII-based hologram with MALT. (b) The observer’s eye samples dense light rays from multiple virtual pinhole arrays in the display stage of pinhole mode with MALT.

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As a result, in the display stage, the hologram will reproduce the recorded dense light rays, as well as the multiple virtual pinhole array. As shown in Fig. 3(b), the observer will sample much more light rays and see more voxels due to the extra virtual pinhole arrays. Thus, the sampling rate or the spatial resolution is enhanced without decreasing the angular resolution by use of MALT. It is noted that the times of spatial resolution enhancement in the proposed method is equal to the moving steps of the MALT. While in previous researches with MALT [11,17,18], there is no proportionality between the number of moving steps and the improvement of spatial resolution. In the lens-based II display [17,18], the spatial resolution is usually determined by the elemental image resolution. The image quality is also influenced by the lens pitch since the viewer samples one patch from each lens. The lens array structure is apparent in the perceived image because the edges of each lens lead to the discontinuity of the 3D image, especially when the fill factor of the lens array is smaller than 1. In this case, the MALT will increase the sampling rate through fast moving the lens array. Within the response time of human eye, the viewer sees multiple images and mix them together. Then, the image becomes continuous and the lens array structure disappears but the resolution enhancement is not equal to the step number.

Simulation and optical reconstruction were performed to verify the spatial resolution enhancement. First, four groups of EIA were captured sequentially by moving the virtual camera array sequentially. Compared with the original position, the virtual camera array was moved by p/2 along the horizontal axis, the vertical axis, and both the horizontal and vertical axis, respectively. Figures 4(a) and 4(c) (Visualization 1) show the simulation and optical reconstruction results without MALT, and Figs. 4(b) and 4(d) (Visualization 2) show the simulation and optical reconstruction results with MALT. The discontinuous structures caused by a large pinhole pitch in Figs. 4(a) and 4(c) limit the image quality, while Figs. 4(b) and 4(d) show better image quality due to increased sampling rate. It is easily confirmed from the enlarged details that four times more voxels are observed due to the use of MALT, thus increasing the spatial resolution by four times. The advantage of the proposed method is that the MALT is no longer needed in the display stage. The composite hologram is used just as the ordinary hologram, and the multiplexed information could be presented in one hologram. One limitation of the proposed method is that the optical capture with MALT is not stable and has a vibration error, especially for moving objects. It is not easy to control the position of the lens array precisely in each step with fast speed. One solution is to use computer technology to render multiple EIAs at the same time, without really moving the lens array. Another simpler solution is to generate the intermediate EIs by repeating or interpolation [16]. In the proposed method, the use of MALT will reduce the brightness of each pinhole since the pinhole number is increased. It is not a serious problem because the brightness can be compensated by increasing the laser power. The speckle noise will not increase apparently because the pinhole pitch is large and the mutual interference is limited.

Fig. 4. The simulation reconstruction results from different viewing directions of the pinhole mode (a) without and (b) with MALT. The optical reconstruction results from different viewing directions of the pinhole mode (c) without (Visualization 1) and (d) with MALT (Visualization 2).

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4. Lens mode of PII-based hologram

In the pinhole mode of PII-based hologram, the observer’s eye can only sample one light ray from each virtual pinhole. From the areas outside of the pinholes, the observer cannot see any light. These dark areas make the 3D image discontinuous. Thus, the fill factor of pinhole mode is low, and abundant pinholes are needed to decrease the visibility of the dark areas, as well as increasing the spatial resolution. It will then increase the burden of EIA rendering. This situation is very similar to the pinhole-type integral imaging or point light source integral imaging. While in integral imaging using a lens array, the fill factor is high since the active area on the lens array is much larger than that on the pinhole array or the point light source array. A lens array with a small number of lenses can also present a continuous 3D image with high spatial resolution. This will help reduce the burden of EIA rendering. Next, we will demonstrate that by the use of random phase, the lens mode of PII-based hologram can be activated.

As shown in Fig. 5(a), in the recording process of pinhole mode, the narrow light rays emitted from the pinhole are recorded holographically on the CGH plane. In the display stage, these reproduced light rays will continue the propagation along the original directions without diffusion. It looks like that they are still coming from the original pinhole. To activate the lens mode, as shown in Fig. 5(b), we add a virtual diffusor with random phase distribution RP(x,y) to the complex amplitude U(x,y), then Eq. (1) is modified to:

(6)$$U^{\prime}(x,y) = RP(x,y) \cdot u(x,y) \cdot Proj[EI(x,y)],\textrm{ }u(x,y) = \frac{1}{R}\exp (i\frac{{2\pi }}{\lambda }R).$$

The introduction of random phase diffusor will cause diffusion on the CGH plane, and the reproduced light rays will continue the propagation with a diffusion angle α. If these light rays are back propagated, they will converge at the original pinhole, but with a larger beam width. It looks like that these light rays are coming from a lens. To ensure the lens size is exactly equal to p, the sampling pitch d of the random phase is given in the following equation:

(7)$$\sin \left( {\frac{\alpha }{2}} \right) = \frac{p}{{2L}} = \frac{\lambda }{{2d}},$$

where λ is the wavelength, and small angle approximation is applied. Since each light ray has a diffusion angle, similar to the lens imaging case, the virtual pinhole array now can be regarded as a virtual lens array. The fill factor is high and the 3D image is continuous, just like the lens-type integral imaging. That is why it is called the “lens mode”. Here, all the light rays are focused (have the minimum spot size) on the recording plane, so the recording plane acts as the reference plane in resolution-priority II (RPII) [12]. According to the guidelines of RPII, the recording plane needs to be located at the central depth plane of the 3D image [12].

Fig. 5. (a) The recording and display principle of pinhole mode. (b) The recording and display principle of lens mode. (c) The simulation reconstruction results of the lens mode. (d) Optical reconstruction results of lens mode (Visualization 3). (e) The intensity distributions along a horizontal line of the optical reconstruction results of three cases: pinhole mode without MALT, pinhole mode with MALT and lens mode.

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Note, here the virtual diffusor cannot be replaced by a real diffusor for three reasons. First, a real diffusor, with a specific diffusing angle, is costly to customize. Second, in the lens mode, the recording plane needs to be located close to the object for good quality. For presenting a 3D image with large depth, multiple recording planes located at different depths are required. However, we cannot place multiple real diffusors at different depths, since they will interfere with each other and destroy the optical wave field. However, in calculation, multiple virtual diffusors can be set to diffuse the corresponding information without mutual interference. Lastly, a real diffusor can only work for real image, but cannot diffuse a virtual image well, because the real diffusor cannot be placed close to a virtual image.

Next, the optical experiments were performed with the EIA used in the above pinhole mode. The recording plane was set 10 mm distant from the pinhole array. According to Eq. (7), the sampling pitch of random phase is d= (0.5*10/0.25)µm=20µm. Figures 5(c) and 5(d) show the simulation and optical reconstruction results of lens mode (Visualization 3). Compared with the pinhole mode, the 3D image is more continuous and smoother. The diffusion on the recording plane enlarges the active area of the virtual pinhole array, and the observer can sample much denser light rays without the need of increasing pinhole number. Theoretically, the fill factor of the 3D image is 100%.

Figure 5(e) shows the intensity distributions along a horizontal line of the optical reconstruction results of three cases: pinhole mode without MALT, pinhole mode with MALT and lens mode. It can be seen that the regular pinhole structure is apparent from the intensity change of the pinhole mode. The intensity change of the lens mode is random due to the speckle noise. In the pinhole mode, the pinhole area or the active area can be identified by setting the threshold value as the mid-value of the intensity. Then, the pinhole radius is calculated as 0.06 mm. The fill factors in the pinhole mode are calculated as 18% and 72% without and with MALT,respectively. In the lens mode, one cannot see any pinhole structure. The intensity change is mainly caused by speckle noise and the 3D image is continuous. The fill factor can be thought to be close to 100%.

5. Resolution enhancement for lens mode with high-resolution EIA encoding

In the lens mode, the spatial resolution is not determined by the pinhole number. Since the display principle of the lens mode is similar to the RPII, the analysis of resolution is similar. The display principle of RPII is that all the elemental lenses image or project the EI onto the reference plane, and the observer samples a specific patch on the reference plane through each lens. Thus, the projected resolution on the reference plane determines the spatial resolution of the 3D image. In the same way, the projected resolution on the recording plane of the lens mode determines the spatial resolution of the 3D image. As shown in Fig. 6(a), the projected resolution R on the recording plane is expressed as:

(8)$$R = \frac{1}{{M{p_x}}} = \frac{g}{{L{p_x}}},$$

where p_x is the pixel pitch of the EIA. According to Eq. (8), one can increase the spatial resolution by using a lens array with long focal length, or by shortening the image depth, or by increasing the total pixel number of EIA. Increasing focal length will decrease the viewing angle. Shortening the image depth will limit the total depth range of 3D image. Increasing the pixel number is a common and effective method for spatial resolution enhancement in II, but the cost for high-resolution display panels is high. It is interesting that the proposed method can encode a high-resolution EIA to a low-resolution hologram. Figure 6(a) shows the encoding process in which the sampling pitches of the EIA and hologram are the same as p_x. In the case of magnification ratio M>1, the projected voxel size Mp_x is larger than the sampling pitch p_x of the hologram. That is, the holographic volume of the hologram is not effectively utilized. As shown in Fig. 6(b), a high-resolution EIA with smaller pixel pitch p′_x can be used for encoding the low-resolution hologram. The acceptable minimum p′_x is given as p′_x=p_x/M. That is, a low-resolution hologram can record the information of a high-resolution EIA with M² times amount of information. Although a high-resolution imaging sensor is needed, the cost is much lower than a high-resolution display.

Fig. 6. (a) Low-resolution EIA to low-resolution hologram recoding (b) high-resolution EIA to low-resolution hologram recoding.

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Next, simulation and optical experiments were performed to verify spatial resolution enhancement. A bee model and a school badge were built in the 3ds Max modeling software, as shown in Fig. 7(a). The camera array consisted of 80${\times}$80 virtual pinhole cameras, and the pitch and focal length were set as p=0.25 mm and g=2.5 mm. The 3D objects were located about 100 mm away from the camera array. Note, here the image quality will degrade due to the large image depth according to Eq. (8). Three groups of EIA were captured with pixel number of 4000${\times}$4000, 8000${\times}$8000 and 16000${\times}$16000. They are encoded to the corresponding holograms with the same resolution of 4000${\times}$4000 and pixel pitch of 5 µm. Figures 7(c)–7(e) (Visualization 4, Visualization 5, Visualization 6) show the simulation reconstruction results at different depths and optical reconstruction results for the three holograms, respectively. From the enlarged details, it is easily confirmed that the image quality is improved with high-resolution EIA encoding. The peak signal-to-noise-ratio (PSNR) of the school badge image in the simulation reconstruction of Figs. 7(c)–7(e) are calculated as 13.32 dB, 14.80 dB and 16.69 dB, respectively. The increase of PSNR verifies the quality improvement. By encoding a high-resolution EIA to a low-resolution hologram, the holographic volume of the hologram is effectively utilized. The spatio-angular resolution tradeoff is relieved a lot since the total required pixel number (the SBP of the hologram) is only 1/ M² of conventional II. The image quality of the optical reconstruction is not as good as the simulation due to the speckle noise [19] and binarization.

Fig. 7. (a) The bee model and the school badge built in the 3ds Max modeling software. (b) The depth range of the models. The simulation reconstruction results at different depths and the optical reconstruction of lens mode with (c) EIA of 4000${\times} $4000 resolution (Visualization 4) and (d) EIA of 8000${\times} $8000 resolution (Visualization 5) and (e) EIA of 16000${\times} $16000 resolution (Visualization 6) encoded to the holograms of 4000${\times} $4000 resolution.

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6. Discussion and conclusion

The spatio-angular resolution tradeoff is an inherent issue in many 3D display techniques such as II and HS. The product of the spatial resolution and angular resolution is a constant. More specifically, in HS, the spatial sampling number S is equal to the hogel number N, and the angular sampling number A is expressed as:

(9)$$A = \frac{{{{\sin }^{ - 1}}(\lambda /{p_x})}}{{{{\sin }^{ - 1}}(\lambda /p)}} \approx \frac{p}{{{p_x}}}.$$

Thus, the product of S and A is Np/p_x, and is equal to the total pixel number of the display device, and determines how many light rays can be produced. The spatial resolution influences the image quality, and the angular resolution determines the number of perspectives can be perceived. In the pinhole-based II and point light source II, the spatial resolution mainly depends on the pinhole number or point light source number, while the angular resolution depends on the EI resolution. Thus, fast moving the pinholes [20] or increasing pinholes with high-resolution hologram printing [21] can increase the spatial resolution without decreasing angular resolution. In the RPII using a lens array, the situation is the opposite. The spatial resolution mainly depends on the EI resolution, while the angular resolution depends on the lens number [22]. Thus, the common method to increase spatial resolution without decreasing angular resolution is to increase the effective pixel number of the display device by time-multiplexing or spatio-multiplexing method [14]. However, all of these resolution enhancement methods increase the complexity and cost of the II display system.

In this paper, it is demonstrated that by proper use of random phase, the “pinhole mode” and the “lens mode” of PII-based hologram can be switched at will. The spatial resolution of the pinhole mode mainly depends on the pinhole number, so the pinhole mode is suitable for a large number of EIs with moderate EI resolution. The spatial resolution of the lens mode mainly depends on the EI resolution, so the lens mode is suitable for a moderate number of EIs with high EI resolution. The fill factor of the pinhole mode is lower than the lens mode. This will lead to discontinuous 3D image. Except for the difference on the resolution and fill factor, the depth performances of the two modes are different. The display principle of the lens mode can be thought as lens imaging, just like the RPII. Thus, the limited depth of field (DOF) of the lens limits the depth range of the lens mode. While in the pinhole mode, the observer views the 3D image by receiving narrow light rays through the virtual pinholes without lens imaging, so the depth range of the pinhole mode is larger than the lens mode. According to the different features of the two display modes, they can be chosen for different applications.

Two different strategies are introduced for the spatial resolution enhancement of the two modes. For the pinhole mode, the MALT is utilized to increase the effective pinhole number or sampling rate. For the lens mode, a high-resolution EIA encoding method is proposed to effectively utilize the holographic volume. Compared with the spatial resolution enhancement methods for II, the proposed methods are cost-effective. In the first method, the MALT is no longer required in the display stage, while the MALT is still required in II display. In the second method, a low-resolution hologram can present the information of a high-resolution EIA, while a high-resolution display panel is needed to present a high-resolution EIA in II display. In addition, the two methods could be combined. The high-resolution EIA encoding could also be added into the first method to enhance the angular resolution of pinhole mode. Similarly, the MALT could also be added into the second method to enhance the sampling rate of lens mode. The function of combing the two methods will be demonstrated in future work. As a result, the spatio-angular resolution tradeoff inherited from PII is relieved a lot in the PII-based hologram.

Funding

National Natural Science Foundation of China (61805065).

Acknowledgments

The authors thank anonymous reviewers for their thoughtful and helpful comments.

Disclosures

The authors declare no conflicts of interest.

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Name	Description
Visualization 1	The optical reconstruction results from different viewing directions of the pinhole mode without MALT
Visualization 2	The optical reconstruction results from different viewing directions of the pinhole mode with MALT
Visualization 3	The optical reconstruction results of lens mode
Visualization 4	optical reconstruction of lens mode with EIA of 40004000 resolution encoded to the holograms of 40004000 resolution
Visualization 5	optical reconstruction of lens mode with EIA of 80008000 resolution encoded to the holograms of 40004000 resolution
Visualization 6	optical reconstruction of lens mode with EIA of 1600016000 resolution encoded to the holograms of 40004000 resolution

Enhanced resolution of holographic stereograms by moving or diffusing a virtual pinhole array

Abstract

1. Introduction

2. Pinhole mode of PII-based hologram

3. Resolution enhancement for pinhole mode with MALT

4. Lens mode of PII-based hologram

5. Resolution enhancement for lens mode with high-resolution EIA encoding

6. Discussion and conclusion

Funding

Acknowledgments

Disclosures

References

Supplementary Material (6)

Cited By

Figures (7)

Equations (9)

Optics Express