## 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 [2–4] 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 [5–7] and the other is the quality of reconstructed images [8,9].

The most widely used method [10–14] 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 [15–20]. 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

*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

*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 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

So Eq. (7) can be simplified as

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

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

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:

*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.

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

The second step can be listed as

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

*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.

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.

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.

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.

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.

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.

## 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.

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 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 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.

## 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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