Spherical crown diffraction model by occlusion utilizing for a curved holographic display

Chengmin Liu; Jun Wang; Yang Wu; Xiangli Lei; Peiding Wang; Han Han; Chun Chen

doi:10.1364/OE.465321

1. Introduction

Holographic display is a promising technology for the reconstruction of three-dimensional (3D) scenes because it can provide all the depth cues that the human eyes can perceive [1,2]. In recent years, there are many researches to improve the quality of reconstructed images [3,4]. However, the conventional planar holograms have some limitations in information capacity and viewing angle. Fortunately, the wide field of view holographic display using curve hologram was reported by O. Soares and J. Fernandes [5], which makes it possible to reconstruct and observe the image from a 360° horizontal direction and opens up a new research direction for holographic 3D display. Since actual optical holography requires strict optical recording conditions, computer-generated holograms can be generated more easily and conveniently using computer technology [6], which offers a new possibility in the field of optical holography.

In recent years, the calculation of light field distribution on the curved surface has made great development. However, the difficulty of calculating and generating curve holograms is much greater than that of planar holograms. Therefore, many fast computation methods for curve holograms are proposed. Some of the methods for obtaining computer-generated cylindrical holograms are proposed [7–14]. A new method of conical computer-generated hologram with an expanded viewing angle is also proposed [15]. On the other hand, spherical holography is also one of the research directions of curve holography. Compared to other curved surfaces, spheres have a symmetrical structure and a full field of view range. Therefore, many studies on spherical holography were carried out. In 2006, a method of fast convolution algorithm based on Fast Fourier Transform (FFT) was proposed by M. Tachiki et al. [16]. In 2013, a method of spherical wave spectrum based on wave propagation defined in spectral-domain was proposed by B. Jackin and T. Yatagai [17]. An acceleration method by using a graphics processing unit was proposed by G. Li et al. [18]. Besides, a fast diffraction calculation by using the phase compensation method is proposed [19]. Recently, some applications based on spherical holography have also been proposed. H. Cao and E. Kim introduced a faster generation method for holographic videos of moving objects in space using a spherical hologram based on a 3D rotational motion compensation scheme [20]. For the quality improvement of spherical holograms, a method to suppress speckle noise by using spherical self-diffraction iteration was proposed [21]. Moreover, the issue of occlusion culling has been discussed also. An occlusion culling method based on spherical harmonic transform was proposed by Y. Sando et al. to generate spherical computer-generated hologram [22]. In planar diffraction, occlusion in plane-to-plane diffractions using a double-phase approach has been studied [23]. However, to our knowledge, the information on the occluded surface has never been focused on, which is also available to be used.

In this paper, a spherical crown diffraction model by occlusion utilizing for curve holographic display is proposed. Compared to the conventional spherical model, the information on the surface that is culled in model of occlusion culling is re-utilized in the proposed model. Firstly, spherical outside-in propagation (OIP) and inside-out propagation (IOP) models are presented by analyzing the meaning of the point spread function (PSF) and obliquity factor in the concentric sphere model. Secondly, a method of occlusion utilizing is proposed by using PSF with an optical-path-select function (OPSF), which is obtained by analyzing the diffraction area of source points and can remain the information of the utilized surface. Thirdly, based on the method of occlusion utilizing, the spherical crown diffraction model is proposed, and an accurate calculation formula is given. In the proposed model, OPSF is applied by further analyzing the diffraction area of the source point in the spherical crown model. In addition, the proposed spherical crown diffraction model by occlusion utilizing has higher optical utilization than that model by occlusion culling. The correctness and effectiveness of the proposed spherical crown diffraction model are demonstrated by numerical reconstruction of the point source method and from different perspectives. In discussion section, an approximation of the spherical crown diffraction model is proposed to reduce the time of hologram generation. In addition, the error and feasibility of this approximation method are analyzed. Besides, the sampling conditions in detail are described in Appendix A. The advantages of the proposed model for view angle are briefly described in Appendix B Moreover, the feasibility is verified by reconstructing images with multiple methods in Appendix C. And the practicability of display of 3D objects is verified by using the layer-oriented method in Appendix D [24,25].

2. Method

2.1 Spherical OIP and IOP models

According to Rayleigh-Sommerfeld (RS) diffraction formula, the complex amplitude of the diffraction field in a spherical surface can be expressed as:

(1)$${U_d} = \frac{1}{{j\lambda }}\int\!\!\!\int\limits_S {{U_o}} \frac{{\exp ({jkd} )}}{d}\cos \alpha ds,$$

where U_o and U_d represent the object field and complex amplitudes of the diffraction field, respectively. The integral field S is the entire spherical surface of the object. k represents the wavenumber. λ is the wavelength of the diffraction wave. α is the angle between the propagation vector from the source point (ϕ_o, θ_o, R_o) to the destination point (ϕ_d, θ_d, R_d) and the normal vector n of the source point (ϕ_o, θ_o, R_o). Therefore, cosα represents the obliquity factor. d represents the distance between the source point (ϕ_o, θ_o, R_o) and the destination point (ϕ_d, θ_d, R_d). The geometric relationship on the sphere is shown in Fig. 1(a). Let P (ϕ_R, θ_R, R) and Q (ϕ_o, θ_o, r) be any point on the outer sphere and the inner sphere, respectively, then d can be approximated as a convolutional form:

(2)$$d = \sqrt {{r^2} + {R^2} - 2rR\cos ({{\phi_R} - {\phi_o}} )\cos ({{\theta_R} - {\theta_o}} )} .$$

In addition, due to the different propagation directions of light, the spherical propagation model can be divided into two models: IOP and OIP models. The geometric relationship between the two models is shown in Figs. 1(b) and 1(c). Different models have different obliquity factors. We can get the expression of the obliquity factor from the geometric relationship in Fig. 1 and the cosine theorem, as follows:

(3)$$\cos \alpha = \left\{ \begin{array}{l} - \frac{{{r^2} + {d^2} - {R^2}}}{{2rd}} \ldots \ldots \ldots IOP\\ \frac{{{R^2} + {d^2} - {r^2}}}{{2rd}} \ldots \ldots {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} OIP \end{array} \right..$$

Substitute Eq. (2) into the above equation, we have

(4)$$\cos a = \left\{ \begin{array}{l} - \frac{{r - R\cos ({\varDelta \phi } )\cos ({\varDelta \theta } )}}{d} \ldots \ldots \ldots IOP\\ \frac{{R - r\cos ({\varDelta \phi } )\cos ({\varDelta \theta } )}}{d} \ldots \ldots \ldots OIP \end{array} \right..$$

Therefore, we can get the PSF of the diffraction formula from Eq. (2) and Eq. (4) as:

(5)$$h({\phi ,\theta } )= \frac{{\exp ({jkd} )}}{{jkd}}\cos \alpha .$$

Then, the RS formula of the spherical diffraction can be expressed as:

(6)$${U_d}({{\phi_d},{\theta_d}} )= \int\!\!\!\int {{U_o}} ({{\phi_o},{\theta_o}} )h({{\phi_d} - {\phi_o},{\theta_d} - {\theta_o}} )d{\phi _o}d{\theta _o}.$$

It is a form of convolution. Using the Fourier transform property of convolution, it can be written as:

(7)$$U{}_d({{\phi_d},{\theta_d}} )= {U_o}\ast h = IFFT[{FFT({{U_o}} )\bullet FFT(h )} ].$$

Finally, we get the complete spherical convolution algorithm formula. Then we will use this formula to establish our proposed model.

Fig. 1. Spherical diffraction model. (a) Geometric relationship of concentric spheres. (b) IOP model; (c) OIP model.

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2.2 Principle of occlusion utilizing

In Eq. (7), the integral is calculated over the entire surface of the spherical object, and the PSF describes the propagation characteristics of the light wave. As shown in Figs. 2(a) and 2(b), the holographic spherical surface is divided into S1 and S2 according to the diffraction relationship from the object surface to the holographic surface. In previous research, S2 is culled while S1 is retained to avoid the overlapping in reconstructed image, and such a model is called the model of occlusion culling [13]. On the contrary, a method of occlusion utilizing is proposed in this paper to retain S1 and remove S2, and such a model is called the model of occlusion utilizing. In our proposed method, the propagation range of the light and the boundary conditions of RS diffraction are considered to utilize this information, and the practical applications are satisfied by limiting the diffraction area. To realize the occlusion utilizing, we need to consider it based on geometric optics, because the process of occlusion utilizing is the process that remains the light ray selectively.

Fig. 2. Illumination of occlusion utilizing. Diffraction area to be utilized on hologram surface in (a) OIP model and (b) IOP model. Realizing occlusion utilizing by the optical path selection in (c) OIP model and (d) IOP model of 2D view.

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To simplify analysis, a 2D view is used to discuss the relationship between the angular coordinates between the source and destination points and the optical path for more intuitive, as shown in Figs. 2(c) and 2(d).

From Fig. 2, we can see that the optical path d is increasing monotonously from (R-r) to (R + r) when ψ is increasing from 0 to π. When |ψ|=ψ_c, the light ray is in the tangent direction. Since the radius and the tangents are perpendicular, the optical path d = (R²-r²)^1/2. Therefore, for |ψ|<ψ_c, the diffraction area is S1 where the diffraction waves can reach directly. Similarly, for |ψ|≥ψ_c, the diffraction area is S2 where the information is utilized in our model. Our method is to utilize the information of utilized surfaces S2, which has never been considered before. The utilized surface is the diffraction area when |ψ|≥ψ_c and d≥(R²-r²)^1/2. we can set an OPSF to limit the diffraction area of the source point. The light ray that reaches the utilized surface is retained, while another light ray is removed, realizing the use of information on the utilized surface.

When a light ray is not selected, set OPSF to 0 so that its corresponding PSF becomes 0; when a ray is selected, set OPSF to 1 so that its corresponding PSF can be maintained. Therefore, the OPSF can be established as follow:

(8)$$OPSF = \left\{ \begin{array}{l} 1\quad \quad d \ge \sqrt {{R^2} - {r^2}} \\ 0\quad \quad d < \sqrt {{R^2} - {r^2}} \end{array} \right..$$

Because the OPSF is used to determine whether the light is selected, each OPSF has its corresponding PSF. The new PSF for occlusion utilizing is $h^{\prime}({\phi ,\theta } )$ as:

(9)$$h^{\prime}({\phi ,\theta } )= h({\phi ,\theta } )\times OPSF = \left\{ \begin{array}{l} h({\phi ,\theta } )\quad \quad d \ge \sqrt {{R^2} - {r^2}} \\ 0\quad \quad \quad \quad d < \sqrt {{R^2} - {r^2}} \end{array} \right..$$

2.3 Spherical crown diffraction model by occlusion utilizing

Based on the principle of occlusion utilizing, we propose a spherical crown diffraction model by occlusion utilizing for curve holographic display. This proposed diffraction model is based on the original spherical holography. Only a part of the spherical crown is taken as the holographic surface, and a part of the corresponding object surface is also taken. In order to show the optical path limitation more vividly, the diffraction areas of the IOP and OIP models of the proposed spherical crown diffraction model are shown in Fig. 3. ϕ_m is the maximum value of ϕ in the latitude direction. P is a point on the object surface. Q₁ and Q₂ are the boundary points on the holographic surface. According to the RS diffraction formula, the distance d between the object point and the point on the corresponding holographic plane needs to be obtained. According to the geometric relationship in the figure, the maximum value of d is easily obtained as R + r, while the minimum value is the distance between P and Q₁. Then in triangle POQ₁, γ is the included angle of POQ₁, and the following formula can be obtained from the cosine theorem:

(10)$${d_m} = {[{{R^2}\textrm{ + }{r^2}\textrm{ - }2Rr\cos (\gamma )} ]^{\frac{1}{2}}}.$$

Since the geometric relationship γ+ϕ+ϕ_m=π, Eq. (10) is simplified as follows:

(11)$${d_m}^{}(\phi )= {[{{R^2} + {r^2} + 2Rr\cos ({\phi + {\phi_m}} )} ]^{\frac{1}{2}}}.$$

In the spherical crown diffraction model, unnecessary diffraction rays should be removed during numerical simulation. According to the relationship obtained above, we can utilize the part of d ≥ d_m(ϕ), light ray needs to be retained. In the same way, the light ray needs to be removed when d < d_m(ϕ). According to the previous section, OPSF can be rewritten as:

(12)$$OPSF^{\prime}(\phi )= \left\{ \begin{array}{l} 1\quad \quad d \ge {d_m }(\phi )\\ 0\quad \quad d < {d_m }(\phi )\end{array} \right..$$

Then, we will get a new PSF $h^{\prime\prime}({\phi ,\theta } )$:

(13)$$h^{\prime\prime}({\phi ,\theta } )= h({\phi ,\theta } )\times OPSF^{\prime}(\phi )= \left\{ \begin{array}{l} h({\phi ,\theta } )\quad \quad d \ge {d_m }(\phi )\\ 0\quad \quad \quad \quad d < {d_m }(\phi )\end{array} \right..$$

Since ϕ corresponding to each d_m(ϕ) is a constant, we can get the following formula:

(14)$$U{}_d({{\phi_d},{\theta_d}} )= \int\!\!\!\int {U{}_o({{\phi_o},{\theta_o}} )h^{\prime\prime}({{\phi_d},{\phi_o},{\theta_d} - {\theta_o}} )} d{\phi _o}d{\theta _o}.$$

We can rewrite Eq. (14) as convolution calculation:

(15)$$U{}_d({{\phi_d},{\theta_d}} )= \int {U{}_o({{\phi_o},{\theta_o}} ){\ast _\theta }h^{\prime\prime}({{\phi_d},{\phi_o},\theta } )} d{\phi _o},$$

where *_θ means the one-dimensional convolution about the azimuth direction. It is calculated by FFT to obtain the following equation:

(16)$$U{}_d({{\phi_\textrm{d}}, {\theta_d}} )= \int {IFFT[{FFT({{U_o}({{\phi_o},{\theta_o}} )} )\bullet FFT({h^{\prime\prime}({{\varphi_d},{\varphi_o},\theta } )} )} ]} d{\phi _o},$$

where FFT_θ is used for the azimuth direction, after calculating every d_m(ϕ) in this way, the diffraction field at different ϕ can be obtained.

Fig. 3. Spherical crown diffraction model. (a) 3D geometric relationship; Diffraction area by optical path limitation in top-view of (b) OIP and (c) IOP model.

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From the previous analysis, Eq. (16) is obtained by one-dimensional FFT in azimuth direction and integral operation in latitude direction. The diffraction formula of spherical crown diffraction model in the latitude direction cannot be transformed into convolutional form because d in OSPF’ is a variable value related to ϕ.

Next, we make an approximation in spherical crown diffraction model. We replace the changed d_m(ϕ) with the maximum d_m-max of d_m, and the d_m-max is a constant. $h^{\prime\prime}({\phi ,\theta } )$ can be approximated as:

(17)$$h^{\prime\prime}{({\phi ,\theta } )_{appr}} = \left\{ \begin{array}{l} h({\phi ,\theta } )\begin{array}{c} \quad \end{array}\begin{array}{c} \quad \end{array}d \ge {d_{m - \max }}\\ 0\begin{array}{c} \quad \end{array}\begin{array}{c} \quad \end{array}\begin{array}{c} \quad \end{array}\begin{array}{c} \quad \end{array}d < {d_{m - \max }} \end{array} \right..$$

We can get the following formula:

(18)$$U{}_d({{\phi_d},{\theta_d}} )= \int\!\!\!\int {U{}_o({{\phi_o},{\theta_o}} )h^{\prime\prime}{{({{\phi_d} - {\phi_o},{\theta_d} - {\theta_o}} )}_{appr}}} d{\phi _o}d{\theta _o}.$$

Using the convolution property of Fourier transform, we can finally get such a formula:

(19)$$U{}_d({{\phi_d},{\theta_d}} )= IFFT[{FFT({{U_o}({{\phi_o},{\theta_o}} )} )\bullet FFT({h^{\prime\prime}{{({\phi ,\theta } )}_{appr}}} )} ].$$

This formula uses two-dimensional FFT, which can save a lot of time compared with Eq. (16). The relevant analysis of approximation error will be discussed in section 4.

2.4 Comparison of occlusion culling and utilizing

Considering the practical application scenarios, the optical diagram of the spherical crown model is constructed. Figure 4(a) shows the possible optical path of a spherical crown diffraction model by occlusion culling. The light source uses a parallel incident planar light wave, which is adjusted by an optical component to change its previous planar light wave to the spherical light wave. Then the spherical light wave is irradiated to the hologram which is loaded on the transmissive spatial light modulator (SLM). Finally, the reconstructed image is displayed on the outside large spherical crown surface. Figure 4(b) shows the possible optical path of a spherical crown diffraction model by occlusion utilizing. Similarly, the light source uses a plane light wave, which is adjusted by an optical component to change its previous plane light wave to the spherical light wave. Then the spherical light wave is irradiated to the hologram which is loaded on the reflective SLM. Finally, the reconstructed image is displayed on the outside large spherical crown surface.

Fig. 4. Optical path diagram of the spherical crown model. Spherical crown diffraction model by (a) occlusion culling and (b) occlusion utilizing.

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Having examined the parametric data of SLM, the optical utilization of transmissive SLM can only reach a maximum of about 30%∼40%, while the optical utilization of reflective SLM is up to 80%. Analyzing multiple sets of parameters, we can know that the optical utilization of reflective SLM is 2∼4 times higher than that of transmissive SLM and the use of reflective SLM can greatly improve the optical utilization. Therefore, the proposed spherical crown diffraction model by occlusion utilizing also has higher optical utilization compared with spherical crown diffraction model by occlusion culling.

3. Numerical simulation

In this section, experiments of numerical simulations are carried out to show the correctness, effectiveness, feasibility, and practicability of the proposed spherical crown diffraction model. In the proposed model, ϕ_m is set to π/3, radii of inner and outer spherical crowns are set to 5 cm, and 50 cm, respectively. According to the sampling conditions:

(20)$${N_\theta } > \frac{{4\pi \times \min ({R,r} )}}{\lambda },{N_\phi } > \frac{{2{\phi _m} \times \min ({R,r} )}}{\lambda }.$$

The detailed derivation process is described in Appendix A, subsection. The wavelength is set to terahertz range in order to obtain a considerable number of sampling points. In the numerical simulation, the computing platform and resource include Python 3.8, Windows 10 operation system, Intel Core i7-8565U CPU @ 1.80 GHz, and 8.00 GB RAM. In addition, the peak signal-to-noise ratio (PSNR) is employed to evaluate the quality of the reconstructed image, which compares the reconstructed results with the original image [26]. The PSNR is defined as:

(21)$$PSNP({f,g} )= 101g\left( {{{255}^2}/\left( {\frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{{({{f_{ij}} - {g_{ij}}} )}^2}} } } \right)} \right),$$

where M and N are the numbers of pixels of the original and reconstructed images in azimuth and latitude directions, respectively. f_ij and g_ij are the amplitude of the original image and the reconstructed image, respectively. Higher PSNR means better reconstruction quality.

3.1 Correctness — reconstruction with point source method

In order to verify the correctness of the proposed spherical crown diffraction model by occlusion utilizing, the complex amplitude of the diffraction field on the hologram is generated by using the diffraction formula of spherical crown diffraction model with one-dimensional FFT, and the object image is reconstructed by using point source method. To have the same sampling interval in the latitude and azimuth directions, the sampling number in the azimuth direction is set to be the same as in the latitude direction. Therefore, the range of θ in the azimuth direction is also taken from -π/6 to π/6. The wavelength is set to 450 µm, and the image resolution of 256×256 is chosen according to Eq. (20). Simulation results are shown in Fig. 5.

Fig. 5. Results of correctness verification of proposed diffraction model of spherical crown diffraction model (a1)(a2) Original images. (b1)(b2) Amplitude of diffraction field generated by the proposed model. (c1)(c2) Phase of diffraction field generated by the proposed model. (d1)(d2) Reconstructed images by using point source method. (e1)(e2) 3D diffraction model of spherical crown.

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As shown in Fig. 5, Figs. 5(a1) and 5(a2) are the original images. Figures 5(b1) and 5(b2) are corresponding the amplitude of diffraction field. Figures 5(c1) and 5(c2) are corresponding the phase of the diffraction field. Then reconstructed images are obtained by using the point source method and shown in Figs. 5(d1) and 5(d2). Figures 5(e1) and 5(e2) show the relationship between the diffraction field and reconstructed images in the 3D diffraction model of spherical crow. The results show that clear reconstructed images can be obtained, and the reconstructed images retain almost all the details of the original images. The correctness of our proposed spherical crown diffraction model is verified by these numerical simulation results.

3.2 Effectiveness — reconstruction with 360°

In order to verify the effectiveness of our proposed spherical crown diffraction model by occlusion utilizing with a 360° visual range, this experiment chooses to reconstruct the image from different angles. Therefore, the wavelength was set to 250 µm and the resolution of the image was chosen to be 3072 × 512 in order to be close to the Nyquist sampling limit. Moreover, to better observe the experimental results, the method of complex amplitude reconstruction is employed. In order to make the results more practical and intuitive, unlike the above flat graphical presentation, we show the results from different angles in the proposed model. The specific verification method is as follows: Firstly, the original image is placed in the proposed model according to the corresponding coordinates, as shown in Fig. 6. After that, different views of 0, -π, -π/2, and π/2, are selected to generate and reconstruct the images. Then, observe whether the results of reconstructed images from different angles satisfy the corresponding position relationship with the original images in the proposed model. The simulation results are shown in Fig. 7.

As shown in Fig. 7, the reconstructed images are shown at angles of 0, -π, -π/2, π/2. We compare the graphical features of the original and reconstructed images from different angles to know that the reconstructed image corresponds to its reconstructed perspective, which proves that the proposed model can be displayed from all angles of 360°. Therefore, the effectiveness of our proposed spherical crown diffraction model is verified.

Fig. 6. Original image.

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Fig. 7. Results of correctness verification by reconstructing images from different angles.

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4. Discussion

In Subsection 2.3, it is mentioned that the diffraction formula of the proposed spherical crown diffraction model using one-dimensional FFT can be approximately transformed into a diffraction formula using two-dimensional FFT. Therefore, the time to generate hologram is effectively reduced after performing an approximation using two-dimensional FFT. In this section, the time for generating hologram is compared with and without approximation, and the quality of reconstructed image is also compared. Furthermore, in order to verify the feasibility of spherical crown diffraction model with approximation, the error of spherical crown diffraction model with approximation is analyzed. We choose images with the resolution of 3072×512 for simulation experiments. We compare the time of hologram generation and quality of reconstructed images with and without the approximation. Simulation results are shown in Fig. 8.

Fig. 8. Comparison of reconstructed image quality and time of hologram generation without and with approximation. (a1)(a2) Original images. (b1)(b2) Reconstructed images without approximation. (c1)(c2) Reconstructed images with approximation.

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Figures 8(a1), 8(b1), and 8(c1) are the original image “Fig-SCU”, reconstructed images without and with approximation, respectively. Figures 8(a2), 8(b2), and 8(c2) show the original image “Fig-SCU+”, reconstructed images without and with approximation, respectively. As can be seen from the pictures, the proposed model with approximation takes less than 0.4 seconds. In contrast, the proposed model without approximation takes about 120 seconds. The differences between the two reconstructed images with and without approximation are only about 2 dB in PSNR, which shows that the effect on image quality is very slight, and the quality could be improved by other optimization methods. Therefore, it gains a speedup of 300X with the cost of only 2 dB in PSNR, and the results verify that the proposed spherical crown diffraction model with approximation that can effectively reduce the time of hologram generation.

In addition, the error of the proposed model with approximation is analyzed. Since d_m(ϕ) is replaced by a constant d_m-max which is the maximum value of d_m(ϕ). There is a distance difference between them, which is the error:

(22)$$error[d ]= {d_{m - \max }} - {d_m}(\phi ).$$

Combining with the formula for calculating d_m(ϕ) in Eq. (11), it can be seen that the d_m(ϕ) can reach the maximum d_m-max when ϕ= 0:

(23)$${d_{m - \max }} = {[{{R^2} + {r^2} + 2Rr\cos ({{\phi_m}} )} ]^{\frac{1}{2}}}.$$

From above, it shows that error[d] varies at different ϕ. And as ϕ varies from 0 to ϕ_m, error[d] varies from 0 to error[d]_max, correspondingly. Here, error[d]_max can be expressed as:

(24)$$error{[d ]_{\max }} = {[{{R^2} + {r^2} + 2Rr\cos ({{\phi_m}} )} ]^{\frac{1}{2}}} - {[{{R^2} + {r^2} + 2Rr\cos ({2{\phi_m}} )} ]^{\frac{1}{2}}}.$$

Moreover, it shows that error is also related to the value of r and R. We set the inner spherical crowns radius r of the model to 3 cm, 4 cm, and 5 cm, respectively. And the outer spherical crown radius varies from 30 cm to70 cm. The results of the error[d]_max variation are shown in Fig. 9(a). The value of error[d]_max is in range of 3∼5 cm with different radii of R, which is acceptable. And the variation of error[d]_max is negligible with different radii of r.

Fig. 9. Results of error analysis. (a) error[d]_max with different radii. (b) PSNR between reconstructed images without and with the approximation at different radii.

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Furthermore, we have compared the quality of the reconstructed images with different radius values. The PSNR is used to discriminate the quality change of the reconstructed images without and with approximation. In order to compare the results more intuitively, the results show the PSNR between the reconstructed images without and with the approximation. The results of PSNR are displayed in Fig. 9(b). With the change of radius R, the PSNR varies very slightly. Therefore, the results further validate the effectiveness and feasibility of the proposed spherical crown diffraction model with approximation.

5. Conclusion

In this paper, a spherical crown diffraction model by occlusion utilizing for curve holographic display is proposed. This is the first time that the occlusion surfaces of spherical holography have been utilized. In addition, the proposed model by occlusion utilizing has higher optical utilization than that model by occlusion culling. In the proposed model, a method of occlusion utilizing is proposed firstly by an OPSF to remain the desired light information. Based on the proposed method of occlusion utilizing, a new OPSF is obtained by further analyzing the diffraction range of source points in the spherical crown model, and then a spherical crown diffraction model and formula for occlusion utilizing are realized. The correctness of the proposed model is verified by reconstructing images with the point source method firstly. Then, reconstruction of the spherical crown diffraction model from different angles is simulated to verify the advantage of the proposed model with a large view angle of 360° in the azimuth direction and a larger view angle in the latitude direction. Moreover, the practicality of the proposed diffraction model with conventional encoding methods and the feasibility of 3D reconstruction are verified. In discussion, an approximation of spherical crown diffraction model is proposed to reduce the time of hologram generation. In addition, the error and feasibility of this approximation method are analyzed. With the development of curved display screens [27] and flexible display materials [28], our proposed model might have a prospective future.

6. Appendix

6.1 Sampling conditions

In order to carry out numerical simulation experiments, we must satisfy the Nyquist sampling theorem in both latitude and azimuth directions. Since the spatial frequency of the objective function U_o(ϕ_o, θ_o) is smaller than h$^{\prime\prime}$ (ϕ, θ), we only consider the maximum spatial frequency of h$^{\prime\prime}$ (ϕ, θ). What’s more, since the change speed of d is much smaller than the change speed of the phase, the Nyquist theorem just needs to be considered in the phase term simply.

(25)$${v_\theta }({\phi ,\theta } )= \frac{1}{{2\pi }}\frac{{\partial h^{\prime\prime}({\phi ,\theta } )}}{{\partial \theta }} \approx \frac{1}{{2\pi }}\frac{{\partial ({kd} )}}{{\partial \theta }},$$

(26)$${v_\phi }({\phi ,\theta } )= \frac{1}{{2\pi }}\frac{{\partial h^{\prime\prime}({\phi ,\theta } )}}{{\partial \phi }} \approx \frac{1}{{2\pi }}\frac{{\partial ({kd} )}}{{\partial \phi }}.$$

Substituting the value of d into the above formula, the maximum values of |v_θ|_max and |v_ϕ|_max can be calculated as follows:

(27)$$|{v_\theta }{|_{\max }} = |{v_\phi }{|_{\max }} = \frac{{k \times \min ({R,r} )}}{{2\pi }}.$$

Here, min() means the operation of selecting a smaller input. Therefore, the spatial-spectral bandwidth BW_θ and BW_ϕ are as follows:

(28)$$B{W_\theta } = B{W_\phi } = 2|{v_\theta }{|_{\max }} = 2|{v_\phi }{|_{\max }} = \frac{{k \times \min ({R,r} )}}{\pi }.$$

In the spherical crown diffraction model, the value of θ is taken from -π to π in azimuth direction. Due to the spatial symmetry of the spherical crown model, the value of ϕ only needs to be taken from -ϕ_m/2 to ϕ_m/2 in latitude direction. Therefore, sampling points N_θ and N _ϕ are as:

(29)$${N_\theta } > \frac{{4\pi \times \min ({R,r} )}}{\lambda },{N_\phi } > \frac{{2{\phi _m} \times \min ({R,r} )}}{\lambda }.$$

6.2 View angle of proposed method

In this subsection, the features of view angle are analyzed on the proposed spherical crown diffraction model. As shown in Fig. 10, R and r are the radii of the outer and inner spherical crowns, respectively. ϕ_m is the maximum value of ϕ in the latitude direction. We divide the spherical crown diffraction model into two directions to observe. Figure 10(a) shows the features of view angle in azimuth direction. It is obvious that the proposed diffraction model has a 360° omni-directional panoramic field of view in the azimuth direction. Figure 10(b) shows the features of view angle of latitude direction. We can get that central angle of the entire spherical crown is 2ϕ_m. We have a fully visible observation range in such a range of 2ϕ_m. For example, the view angle of latitude direction is 120° if ϕ_m is set to 60°_.

Fig. 10. View angle of spherical crown diffraction model in (a) azimuth direction and (b) latitude direction.

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In practical application scenarios, due to the space constraints of actual objects, such as desktops or wall corners, an entirely spherical display cannot be effectively achieved. However, the flexibility of the proposed spherical crown diffraction model can solve this problem. In a word, this proposed model not only retains the conventional spherical holographic 360° view in the azimuth direction, but also has more flexibility. And it also has the advantage of larger viewing angle in latitude direction compared to cylindrical holography.

6.3 Feasibility —for display

In order to verify the feasibility of the proposed spherical crown diffraction model for display, random phase is necessary to be added to the object to simulate the diffuse reflection and achieve the merit of spherical crown diffraction model in field of view. Considering the modulation of SLM for amplitude-only or phase-only holograms, the amplitude-truncation (AT), and double-phase (DPH) methods [29] are applied to generate phase-only holograms. Then time multiplexing (TM) method [30] can be used to reduce speckle noise and improve the quality of reconstructed images. Meanwhile, we choose the image resolution of 3072×512 near the Nyquist sampling limitation and set the wavelength to 250 µm. The simulation results of reconstructed images are shown in Fig. 11.

Fig. 11. Results of feasibility verification of proposed spherical crown diffraction model for display. (a) Original image. Reconstructed image by AT (b) without and (c) with a random phase. (d) Reconstructed image by AT with random phase and TM with 15 times multiplexing. Reconstructed images (e) with and (f) without random phase by DPH.

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The original image is shown in Fig. 11(a). The reconstructed image of the phase-only hologram using AT is shown in Fig. 11(b). It only contains the outline of the object and is not of high quality. The reconstructed image of phase-only hologram using AT with adding random phase to the object is shown in Fig. 11(c). The reconstructed image is low quality and presences more scattering noise. Therefore, Fig. 11(d) shows the reconstructed image using TM with 15 times multiplexing to enhance its quality. We can see that there is 2 dB improvement in the PSNR of the reconstructed image after using TM, which indicates that using this method can enhance the image quality. Figures 11(e) and 11(f) show the reconstructed images of encoded holograms by DPH with and without the addition of random phase. The PSNRs of Figs. 11(b)–11(f) are 15.37 dB, 15.56 dB, 17.25 dB, 18.32 dB, and 19.46 dB, respectively. The results show that the quality of reconstructed images in the proposed model can be improved by the TM method and DPH method, and quality of the reconstructed images with random phase is acceptable, which verifies the feasibility of the proposed spherical crown diffraction model for display.

6.4 Practicability — display of 3D objects

In order to verify the practicability of the proposed spherical crown diffraction model for display of 3D objects, holograms and reconstructed images of 3D objects are conducted by using the 3D layer-oriented method as shown in Fig. 12. In order to simplify the simulation experiment and make the results clearer, three different letters “K”, “M”, and “Q” are objects on the spherical crown layer at the depths of 15 cm, 30 cm, and 45 cm, respectively. The complex amplitude of the diffraction field on the hologram is generated on the spherical crown layer at the depth of 5 cm. And the amplitude of diffraction fields is shown in Figs. 13(a)–13(c), respectively. The final hologram is obtained by summing up the holograms of each layer, and the formula is as:

(30)$$\textrm{hologra}{\textrm{m}_{\textrm{sum}}}\textrm{ = }\sum\limits_\textrm{i} {\textrm{hologram}} \textrm{(i)}\textrm{.}$$

Figures 13(d)–13(f) are reconstructed images of the hologram_sum by occlusion utilizing at depths of 15 cm, 30 cm, and 45 cm with the spherical crown diffraction model, respectively. According to diffraction theory, as the diffraction distance increases, the less energy is recorded at the diffraction surface. We can see that the intensity of reconstructed images decreases with the increase of distance from the object to the holographic surface, which is expressed as the decrease of brightness. This result is in line with expectations. From Figs. 13(d)–13(f), we can see that only the pattern at a certain depth can be clearly reconstructed when focusing at that depth. Figure 14 shows the amplitude of diffraction fields and reconstructed images of the hologram_sum by occlusion culling at depths of 15 cm, 30 cm, and 45 cm with the spherical crown diffraction model, respectively. Comparing the results of the occlusion utilizing and the occlusion culling, there is almost no difference in their reconstruction results. Therefore, the practicability of the proposed spherical crown diffraction model for display of 3D objects is further verified.

Fig. 12. 3D hierarchical graph of spherical crown diffraction model.

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Fig. 13. Hologram generation and reconstruction by using 3D layer-oriented method in model of occlusion utilizing. Amplitude of diffraction fields generated with depths of (a) 15 cm, (b) 30 cm, and (c) 45 cm. Reconstructed images of hologram_sum at depths of (d) 15 cm, (e) 30 cm, and (f) 45 cm.

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Fig. 14. Hologram generation and reconstruction by using 3D layer-oriented method in model of occlusion culling. Amplitude of diffraction fields generated with depths of (a) 15 cm, (b) 30 cm, and (c) 45 cm. Reconstructed images of hologram_sum at depths of (d) 15 cm, (e) 30 cm, and (f) 45 cm.

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Funding

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

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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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Spherical crown diffraction model by occlusion utilizing for a curved holographic display

Abstract

1. Introduction

2. Method

2.1 Spherical OIP and IOP models

2.2 Principle of occlusion utilizing

2.3 Spherical crown diffraction model by occlusion utilizing

2.4 Comparison of occlusion culling and utilizing

3. Numerical simulation

3.1 Correctness — reconstruction with point source method

3.2 Effectiveness — reconstruction with 360°

4. Discussion

5. Conclusion

6. Appendix

6.1 Sampling conditions

6.2 View angle of proposed method

6.3 Feasibility —for display

6.4 Practicability — display of 3D objects

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (14)

Equations (30)

Optics Express