Abstract

Heavy computational complexity and imprecise reconstruction of objects are crucial problems in computer-generated holograms. In this paper, we propose a non-uniform sampling based on novel compressed look up table method to generate holograms. The method consists of two steps: in the first step, the non-uniform basic modulation factors are precalculated and stored in look-up-table. Secondly, fringe patterns for other points are obtained by simply shifting and multiplying the pre-calculated non-uniform basic modulation factors, and the final computer-generated hologram is obtained by adding them all together. The proposed method eliminates the redundant information properly and modulates the reconstructed images precisely. Numerical simulation results show proposed method reduces the memory usage, speeds up computation time and the quality of reconstructed images do not degrade evidently compared with uniform sampling method. Optical experiments results are in good agreement with numerical simulation results. The proposed method is simple, effective and could be applied in the holographic field in the future.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Corrections

20 December 2019: Typographical corrections were made to Figs. 4 and 8 and the captions of Tables 4–8.

1. Introduction

Holographic display [1], providing all depth cue needed by the human visual system, is regarded as the ultimate three-dimension (3D) display technology [24] and has attracted more attentions in recent years. However, there are two main problems which restrict the development of holographic display. One is the heavy computational load [57] and the other is the quality of reconstructed images [8,9].

The most widely used method [1014] to generate computer-generated hologram (CGH) is point-based method, where a 3D object is decomposed as a collection of self-luminous points and the hologram can be generated by the interference of all light waves from point sources. However, it suffers from a heavy computational load. In order to solve the problem, several approaches are proposed to speed up the computation [1520]. Look-up-table (LUT) method [6], where the fringe patterns (FPs) of all the object points are pre-calculated and stored in a table, was proposed. When calculating the CGHs, it just needs to read out corresponding FPs and add them up instead of calculation online. Novel look-up-table (N-LUT) method was proposed to reduce the memory usage of LUT greatly [7] by only pre-calculating and storing the FPs of the center object points on each sliced plane. In Split look-up-table (S-LUT) method, the FPs of point sources on each slice can be generated by the split horizontal and vertical modulation factors [10], which reduces memory usage greatly compared with N-LUT. Compressed look up table (C-LUT) [11] method and Accurate compressed look-up-table (AC-LUT) method [21] has been developed to reduce the large memory usage of S-LUT, where the horizontal and vertical modulation factors contain no depth information, so the memory usage does not increase with the increment of the depth layers. However, in these methods, CGHs are sampled at twice the maximum spatial frequency of holograms based on the Shannon–Nyquist sampling theorem, which means uniform sampling (US). In recent years, non-uniform sampling (NUS) method has been explored to speed up the computation. NUS algorithm for computing holograms has been explored by Pappu [22], where different segments of the hologram plane are sampled at corresponding twice maximum spatial frequency. However, the effectiveness of this method is limited by the calculated scenes because the spatial frequency is determined by the recorded objects. Tunable non-uniform sampling (TNUS) method [23], which develops a novel algorithm to sample on hologram plane and combines with N-LUT method, is proposed to speed up CGH generation and precisely modulate the reconstructed intensities of phase-only CGH. However, the memory usage of this algorithm is huge, which also limits the computation speed. Therefore, the memory usage needs to be reduced further.

In this paper, we propose a non-uniform sampling based on novel compressed look up table method (NUS-NCLUT) to get high quality reconstructed images with less memory usage and faster speed. In proposed method, we pre-calculate the non-uniform basic horizontal and vertical modulation factor and store them in LUT. Then, we generate the CGHs just by shifting the non-uniform basic horizontal and vertical modulation factors and multiplying them. Numerical simulations and optical experiments are performed to verify the feasibility of the proposed method.

2. Principle and method

In point- based method, the 3D object is considered as a collection of self-luminous points, which emit spherical waves irradiating uniformly on the hologram plane, and the wavefront distribution on the hologram plane after transmission is given by

$$H(x{^{\prime}_p}, y{^{\prime}_q}) = \sum\limits_{j = 0}^{N - 1} {{A_j}\exp [i(k{r_j} + {\phi _j})]}$$
Where
$$\varphi (x{^{\prime}_p}, y{^{\prime}_q}) = k{r_j}$$
$${r_j} = {({{{(x{^{\prime}_p} - {x_j})}^2} + {{(y{^{\prime}_q} - {y_j})}^2} + {{(d - {z_j})}^2}} )^{1/2}}$$
$H(x{^{\prime}_p}, y{^{\prime}_q})$ is the complex amplitude on hologram plane, N is the number of object points. $({x_j},{y_j},{z_j})$ and ${A_j}$ are the coordinate and amplitude of object point j, respectively. $\lambda$ is wavelength and $k = 2\pi /\lambda$ is wave number. d is the distance between the object plane and hologram plane. ${\phi _j}$ is the random phase, which is distributed between $0$ and $2\pi$.

The spatial frequency distribution of the FPs can be extracted from Eqs. (2) and (3) as

$${f_x} = \frac{1}{{2\pi }}\left|{\frac{{\partial \varphi }}{{\partial x}}} \right|= \frac{{|{x - {x_0}} |}}{{\lambda r}}, {f_y} = \frac{1}{{2\pi }}\left|{\frac{{\partial \varphi }}{{\partial y}}} \right|= \frac{{|{y - {y_0}} |}}{{\lambda r}}$$
where ${f_x}$ and ${f_y}$ corresponds to discrete spatial frequency on x and y direction, respectively. According to sampling theory, the sampling interval $\Delta x$ and $\Delta y$ on the hologram plane can be easily given by
$$\Delta x \le \frac{1}{{2{f_x}}}, \Delta y \le \frac{1}{{2{f_y}}}$$
Equation (5) indicates the sampling intervals in the different segments on the hologram plane are different: larger in the central part, while smaller in the border. If the CGHs are sampled uniformly, the sampling rates is twice maximum spatial frequency on the hologram plane. This result demonstrates the total number of NUS on the hologram plane is always less than the total number of US. Therefore, the NUS can reduce the sampling number of the CGHs, which will speed up the calculation and reduce the memory usage when the FPs need to be stored.

In Fresnel region [24], Eq. (1) can be written as

$$H({x^{\prime}}_p,{y^{\prime}}_q) = \sum\limits_{j = 1}^N {{A_j}} \exp \{ ik[(d - {z_j}) + \frac{{{{({x^{\prime}}_p - {x_j})}^2} + {{({y^{\prime}}_q - {y_j})}^2}}}{{2(d - {z_j})}}]\}$$
We split the vertical and horizontal information, and the Eq. (6) can be written as:
$$H({x^{\prime}}_p,{y^{\prime}}_q) = \sum\limits_{j = 1}^N {{A_j}} \exp [ik[(d - {z_j})]\cdot \exp [ik\frac{{{{({x^{\prime}}_p - {x_j})}^2}}}{{2(d - {z_j})}}]\cdot \exp [ik\frac{{{{({y^{\prime}}_q - {y_j})}^2}}}{{2(d - {z_j})}}]$$
We define $H({x^{\prime}}_p,{x_j},{z_j},\lambda ) = \exp [ik{({x^{\prime}}_p - {x_j})^2}/2(d - {z_j})]$ as the horizontal modulation factor, $V({y^{\prime}}_q,{y_j},{z_j},\lambda ) = \exp [ik{({y^{\prime}}_q - {y_j})^2}/2(d - {z_j})]$ as the vertical modulation factor and $L({z_j},\lambda ) = \exp [ik(d - {z_j})]$ as the longitudinal modulation factors.

So Eq. (7) can be simplified as

$$H({x^{\prime}}_p,{y^{\prime}}_q) = \sum\limits_{j = 1}^N {{A_j}} L({z_j},\lambda )(H({x^{\prime}}_p,{x_j},{z_j},\lambda )V({y^{\prime}}_q,{y_j},{z_j},\lambda ))$$
For ${N_{xy}}$ object points falling on the same layer of the 3D object, they have the same longitudinal modulation factors $L({z_j},\lambda )$. So Eq. (8) can be written as
$$H({x^{\prime}}_p,{y^{\prime}}_q) = \sum\limits_{{j_z} = 1}^{{N_z}} {[\sum\limits_{{j_{xy}} = 1}^{{N_{xy}}} {{A_{{j_{xy}}}}(H({x^{\prime}}_p,{x_{{j_{xy}}}},{z_j},\lambda )V({y^{\prime}}_q,{y_{{j_{xy}}}},{z_j},\lambda ))} } ]L({z_{{j_z}}},\lambda )$$
Where ${j_z}( = 1,2, \cdots ,{N_z})$ is the sequence number in the 2D image plane of the 3D object, ${j_{xy}} ( = 1,2, \cdots ,{N_{xy}})$ is the sequence number of points in each 2D image plane.

For ${N_x}$ object points falling on the same vertical line of each 2D image plane, they have the same horizontal modulation factor $H({x^{\prime}}_p,{x_m},{z_j},\lambda )$, so Eq. (9) can be written as

$$H({x^{\prime}}_p,{y^{\prime}}_q) = \sum\limits_{{j_z} = 1}^{{N_z}} {\{ \sum\limits_{{j_x} = 1}^{{N_x}} {[\sum\limits_{{j_y} = 1}^{{N_y}} {{A_{{j_y}}}V({y^{\prime}}_q,{y_{{j_y}}},{z_j},\lambda )} ]} H({x^{\prime}}_p,{x_{{j_x}}},{z_j},\lambda )\}} L({z_{{j_z}}},\lambda )$$
Where ${j_x}, {j_y} ( = 1,2, \cdots {N_x}, = 1,2, \cdots {N_y})$ are the sequence number of points on the vertical and horizontal line in each 2D image plane.

We define $H({x^{\prime}}_p,{x_m},{z_j},\lambda ) = \exp [ik{({x^{\prime}}_p - {x_\textrm{m}})^2}/2(d - {z_j})]$ as the basic horizontal modulation factor, $V({y^{\prime}}_q,{y_m},{z_j},\lambda ) = \exp [ik{({y^{\prime}}_q - {y_m})^2}/2(d - {z_j})]$ as the basic vertical modulation factor. ${x_m}$ and ${y_m}$ are the middle points of the row and the column in every depth layer of 3D object, respectively. Therefore, other horizontal and vertical modulation factors in each depth layer can be obtained by simply shifting the basic horizontal and vertical modulation factors.

Therefore, Eq. (10) can be written as

$$H({x^{\prime}}_p,{y^{\prime}}_q) = \sum\limits_{{j_z} = 1}^{{N_z}} {\{ \sum\limits_{{j_x} = 1}^{{N_x}} {[\sum\limits_{{j_y} = 1}^{{N_y}} {{A_{{j_y}}}V({y^{\prime}}_q - {y_j},{y_m},{z_j},\lambda )} ]} H({x^{\prime}}_p - {x_j},{x_m},{z_j},\lambda )\}} L({z_{{j_z}}},\lambda )$$
In offline computation, we can pre-calculate and store the basic modulation factors $H({x^{\prime}}_p,{x_m},{z_j},\lambda )$ and $V({y^{\prime}}_q,{y_m},{z_j},\lambda )$ in every 2D image plane.

In order to ensure the resolution of the hologram, the resolution of basic horizontal and vertical modulation factors for certain reconstructed distance and wavelength is given as:

$$\begin{array}{l} \textrm{Resolution of }H({x^{\prime}}_p,{x_m},{z_j},\lambda ) : p + {N_x}\Delta x\\ \textrm{Resolution of }V({y^{\prime}}_q,{y_m},{z_j},\lambda ) : q + {N_y}\Delta y \end{array}$$
where p and q are the horizontal and vertical resolution in hologram plane, respectively. $\Delta x$ and $\Delta y$ denote the ratio of the object sampling interval to the pixel size of the CGH in horizontal and vertical direction.

It is worthy noting that the CGHs are loaded on spatial light modulators (SLM), which requires its input signal to be sampled at the fixed frequency. To match the generated CGH with the SLMs, we build the non-uniform basic modulation factors as Fig. 1 shows. For the middle points in every depth layer of 3D object, we compute the horizontal and vertical spatial frequency and sampling interval of the first pixel $(r = 1)$ in CGHs through Eqs. (4) and (5). Assuming n pixel intervals satisfy sampling theory, then we study the $r + n$th pixel and repeat the above steps until $r > p + {N_x}\Delta x\textrm{ or }r > q + {N_y}\Delta y$. Finally, the non-uniform basic modulation factors are constructed. In this way, only partial pixels need to be calculated and stored, which can save the memory usage because the non-uniform basic horizontal and vertical modulation factor in LUT are sparse matrices. On the online computation, we pad zero to the vacant position in the non-uniform basic modulation factors to fill the predetermined size of the CGH.

 figure: Fig. 1.

Fig. 1. The flow chart of the non-uniform basic modulation factors generation.

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Therefore, as shown in Fig. 2, the proposed algorithm can be divided into two steps. The first step is to calculate the non-uniform basic modulation factors $H({x^{\prime}}_p,{x_m},{z_j},\lambda )$ and $V({y^{\prime}}_q,{y_m},{z_j},\lambda )$, and store them in LUT during the offline computation. The second step is to read out the non-uniform basic modulation factors from LUT and generate the holograms.

The first step can be listed as

$$\begin{array}{l} //\textrm{offline computation},\textrm{ to build a LUT}\\ \textrm{For }{x_m}\textrm{ of 2D image planes and }{\textrm{x}_p}^{\prime} + {N_x}{\Delta }x\textrm{ of hologram satisfy non-uniform sampling}\\ \ \ \ \ \ \ H({x^{\prime}}_p,{x_m},{z_j},\lambda ) = \exp [ik{({x^{\prime}}_p - {x_\textrm{m}})^2}/2(d - {z_j})]\\ \textrm{End}\\ \textrm{For }{y_m}\textrm{ of 2D image planes and }{\textrm{y}_q}^{\prime} + {N_x}{\Delta }y\textrm{ of hologram satisfy non-uniform sampling}\\ \ \ \ \ \ \ V({y^{\prime}}_q,{y_m},{z_j},\lambda ) = \exp [ik{({y^{\prime}}_q - {y_m})^2}/2(d - {z_j})]\\ \textrm{End} \end{array}$$

The second step can be listed as

$$\begin{array}{l} \textrm{/ /online computation},\textrm{ to read out the data from LUT and generate the hologram }\\ \textrm{For each }{\textrm{z}_j}\\ \ \ \ \ \ \ \ \textrm{For each }{\textrm{x}_j}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{For each }{\textrm{y}_q}^{\prime}\ \textrm{of hologram and each }{\textrm{y}_j}\textrm{ that }{\textrm{A}_j} \ne 0\textrm{ and }\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{have the same }{\textrm{x}_j}\textrm{(j = 0,1} \ldots {\textrm{N}_y} - 1)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{V} = {A_j}*V({y^{\prime}}_q - {y_j},{y_m},{z_j},\lambda ) + V;\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{End}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{For each }{\textrm{x}_p}^{\prime},{y_q}^{\prime}\textrm{ of hologram}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ HV = V*H({x^{\prime}}_p - {x_j},{x_m},{z_j},\lambda ) + HV;\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{End}\\ \ \ \ \ \ \ \textrm{End}\\ \ \ \ \ \ \ \textrm{For each }{\textrm{x}_p}^{\prime},{y_q}^{\prime}\textrm{ of hologram}\\ \ \ \ \ \ \ \ \ \ \ \ \ H({x_p}^{\prime},{y_q}^{\prime}) = HV*L({z_j},\lambda ) + H({x_p}^{\prime},{y_q}^{\prime});\\ \ \ \ \ \ \ \textrm{End}\\ \textrm{End} \end{array}$$

 figure: Fig. 2.

Fig. 2. Diagram of the proposed method to generate the CGH.

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Peak signal-to-noise ratio (PSNR) and speckle contrast (SC) are used to evaluate the image quality. The expressions of the two parameters are shown as

$$PSNR = 10\lg \left( {\frac{{{{({{2^n} - 1} )}^2}}}{{MSE}}} \right)$$
$$SC = \frac{{\sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{({p_i} - \mathop I\limits^ - )}^2}} } }}{{\mathop I\limits^ - }}$$
Where N is the number of pixels, ${p_i}$ is the intensity of each pixel in the image, $\mathop I\limits^ - $ is the average intensity of all pixels, and $MSE$ is the mean square error between the ideal image and the reconstructed image.

3. Numerical simulations and emulation

To demonstrate the feasibility of proposed method, we perform numerical simulations. Our program is run by a computer with MATLAB on Core i7-7700, 3.6 GHz, and 8G RAM. The parameters we used are listed in Table 1.

Tables Icon

Table 1. CGH computation parameters

As shown in Table 2, in TNUS method, the memory usage keeps in the order of megabytes (MBs). In S-LUT, C-LUT and AC-LUT methods, the memory usage reduces further but still keeps to the order of Mbs. While in NUS-NCLUT method, the memory usage reduces to the order of Kilobytes (Kbs). Compared with TNUS method, the memory usage reduces 431, 463 and 482 times in proposed method when the reconstructed distance is 200 mm and the wavelength is 639 nm, 532 nm, 473 nm, respectively.

Tables Icon

Table 2. Memory usage by using different methods

As shown in Fig. 3, although TNUS method applies NUS, large memory usage restricts the online computation time. NUS-NCLUT method is the fastest among all methods because of the least memory usage.

 figure: Fig. 3.

Fig. 3. Comparison of online computation time by using TNUS, S-LUT, C-LUT, AC-LUT and NUS-NCLUT methods.

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We compare the numerical simulation results by using proposed method and AC-LUT method and the results are shown in Fig. 4. PSNR and SC of numerical simulation results are given in Table 3. From Table 3 we can conclude that there is little difference of PSNR and SC between reconstructed images by using proposed method and AC-LUT method, which demonstrates proposed method can reconstruct the images successfully.

 figure: Fig. 4.

Fig. 4. Numerical simulation results, where Figs. 4(a)-(c) are ideal images, Figs. 4(d)-(f) and Figs. 4(g)-(i) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.

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Tables Icon

Table 3. PSNR and SC of simulated reconstructed gray images

Then proposed method is tested with color image. We reconstruct the colorful scene, where a red teapot and a green pyramid locate on a chess board. From Fig. 5 and Table 4, we can conclude that proposed method can reconstruct the scene with right color and shape. Numerical simulation results verify the feasibility of proposed method.

 figure: Fig. 5.

Fig. 5. Numerical simulation color results, where Fig. 5(a) is ideal image, Figs. 5(b) and 5(c) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.

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Table 4. PSNR and SC of simulated reconstructed color images

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Table 5. PSNR and SC of optical reconstructed gray images

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Table 6. PSNR and SC of optical reconstructed color images

To verify proposed method further, we reconstruct the 3D image. In simulation, we reconstruct ‘D’ and ‘A’ whose reconstructed distance is 200mm and 220mm, respectively. From Fig. 6, we can see that both characters are reconstructed faithfully with specific distance. These results demonstrate the feasibility of proposed method, which can reconstruct 3D images with correct depth information.

 figure: Fig. 6.

Fig. 6. Numerical simulation 3D results by using NUS-NCLUT method, where Figs. 6(a) and 6(b) are focused on 200 mm and 220 mm, respectively.

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4. Optical experiments

To demonstrate the feasibility of proposed method, we also perform the optical experiments. The size of object is 400 × 400. The resolution of hologram is 1080×1080, and the pixel size is 8µm. The reconstructed images are captured by a CCD (Lumenera camera INFINITY 4-11C). The green laser with wavelength 532nm are utilized and the reconstructed distance is 200 mm. In the optical experiments, the zero-order beam elimination method [24] is adopted to improve the quality of the reconstructed images. Besides, the temporal multiplexing method is utilized to generate the color holographic display, where red (639 nm), green (532 nm), and blue (473 nm) components are reconstructed and formed into color objects by using time integration. The schematic of the optical experimental setup for reconstruction is shown in the Fig. 7.

 figure: Fig. 7.

Fig. 7. Setup of the holographic display system: SLM is the spatial light modulator, PC is the personal computer, L1 and L2 are the Fourier transform lens.

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The monochrome results of optical experiments are shown in Fig. 8. From Fig. 8 and Table 5, we can conclude that proposed method can obtain high quality images. The optical experimental results match with numerical reconstructed results well.

 figure: Fig. 8.

Fig. 8. Optical experimental results, where Figs. 8(a)-(c) are ideal images, Figs. 8(d)-(f) and Figs. 8(g)-(i) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.

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Figure 9 shows the optical reconstructed color scenes. From Fig. 9 and Table 6, we can conclude that the optical experiments reconstruct the red teapot, green pyramid and chess board chequered with black and white successfully. The optical experimental results show that proposed method can reconstruct color scene successfully.

 figure: Fig. 9.

Fig. 9. Optical experimental color results, where Fig. 9(a) is ideal image, Figs. 9(b) and 9(c) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.

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Figure 10 shows the optical reconstructed 3D scene by using proposed method. When the CCD focus on one character, the character becomes clear and in good shape while the other becomes blur. The optical experimental results show that proposed method keeps the depth information well and the optical experimental results are in accord with numerical simulation results.

 figure: Fig. 10.

Fig. 10. Optical experimental 3D results by using NUS-NCLUT method, where Figs. 10(a) and 10(b) are focused on 200 mm and 220 mm, respectively.

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5. Conclusion

We have proposed a NUS-NCLUT method based on sampling theory and LUT to reduce the memory usage without sacrificing the quality of reconstructed images. In proposed method, we build the NUS coordinate firstly according to sampling theory. Then we calculate the basic horizontal and vertical factors in NUS coordinate and store them in LUT. Therefore, in online computation, we obtain fringe patterns for other points by simply shifting and multiplying the pre-calculated basic modulation factors, and the final CGH is generated by adding them all together. Numerical simulations and optical experiments are performed to verify proposed method, and the results match well with each other. It is predicted that our method is a promising method for realizing 3D holographic display with less memory usage, faster computation speed, high quality reconstructed images and will have great potential to be applied in the holographic display or various other optical diffraction areas in the future.

Funding

Key Technologies Research and Development Program (2017YFB1002900); National Natural Science Foundation of China (61975014, 61575024, 61420106014); Newton Fund.

Disclosures

The authors declare no conflicts of interest.

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References

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  1. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
    [Crossref]
  2. J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array ofspatial light modulators,” Opt. Express 16(16), 12372–12386 (2008).
    [Crossref]
  3. T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
    [Crossref]
  4. F. Yaraş, H. Kang, and L. Onural, “Circular holographic video display system,” Opt. Express 19(10), 9147–9156 (2011).
    [Crossref]
  5. A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
    [Crossref]
  6. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
    [Crossref]
  7. S. C. Kim and E. S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47(19), D55–D62 (2008).
    [Crossref]
  8. T. Shimobaba and T. Ito, “Random phase-free computer-generated hologram,” Opt. Express 23(7), 9549–9554 (2015).
    [Crossref]
  9. H. Pang, J. Wang, A. Cao, and Q. Deng, “High-accuracy method for holographic image projection with suppressed speckle noise,” Opt. Express 24(20), 22766 (2016).
    [Crossref]
  10. Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. Tanjung, C. Tan, and T. C. Chong, “Fast CGH computation using SLUT on GPU,” Opt. Express 17(21), 18543–18555 (2009).
    [Crossref]
  11. J. Jia, Y. Wang, J. Liu, X. Li, Y. Pan, Z. Sun, B. Zhang, Q. Zhao, and W. Jiang, “Reducing the memory usage for effective computer-generated hologram calculation using compressed look-up table in full-color holographic display,” Appl. Opt. 52(7), 1404–1412 (2013).
    [Crossref]
  12. S. C. Kim and E. S. Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48(6), 1030–1041 (2009).
    [Crossref]
  13. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computergenerated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010).
    [Crossref]
  14. S. C. Kim, J. H. Yoon, and E. S. Kim, “Fast generation of three-dimensional video holograms by combined use of data compression and lookup table techniques,” Appl. Opt. 47(32), 5986–5995 (2008).
    [Crossref]
  15. S. C. Kim, J. M. Kim, and E. S. Kim, “Effective memory reduction of the novel look-up table with onedimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20(11), 12021–12034 (2012).
    [Crossref]
  16. S. C. Kim, X. B. Dong, M. W. Kwon, and E. S. Kim, “Fast generation of video holograms of three-dimensional moving objects using a motion compensation-based novel look-up table,” Opt. Express 21(9), 11568–11584 (2013).
    [Crossref]
  17. S. C. Kim, X. B. Dong, and E. S. Kim, “Accelerated one-step generation of full-color holographic videos using a color-tunable novel-look-up-table method for holographic three-dimensional television broadcasting,” Sci. Rep. 5(1), 14056 (2015).
    [Crossref]
  18. T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using runlength encoding based recurrence relation,” Opt. Express 23(8), 9852–9857 (2015).
    [Crossref]
  19. S. Jiao, Z. Zhuang, and W. Zou, “Fast computer generated hologram calculation with a mini look-up table incorporated with radial symmetric interpolation,” Opt. Express 25(1), 112–123 (2017).
    [Crossref]
  20. Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
    [Crossref]
  21. C. Gao, J. Liu, X. Li, G. Xue, J. Jia, and Y. Wang, “Accurate compressed look up table method for CGH in 3D holographic display,” Opt. Express 23(26), 33194–33204 (2015).
    [Crossref]
  22. R. S. Pappu, “Nonuniformly sampled computer-generated holograms,” Opt. Eng. 35(6), 1538–1544 (1996).
    [Crossref]
  23. Z. Zhang, J. Liu, J. Jia, X. Li, J. Han, B. Hu, and Y. Wang, “Tunable nonuniform sampling method for fast calculation and intensity modulation in 3D dynamic holographic display,” Opt. Lett. 38(15), 2676–2679 (2013).
    [Crossref]
  24. H. Zhang, J. Xie, J. Liu, and Y. Wang, “Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection,” Appl. Opt. 48(30), 5834–5841 (2009).
    [Crossref]

2017 (2)

S. Jiao, Z. Zhuang, and W. Zou, “Fast computer generated hologram calculation with a mini look-up table incorporated with radial symmetric interpolation,” Opt. Express 25(1), 112–123 (2017).
[Crossref]

Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
[Crossref]

2016 (1)

2015 (4)

2013 (3)

2012 (2)

S. C. Kim, J. M. Kim, and E. S. Kim, “Effective memory reduction of the novel look-up table with onedimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20(11), 12021–12034 (2012).
[Crossref]

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (3)

2008 (3)

2005 (1)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

1996 (1)

R. S. Pappu, “Nonuniformly sampled computer-generated holograms,” Opt. Eng. 35(6), 1538–1544 (1996).
[Crossref]

1993 (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
[Crossref]

1992 (1)

A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Asundic, A. K.

Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
[Crossref]

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

Cao, A.

Chong, T. C.

Deng, Q.

Dong, X. B.

S. C. Kim, X. B. Dong, and E. S. Kim, “Accelerated one-step generation of full-color holographic videos using a color-tunable novel-look-up-table method for holographic three-dimensional television broadcasting,” Sci. Rep. 5(1), 14056 (2015).
[Crossref]

S. C. Kim, X. B. Dong, M. W. Kwon, and E. S. Kim, “Fast generation of video holograms of three-dimensional moving objects using a motion compensation-based novel look-up table,” Opt. Express 21(9), 11568–11584 (2013).
[Crossref]

Finke, G.

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

Gao, C.

Hahn, J.

Han, J.

Hennelly, B.

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

Hu, B.

Ito, T.

Jia, J.

Jiang, W.

Jiao, S.

Kakue, T.

Kang, H.

Kim, E. S.

Kim, H.

Kim, J. M.

Kim, S. C.

Kozacki, T.

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

Kujawinska, M.

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

Kwon, M. W.

Lee, B.

Leigh, J. J. S.

A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Li, X.

Liang, X.

Lim, Y.

Liu, J.

Lucente, M.

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
[Crossref]

Masuda, N.

Nakayama, H.

Nishitsuji, T.

Onural, L.

Pan, Y.

Pang, H.

Pappu, R. S.

R. S. Pappu, “Nonuniformly sampled computer-generated holograms,” Opt. Eng. 35(6), 1538–1544 (1996).
[Crossref]

Park, G.

Shimobaba, T.

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

Solanki, S.

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

Stein, A. D.

A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Sun, Z.

Tan, C.

Tanjung, R. B.

Wang, J.

Wang, Y.

Wang, Z.

A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Xie, J.

Xu, X.

Xue, G.

Yaras, F.

Yoon, J. H.

Yu, Y.

Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
[Crossref]

Zaperty, W.

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

Zeng, Z.

Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
[Crossref]

Zhang, B.

Zhang, H.

Zhang, Z.

Zhao, Q.

Zheng, H.

Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
[Crossref]

Zhuang, Z.

Zou, W.

Appl. Opt. (5)

Comput. Phys. (1)

A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Computer (1)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

J. Disp. Technol. (1)

T. Kozacki, M. Kujawińska, G. Finke, W. Zaperty, and B. Hennelly, “Holographic capture and display systems in circular configurations,” J. Disp. Technol. 8(4), 225–232 (2012).
[Crossref]

J. Electron. Imaging (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
[Crossref]

Opt. Eng. (1)

R. S. Pappu, “Nonuniformly sampled computer-generated holograms,” Opt. Eng. 35(6), 1538–1544 (1996).
[Crossref]

Opt. Express (11)

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using runlength encoding based recurrence relation,” Opt. Express 23(8), 9852–9857 (2015).
[Crossref]

S. Jiao, Z. Zhuang, and W. Zou, “Fast computer generated hologram calculation with a mini look-up table incorporated with radial symmetric interpolation,” Opt. Express 25(1), 112–123 (2017).
[Crossref]

C. Gao, J. Liu, X. Li, G. Xue, J. Jia, and Y. Wang, “Accurate compressed look up table method for CGH in 3D holographic display,” Opt. Express 23(26), 33194–33204 (2015).
[Crossref]

T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computergenerated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010).
[Crossref]

S. C. Kim, J. M. Kim, and E. S. Kim, “Effective memory reduction of the novel look-up table with onedimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20(11), 12021–12034 (2012).
[Crossref]

S. C. Kim, X. B. Dong, M. W. Kwon, and E. S. Kim, “Fast generation of video holograms of three-dimensional moving objects using a motion compensation-based novel look-up table,” Opt. Express 21(9), 11568–11584 (2013).
[Crossref]

F. Yaraş, H. Kang, and L. Onural, “Circular holographic video display system,” Opt. Express 19(10), 9147–9156 (2011).
[Crossref]

J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array ofspatial light modulators,” Opt. Express 16(16), 12372–12386 (2008).
[Crossref]

T. Shimobaba and T. Ito, “Random phase-free computer-generated hologram,” Opt. Express 23(7), 9549–9554 (2015).
[Crossref]

H. Pang, J. Wang, A. Cao, and Q. Deng, “High-accuracy method for holographic image projection with suppressed speckle noise,” Opt. Express 24(20), 22766 (2016).
[Crossref]

Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. Tanjung, C. Tan, and T. C. Chong, “Fast CGH computation using SLUT on GPU,” Opt. Express 17(21), 18543–18555 (2009).
[Crossref]

Opt. Lasers Eng. (1)

Z. Zeng, H. Zheng, Y. Yu, and A. K. Asundic, “Off-axis phase-only holograms of 3D objects using accelerated point-based Fresnel diffraction algorithm,” Opt. Lasers Eng. 93, 47–54 (2017).
[Crossref]

Opt. Lett. (1)

Sci. Rep. (1)

S. C. Kim, X. B. Dong, and E. S. Kim, “Accelerated one-step generation of full-color holographic videos using a color-tunable novel-look-up-table method for holographic three-dimensional television broadcasting,” Sci. Rep. 5(1), 14056 (2015).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. The flow chart of the non-uniform basic modulation factors generation.
Fig. 2.
Fig. 2. Diagram of the proposed method to generate the CGH.
Fig. 3.
Fig. 3. Comparison of online computation time by using TNUS, S-LUT, C-LUT, AC-LUT and NUS-NCLUT methods.
Fig. 4.
Fig. 4. Numerical simulation results, where Figs. 4(a)-(c) are ideal images, Figs. 4(d)-(f) and Figs. 4(g)-(i) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.
Fig. 5.
Fig. 5. Numerical simulation color results, where Fig. 5(a) is ideal image, Figs. 5(b) and 5(c) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.
Fig. 6.
Fig. 6. Numerical simulation 3D results by using NUS-NCLUT method, where Figs. 6(a) and 6(b) are focused on 200 mm and 220 mm, respectively.
Fig. 7.
Fig. 7. Setup of the holographic display system: SLM is the spatial light modulator, PC is the personal computer, L1 and L2 are the Fourier transform lens.
Fig. 8.
Fig. 8. Optical experimental results, where Figs. 8(a)-(c) are ideal images, Figs. 8(d)-(f) and Figs. 8(g)-(i) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.
Fig. 9.
Fig. 9. Optical experimental color results, where Fig. 9(a) is ideal image, Figs. 9(b) and 9(c) are the reconstructed images by using NUS-NCLUT and AC-LUT methods, respectively.
Fig. 10.
Fig. 10. Optical experimental 3D results by using NUS-NCLUT method, where Figs. 10(a) and 10(b) are focused on 200 mm and 220 mm, respectively.

Tables (6)

Tables Icon

Table 1. CGH computation parameters

Tables Icon

Table 2. Memory usage by using different methods

Tables Icon

Table 3. PSNR and SC of simulated reconstructed gray images

Tables Icon

Table 4. PSNR and SC of simulated reconstructed color images

Tables Icon

Table 5. PSNR and SC of optical reconstructed gray images

Tables Icon

Table 6. PSNR and SC of optical reconstructed color images

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

H ( x p , y q ) = j = 0 N 1 A j exp [ i ( k r j + ϕ j ) ]
φ ( x p , y q ) = k r j
r j = ( ( x p x j ) 2 + ( y q y j ) 2 + ( d z j ) 2 ) 1 / 2
f x = 1 2 π | φ x | = | x x 0 | λ r , f y = 1 2 π | φ y | = | y y 0 | λ r
Δ x 1 2 f x , Δ y 1 2 f y
H ( x p , y q ) = j = 1 N A j exp { i k [ ( d z j ) + ( x p x j ) 2 + ( y q y j ) 2 2 ( d z j ) ] }
H ( x p , y q ) = j = 1 N A j exp [ i k [ ( d z j ) ] exp [ i k ( x p x j ) 2 2 ( d z j ) ] exp [ i k ( y q y j ) 2 2 ( d z j ) ]
H ( x p , y q ) = j = 1 N A j L ( z j , λ ) ( H ( x p , x j , z j , λ ) V ( y q , y j , z j , λ ) )
H ( x p , y q ) = j z = 1 N z [ j x y = 1 N x y A j x y ( H ( x p , x j x y , z j , λ ) V ( y q , y j x y , z j , λ ) ) ] L ( z j z , λ )
H ( x p , y q ) = j z = 1 N z { j x = 1 N x [ j y = 1 N y A j y V ( y q , y j y , z j , λ ) ] H ( x p , x j x , z j , λ ) } L ( z j z , λ )
H ( x p , y q ) = j z = 1 N z { j x = 1 N x [ j y = 1 N y A j y V ( y q y j , y m , z j , λ ) ] H ( x p x j , x m , z j , λ ) } L ( z j z , λ )
Resolution of  H ( x p , x m , z j , λ ) : p + N x Δ x Resolution of  V ( y q , y m , z j , λ ) : q + N y Δ y
/ / offline computation ,  to build a LUT For  x m  of 2D image planes and  x p + N x Δ x  of hologram satisfy non-uniform sampling             H ( x p , x m , z j , λ ) = exp [ i k ( x p x m ) 2 / 2 ( d z j ) ] End For  y m  of 2D image planes and  y q + N x Δ y  of hologram satisfy non-uniform sampling             V ( y q , y m , z j , λ ) = exp [ i k ( y q y m ) 2 / 2 ( d z j ) ] End
/ /online computation ,  to read out the data from LUT and generate the hologram  For each  z j               For each  x j                               For each  y q   of hologram and each  y j  that  A j 0  and                                have the same  x j (j = 0,1 N y 1 )                                           V = A j V ( y q y j , y m , z j , λ ) + V ;                               End                             For each  x p , y q  of hologram                                     H V = V H ( x p x j , x m , z j , λ ) + H V ;                             End             End             For each  x p , y q  of hologram                         H ( x p , y q ) = H V L ( z j , λ ) + H ( x p , y q ) ;             End End
P S N R = 10 lg ( ( 2 n 1 ) 2 M S E )
S C = 1 N i = 1 N ( p i I ) 2 I

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