## Abstract

Recently, great progress has been made in the research of cylindrical holography as a promising technique of 360° display. However, there is an unsolved issue of occlusion culling, which is critical to cylindrical holography and degrades the reconstructed images due to overlapping. To our knowledge, the occlusion issue in cylindrical holography has never been deeply discussed. In this paper, a method of occlusion culling is proposed for computer-generated cylindrical holograms based on a horizontal optical-path-limit function. In cylindrical diffraction, the propagation characteristics of light waves can be described by the point spread function, which is mainly obtained by analyzing the meaning of the obliquity factor in the concentric cylinder model. Different from the planar diffraction, the diffraction area of each source point is limited within the tangents in cylindrical diffraction. Therefore, a horizontal optical path limit function that acts directly on the point spread function for occlusion culling is established. Besides, the proposed method can be applied to the three-dimensional object by using the layer-oriented method. Moreover, the effectiveness of the proposed occlusion culling method is verified by the numerical simulation results and error analysis of the reconstructed images.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1 Introduction

Holography has been widely regarded as an ideal three-dimensional (3D) display technology because it can provide human eyes the full parallax and depth information [1]. Own to great flexibility of recording both real and virtual objects as well as high efficiency of containing both amplitude and phase information of the object, computer-generated-hologram (CGH) [2] is regarded as an important technology to realize the holographic display. However, the viewing zone of conventional planar holograms is limited by their spatial frequency and physical size [3], which is known as the narrow field of view (FOV) [4], and it becomes an important issue in CGH research. Fortunately, the issue of FOV [5] can be solved by cylindrical holography. In cylindrical holography, the object waves are captured in 360° recordings [6–8]. Therefore, the reconstructed object can be observed from any direction, which is of great significance to the development of holographic 3D imaging and display.

In recent years, researches have devoted many efforts to the cylindrical holography. Yamaguchi et al. proposed a fast calculation method for cylindrical CGH that is viewable in 360° [9]. Jackin et al. introduced another high-speed calculation method based on FFT, in which the angular spectrum diffraction formula of the cylinder model is proposed and the transfer function is found [10]. Sando et al. calculated the 3D Fourier spectrum to record the object waves in all directions and used Bessel function expansion to reduce the calculation time and memory usage for cylindrical CGH [11–13]. Zhao et al. proposed a fast calculation method for cylindrical CGH based on wave propagation in spectral domain [14]. Wang et al. proposed a fast method of generating cylindrical holograms based on FFT and convolution and accomplished the unification of inside-out propagation (IOP) and outside-in propagation (OIP) models [15]. Obviously, great progress has been made in the research of cylindrical holography recently. However, there is an unsolved issue of occlusion culling, which is critical to the visibility and recognizability of cylindrical holography. In this paper, the main concern is to cull the self-occlusion of cylindrical back-face observation surface, which will degrade the reconstructed images by overlapping. To our knowledge, the occlusion issue in cylindrical holography has never been deeply discussed.

In this paper, a method of occlusion culling is proposed for computer-generated cylindrical hologram (CGCH) based on horizontal optical-path-limit function (HOLF) in Rayleigh-Sommerfeld (RS) diffraction [16]. In the proposed method, the RS diffraction formula is applied to establish a point spread function (PSF) for the cylindrical diffraction model with considering the obliquity factor based on the source point, firstly. Secondly, a specific HOLF embedded in the PSF is proposed for occlusion culling by analyzing the diffraction area of source points. The principle of the proposed occlusion culling is further explained here. In our proposed method, due to the occlusion limitation, light waves can’t spread to cylindrical back-face observation surface. Therefore, the diffraction area of each source point is no longer the entire cylindrical observation surface but within the tangents of the point. Hence, the HOLF is established based on the horizontal optical-path of wave propagation to limit the diffraction area, which is the crux to achieve occlusion culling. Since the HOLF acts directly on the PSF, there is little additional calculation time, and therefore, the proposed method can achieve almost the same calculation speed as the conventional method based on convolution does. Furthermore, the proposed method can be applied to the hologram generation and reconstruction of 3D objects by using the layer-oriented method [17,18]. Numerical simulation is carried out to verify the correctness and effectiveness of the proposed method for occlusion culling.

## 2. Proposed method of occlusion culling for CGCH

#### 2.1 Feasibility of cylindrical RS diffraction formula

The RS diffraction formula is deduced from Green's theorem [19] and Kirchhoff integral theorem [20]. The premise for deriving RS diffraction is the following boundary conditions:

- (1) Complex amplitude
*U*of all points on opaque diffraction screen*S*is 0._{1} - (2) The complex amplitude
*U*of the eyelet*∑*is not influenced by the presence of the opaque diffraction screen*S*_{1.}

As shown in Fig. 1(a), the previous RS diffraction formula was deduced based on the planar diffraction model, *∑* is a small eyelet and *S _{1}* is a planar opaque screen. In the cylindrical diffraction model, the eyelet ∑ can still be regarded as a plane, but the planar screen

*S*is replaced by a cylindrical screen as shown in Fig. 1(b). Under conditions of Sommerfeld radiation [21], the complex amplitude of the diffraction field is given by:

_{1}*G*is Green function chosen under the Sommerfeld condition, and

*n*refers to the unit vector in the direction of the external normal. According to the boundary condition (1), in RS diffraction, the contribution of screen

*S*to the integral is 0, so Eq. (1) can be simplified to:

_{1}*U*and

_{d}*U*are complex amplitudes of diffraction and object field, respectively. Integral range

_{s}*C*is the entire cylindrical object surface, and

*d*is the distance between the source point (

*R*) and the destination point (

_{s}, θ_{s}, Z_{s}*R*).

_{d}, θ_{d}, Z_{d}*α*represents the angle between the inverse propagation vector from the point (

*R*) to the point (

_{d}, θ_{d}, Z_{d}*R*) and the external normal

_{s}, θ_{s}, Z_{s}**of the point (**

*n**R*), cos

_{s}, θ_{s}, Z_{s}*a*is determined as obliquity factor.

#### 2.2 Derivation of cylindrical RS diffraction formula

In this section, the derivation process of the obliquity factor cos *a* will be discussed. According to the direction of light waves propagation, the cylindrical diffraction is divided into two models: outside-in propagation (OIP) and inside-out propagation (IOP). *P _{o}* is a point on the outer cylinder, while

*P*is a point on the inner cylinder.

_{i}*Q*is the center of the horizontal cross section containing

*P*,

_{i}*O’*is the center of the horizontal cross section containing

*P*, and

_{o}*E*is the vertical projection of

*P*on this cross section. The derivation is performed in cylindrical coordinates. |

_{i}*Z*-

_{d}*Z*| and |

_{s}*θ*-

_{d}*θ*

_{s}| represent the differences of the vertical and angular coordinate between the source and destination points, respectively. And they are corresponding to the length of line segment

*P*and the size of ∠

_{i}E*P*, respectively. The two cylindrical diffraction models of OIP and IOP are shown in Figs. 2(a) and 2(b), respectively. Furthermore, in order to facilitate the analysis of the geometric relationship, local diagrams are extracted as shown in Figs. 2(c) and 2(d).

_{o}O'EIn the triangle O'P_{o}E, the horizontal optical path of wave propagation |P_{o}E| is deduced as:

*d*is calculated by: According to the cosine theorem, the obliquity factor of OIP and IOP models are given by:

*h*(

*θ*,

*z*) is defined as: then the cylindrical RS diffraction formula is expressed as:

#### 2.3 Principle of occlusion culling

In Eq. (9), the integral limit is the entire cylindrical object surface, indicating that diffraction waves from the source point can act on all points on the cylindrical observation surface, and the propagation characteristics of light waves are described by the PSF. In our proposed method, considering the factors of cylindrical occlusion and the boundary conditions of RS diffraction, the diffraction area is limited to better meet practical application. In fact, the process of occlusion culling is to retain light waves selectively, which is thought under the fixed directions based on geometric optics. Firstly, whether the diffraction light ray passes other points on the cylindrical observation surface before reaching the destination point should be judged. If a light ray passes through more than one point on the cylindrical observation surface, then this light ray should be eliminated. Hence, in the OIP model, the diffraction area of each source point should be limited within the corresponding tangents. Since light is inverse in both two models, the occlusion in the IOP model should also be considered. In the IOP model, because the cylindrical object surface is restricted to be opaque material by the boundary conditions of RS diffraction, light waves can not spread to the cylindrical back-face observation surface. The diffraction area of each source point should also be within the tangents. Due to the symmetry of the cylinder, a column of source points on the object is selected to analyze the diffraction area, which can represent the law of all points. A line light source is used to illuminate the selected source points. The diffraction area is shown in Fig. 3:

As shown in Fig. 4, it’s obvious that the diffraction area of a source point is independent of its z-axis coordinate. Therefore, in the following analysis, the 2D top view is used to discuss the change rule of horizontal optical path with the difference of the angular coordinates between the source and destination points.

*P _{i}* is a point on the inner cylinder and

*P*is a point on the outer cylinder.

_{o}*Q*is the critical point, that is, the tangents point of

_{c}*P*or

_{i}*P*to the cylindrical observation surface.

_{o}*Q*is the destination point of diffraction waves. When the point

_{i}*Q*gradually moves from

_{i}*Q*to

_{1}*Q*in the direction of the brown arrow, the difference of the angular coordinates between the source and destination points |

_{2}*θ*| gradually increases from 0 to π. According to Eq. (4), the horizontal optical path

_{d}-θ_{s}*d*of wave propagation will increases from (

_{op}*R*-

*r*) to (

*R*+

*r*). When

*Q*is at the critical point

_{i}*Q*, the light ray is in the tangent direction. Since the tangents and radius are perpendicular,

_{c}*d*equals to (

_{op}*R*-

^{2}*r*)

^{2}^{1/2}. For |

*θ*-

_{Qi}*θ*|<|

_{Q1}*θ*-

_{Qc}*θ*|,

_{Q1}*d*is smaller than (

_{op}*R*-

^{2}*r*)

^{2}^{1/2}, and

*Q*is in the area

_{i}*S*that diffraction waves can reach. Similarly, for |

_{retain}*θ*-

_{Qi}*θ*|>|

_{Q1}*θ*-

_{Qc}*θ*|,

_{Q1}*d*is greater than (

_{op}*R*-

^{2}*r*)

^{2}^{1/2}, and

*Q*is in the area

_{i}*S*that diffraction waves can’t reach. In a word, we figure out a unified criteria that can filter out redundant diffraction waves according to the horizontal optical path in order to achieve occlusion culling.

_{remove}As shown in Fig. 5, the diffraction area of the source point should be limited, which can be achieved by establishing an HOLF embedded in the PSF. When a light ray should be retained, its HOLF equals to 1, which can keep its corresponding PSF. When a light ray should be removed, that is the horizontal optical path of the light ray is 0, thereby its corresponding PSF is changed to 0, which can be achieved by setting HOLF to 0. Therefore, the HOLF is established as follows:

*h*(

*θ*,

*z*). Therefore, the new PSF is

*h'*(

*θ*,

*z*) as follows:

#### 2.4 Sampling conditions

In order to reasonably simulate experiments, the Nyquist theorem must be satisfied in both the azimuthal and the vertical directions. Since the spatial frequency of the object function *U _{s}*(

*θ*,

_{s}*z*) is not high relative to

_{s}*h'*(

*θ, z*), we can only consider the maximum value of the spatial frequency of

*h'*(

*θ, z*). Besides, due to the spatial transformation speed of [(

*cosa*×

*HOLF*(

*θ, z*))/(

*jλd*)] is far less than

*exp*(

*jkd*) in

*h'*(

*θ, z*), the derivation can be simplified with only the exponential spatial frequency:

*θ=r*/

*R*, z=0,

*f*(

_{θ}*θ, z*) can reach the maximum value and when cos

*θ=*1, z =

*H*/2,

*f*(

_{z}*θ, z*) can reach the maximum.

*N*and

_{θ}*N*in both the azimuthal and vertical directions are as:

_{z}## 3. Simulation results and analysis

#### 3.1 Effectiveness of occlusion culling

To prove the correctness of the cylindrical diffraction model and the effectiveness of the proposed method of occlusion culling, holograms and reconstructed images are simulated with and without occlusion culling by using *h* (*θ*, *z*) and *h'*(*θ*, *z*), separately. The radii of the inner and outer cylinders are set to be 1 cm and 10 cm, respectively, and the height of cylinders is 15 cm. According section 2.4, in the wavelength range of visible light, a considerable number of sampling points will be required. Therefore, we set a larger wavelength of 300 µm. And according to Eq. (15), we chose a resolution of pictures of 512×512 near the Nyquist limit.

The simplest experiment is Young’s interference experiment, in which there are only two points in the object surface located at (-π/32, 0) and (π/32, 0). The diffraction field and its normalized amplitude and the reconstruction are shown in Fig. 6. Obviously, when the PSF is *h*(*θ*, *z*), since each light ray acts on two points on the front-face and back-face surface, there are two amplitude peaks in the diffraction field, which will cause overlapping and degrade the reconstructed images. In contrast, when the PSF is *h'*(*θ*, *z*), there is only one amplitude peak in diffraction field, and the two-point image can be well reconstructed due to the proposed occlusion culling method. Therefore, the correctness of the cylindrical diffraction model and the effectiveness of the proposed occlusion culling has been well confirmed.

In order to have a more intuitive experience visually, we use an image with simple gray distribution to generate and reconstruct the holograms using the IOP and OIP models, respectively. Besides, we compare the reconstruction results and the original image using the Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index (SSIM) to evaluate the quality of the reconstructed images [22]. The evaluation calculation formulas are as:

*M*and

*N*are the number of pixels of the original image and reconstructed image in azimuthal and vertical directions.

*f*and

_{ij}*g*are the amplitude of original image and the reconstructed image, respectively.

_{ij}*μ*, µ

_{f}*and*

_{g}*σ*are the mean luminance and the standard deviation of the original image and reconstructed image, respectively. And

_{f}, σ_{g}*C*

_{1},

*C*

_{2}and

*C*

_{3}are constants to avoid a null denominator. It can be inferred that reconstruction quality is better when the PSNR is higher and the SSIM is closer to 1.

As shown in Fig. 7, the simulation results meet our expectations. The process of occlusion culling can eliminate overlapping and improve the quality of the reconstructed images effectively.

In the above results, the pictures are flat expanded view. To simulate the scene realistically, we give results of reconstructed images from different perspectives. As shown in Fig. 8, the reconstructed image is observed at 0°, 90°, 180°, 270°, which proves that the reconstructed image from the hologram is viewable from all perspectives of 360° after the proposed method of occlusion culling is applied.

In the above experiments, in order to verify the correctness of the theoretical results, we directly use the complex amplitude reconstruction method, which means that the hologram is not encoded and all complex amplitude information is retained. However, most existing spatial light modulators can only match amplitude-only or phase-only holograms. Therefore, in order to verify the applicability of our method, we use the double-phase [23] method to encode the holograms. The radius of outer cylinder is set to 200 mm, the wavelength is 200 µm, and resolution of the test images is 1024×1024. Similarly, the PSNR and SSIM of the reconstructed images with and without occlusion culling are measured. The reconstructed results with and without occlusion culling using OIP and IOP models are shown in Fig. 9.

Obviously, overlapping exists in holograms and reconstructed images without occlusion culling. It can be understood that there are two images with opposite central optical axes, which blurs the details and distorts the structure. Therefore, culling occlusion to avoid overlapping is vital in the reconstruction of holograms.

Next, we add random phase to the original image to simulate the diffuse reflection of the object. Similarly, in order to meet the actual application, the phase-only holograms are used to reconstruct. The reconstructed results are shown in Figs. 10(a)-(h). There is obvious speckle noise in the reconstructed images, which will degrade the quality. Of course, this problem is common to plane and cylindrical diffraction models. Therefore, time multiplexing method is used to reduce speckle noise in reconstructed images [24]. The results are shown in Figs. 10(i)–10(p).

#### 3.2 Calculation time of the occlusion culling method

Owing to the HOLF is directly multiplied by the PSF. Since the FFT and convolution can still be used when calculating the diffraction field, there is little additional calculation time. To compare the calculation time of generating holograms with and without occlusion culling method, simulation is carried out with different resolutions, the average time of 5 experiments using each resolution are shown in Fig. 11. It shows that the calculation time with and without occlusion culling is almost the same, which confirms our analysis.

#### 3.3 Hologram generation and reconstruction of 3D objects

The above method can record singer-layer cylindrical information, and it is only suitable for simple single-layer cylindrical object. To show the universality of the proposed method, it is applied to the hologram generation and reconstruction of 3D objects by using 3D layer-oriented method. The schematic diagram of the 3D layer-oriented method is shown in Fig. 12. To simplify the simulation, three images are placed at different depths to simulate object information obtained from different layers. The final hologram is the sum of the holograms of all layers, as:

In the simulation, the radii of the object planes are set to be 100 mm, 150 mm, 200 mm, respectively and radius of the hologram surface is 10 mm. The images with resolution of 2048×2048 are used to generate and reconstruct holograms with the wavelength of 75 µm, which are shown in Fig. 13. When the holograms are reconstructed, the reconstructed images of the cylindrical observation surfaces with radius 100 mm, 150 mm, 200 mm are observed, respectively. Similarly, in order to test the effect of occlusion culling, two PSFs *h*(*θ*, *z*) and *h'*(*θ*, *z*) are applied when generating and reconstructing holograms. The reconstructed images with and without occlusion culling are shown in Fig. 14. It turns out that only the cylindrical layer-oriented method without occlusion culling cannot successfully reconstruct images and focus on corresponding objects. While the results with occlusion culling indicate that the layer-oriented method is reproduced well when the PSF is *h'*(*θ*, *z*).

## 4. Conclusion

In this paper, a calculation method for occlusion culling based on HOLF is proposed. In the proposed method, the cylindrical RS diffraction formula is deduced by analyzing the geometric relationship of wave propagation and considering the obliquity factor based on the source point, firstly. Then, the HOLF embedded in the PSF is proposed for occlusion culling by limiting the diffraction area of the source point. The effectiveness of the proposed occlusion culling method is verified by the simulation results of Young’s interference. Besides, simulations of different grayscale images are carried out to further verify the significant effect of eliminating overlapping. Furthermore, our method can accomplish the recording and display of 3D objects by combining with 3D layer-oriented method. Meaningfully, our method breaks through the angle limitation of the concentric cylinder model, which can achieve cylindrical holography to solve the issue of the narrow FOV in CGH. It will be a promising technology with the development of the curved display screen [25] and flexible display materials [26] in the future.

## Funding

National Natural Science Foundation of China (U1933132); Chengdu Science and Technology Program (2019-GH02-00070-HZ).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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