## Abstract

In the conventional weighted Gerchberg-Saxton (GS) algorithm, the feedback is used to accelerate the convergence. However, it will lead to the iteration divergence. To solve this issue, an adaptive weighted GS algorithm is proposed in this paper. By replacing the conventional feedback with our designed feedback, the convergence can be ensured in the proposed method. Compared with the traditional GS iteration method, the proposed method improves the peak signal-noise ratio of the reconstructed image with 4.8 dB on average. Moreover, an approximate quadratic phase is proposed to suppress the artifacts in optical reconstruction. Therefore, a high-quality image can be reconstructed without the artifacts in our designed Argument Reality device. Both numerical simulations and optical experiments have validated the effectiveness of the proposed method.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Holographic display, as one of the ideal three-dimensional display technologies, has been given strong attention in several decades since it can provide the depth clues of the 3-D scene and does not cause the vergence-accommodation conflict [1,2]. Especially in advances in the computer’s power and abilities, the computer-generation hologram (CGH) has more and more attention on account of the flexibility in comparison to conventional optical holography [3–5]. Generally, the spatial light modulator (SLM) is adopted to load the CGH. Because of the higher diffraction efficiency of the phase-only SLM compared to the amplitude-only SLM, the phase-only hologram (POH) has been widely concerned [6–12].

However, the low quality of the reconstructed image is known as one of the major issues that restrict the application in holographic display. At present, various non-iterative methods have been proposed to enhance the reconstruction quality of the POH, such as the double-phase (DPH) method [13–17], error diffusion method [18,19], and time-division multiplexing method [20–23]. These methods have significantly improved the reconstruction quality. Whereas, there are still some problems. The DPH method sacrifices the spatial bandwidth product of the hologram, and the reconstructed image has unsatisfied diffraction efficiency by using the error diffusion method. The time-division multiplexing method needs a SLM with a high frame rate, which increases the costs of the system.

Except for the non-iterative methods mentioned above, the iterative optimize algorithm as another approach to improve the reconstruction quality of the POH, has been extensively studied [24–28]. The best famous is the Gerchberg-Saxton (GS) algorithm [24], which is an iterative phase retrieval algorithm. The desired phase profile can be obtained by using the restriction of the target image in the iteration process. Hence, the optimized hologram can reconstruct the high-quality target images. Since it is an iterative algorithm, the convergency value only can be obtained under multiple iterations conditions. To speed up the convergence, some modified weighted GS (WGS) methods have been proposed [29–31]. These methods introduce proportional feedback in the image plane to accelerate convergence and improve reconstructed quality. However, all of the WGS methods have a common numerical calculation problem, which results in the unstable feedback. This matter might lead to the algorithm divergence. Furthermore, iterative algorithm generally adopts the random phase as the initial value for iteration. In the latest work, it is demonstrated that it has a high-quality reconstructed image with reduced speckle noise when the quadratic phase is employed as the initial value [32–35]. Nevertheless, some artifacts appeared in optical reconstruction if the quadratic phase is used. These artifacts overlap the reconstructed primary image, which lead to the increased background noise. Especially in Augmented Reality (AR) device, it seriously affects the view experience.

To address these two issues mentioned above, we firstly analyze the reason for the instability of convergence in the conventional WGS method. On the basis of the reason for the problem, the adaptive weighted Gerchberg-Saxton iteration method (AWGS) is proposed, which can ensure the stability of computational convergence. Moreover, in order to solve the artifacts caused by the quadratic phase, we propose an approximate phase as the initial value for iteration to suppress the appearance of the artifacts in optical reconstruction. Meanwhile, the proposed method is applied in a designed AR device, and our proposed method can reconstruct a high-quality image without the artifacts. The validity of our method is demonstrated by both numerical simulation and optical experiments.

## 2. Method

#### 2.1 Principle of adaptive weighted GS method

The GS algorithm as a phase retrieval algorithm, is proposed to obtain the designed complex amplitude distribution by using the known amplitude distribution. Generally, the GS algorithm based on the forward propagation and backward propagation of the light field is performed by the Fast Fourier Transformation (FFT). In the iteration process, the amplitude of the hologram is replaced by laser illumination intensity. In the image plane, the amplitude is replaced by target image intensity. Meanwhile, the GS algorithm is an iterative method that can be seen as a steepest descent method, reaching extreme value along the negative direction of gradient the function. Nevertheless, it requires multiple iteration propagation to converge.

To speed up its convergence, the WGS iteration algorithm is proposed recently [30,31]. This method constructs a ratio between the amplitude of the target image and the amplitude of the reconstructed image, which is regarded as feedback to restrict the amplitude of the image plane. The weighted ratio $w$ in the WGS method is defined as:

where $A_t$ and $A_r$ are target image intensity and reconstructed image intensity, respectively. $a_k$ is a parameter to enhance the convergence. In this method, to avoid dividing by zero, it needs to add a small number $\varepsilon$ to the denominator.However, the WGS method has a problem of numerical calculation. It causes the feedback failure when the intensity value of a pixel in the reconstructed image is very small, that it is much smaller than the numerator, since the calculated feedback ratio of this pixel is a big number. Therefore, it leads to the divergence of the WGS method. To clarify the problem more clearly, we calculate the maximum value of $w$ with the different iteration numbers as shown in Fig. 1. The blue and orange curves represent the weight $w$ and normalized root mean squared error (RMSE), respectively. Since the calculated weighted values are lager, a logarithmic transformation $log_2(1+max(w))$ is performed, where the operation $max()$ means to obtain the maximum value. The normalized RMSE is defined as:

It can be seen the RMSE is gradually decreasing when there are fewer iteration numbers. However, the weighted values suddenly sharp increase when the iteration numbers increase, which causes the RMSE to rise and the algorithm fails to converge. Meanwhile, the quadratic phase is introduced as the initial value to execute the iteration process. The quadratic phase is expressed as follows:

Here, $a$ and $b$ are parameters of the quadratic phase, and $p$ and $q$ are image pixel coordinates in $x$, $y$ direction. There are more details in Refs. [34]. The simulation results are given in Figs. 1(c), 1(f), and 1(i). Similarly, the WGS method also fails to convergence. Whether the random phase or the quadratic phase is used as the initial value, the WGS method has the numerical calculation problem, which results in the divergence.

To solve this issue, we propose the adaptive weighted GS method (AWGS). The proposed method constructs an exponential term to form feedback, not a ratio, which can avoid the division operation. The proposed feedback $w_{pro}$ is defined as :

This form does not have the problem of numerical calculation in which the divisor is far less than the number of dividends. To verify the feasibility of the proposed method, the same simulation experiments are performed, and the results are shown in Fig. 2 (The details of the proposed method are described in Sec. 2.2). It shows that the feedback weight value by the AWGS method does not change dramatically in Fig. 2. It should be noted that the feedback value is not logarithm transformed. Furthermore, as the iteration numbers increase, the feedback value converges to a stable value of 1, since $e^{A_t-A_r} \approx 1$ when $A_t$ and $A_r$ are approximately equal. It also illustrates the high quality of the reconstructed image and the AWGS method is effective for the random phase and the quadratic phase. Therefore, the proposed method can guarantee its convergence and low RMSE.

#### 2.2 Suppression of artifacts

In section 2.1, the random phase and quadratic phase are used as the initial value in the GS method, respectively. In the previous works, it has been demonstrated that the quadratic phase is better than the random phase in suppressing the speckle noise [32–35]. Nevertheless, significant artifacts appear in the optically reconstructed image if the quadratic phase is employed as the initial value. These artifacts can overlap with the primary image, affecting the quality of the reconstructed image and the viewing experience. The cause of the appearance of the artifacts is the power of convergence of light rays of the quadratic phase. At this point, we propose an approximation algorithm that uses a set of plane wave with discrete directions to approximate the quadratic phase for suppressing the appearance of artifacts, which can weaken the effect of the quadratic phase.

We consider only the $x$ direction to simplify the analysis. For the Fourier hologram, the maximum spatial frequency $f_{max}$ that a hologram can record is:

where $M$, ${\Delta }_{SLM}$, $\lambda$, and $f_{focal}$ are the maximum number of pixels in the target image, sampling interval of the spatial light modulator (SLM), wavelength and focal length of Fourier lens, respectively. To satisfy the Nyquist theorem [36] and implement FFT, the sampling interval of the spatial domain is deduced as:In order to use a set of discrete plane wave, $n$ frequency points are sampled in the bandwidth range $-f_{max}$ to $f_{max}$. It is important to note that $n$ is less than $M$. Then, repeat for each frequency point $M/n$ times, and get $f_x$. In this way, combined with Eq. (6), we can obtain an approximate quadratic phase in $x$ direction, which is expressed as follows:

where $p$ is the pixel coordinates in $x$ direction. The operation $linspace$ $(x_1, x_2, n)$ generates $n$ evenly spaced points between $x_1$ and $x_2$, and the operation $repeat$ $(array, n)$ repeats elements of an array $n$ times. For example, $repeat$ $([1, 3], 2)$ can generate an $array$ $[1, 1, 3, 3]$. Similarly, we can get frequency distribution in $y$ direction, and the final expression is: where $q$, $\Delta {Y}$, and $f_y$ are the pixel coordinates, sampling interval of the spatial domain, and frequency distribution in $y$ direction, respectively. The generated $\varphi (x,y)$ with different $n$ is shown in Figs. 3(b) to 3(e). As $n$ increases, the $\varphi (x,y)$ will turn into the quadratic phase. When $n$ is set to image size, the $\varphi (x,y)$ coincident with the quadratic phase, as shown in Figs. 3(e) and 3(f). It confirms the validity of the proposed method.The flow chart of the whole AWGS method is shown in Fig. 4. The $\varphi (x,y)$ is employed as the initial value to start the iteration. In the iteration process, the amplitude constraint in the hologram plane is unit amplitude, and the amplitude constraint in the target image is expressed as:

## 3. Simulation

In this section, we perform the numerical simulation to verify our proposed method. Besides the RMSE, the peak signal-noise ratio (PSNR) is used as the second evaluation function of the reconstructed image quality. The definition of PSNR for an 8-bit gray image is:

Higher PSNR implies the better quality of the reconstructed image. The gray image Motorbike is used as a test image, as shown in Fig. 5(a). The size is 800 $\times$ 800 and enlarged to 1024 $\times$ 1024 by zero-padding. The sampling intervals of the hologram and wavelength of the laser light are 8 $\mu {m}$ $\times$ 8 $\mu {m}$ and 671 $nm$, respectively. $n$ is set to 200. The focal length of the Fourier lens is 300 $mm$. The simulation results are shown in Fig. 5. In Fig. 5(b), the proposed AWGS method shows better results compared to the GS iteration method with the traditional quadratic phase. The traditional method reaches the convergence at 10 iterations, whereas the AGWS method has a larger optimization space. The AWGS method can converge to better extreme value as the numbers of iteration increases. Besides the gray image Motobike, the binary images are applied as the target images, which are shown in Figs. 5(c) and 5(e). The involved parameters are the same as the Motobike. The simulation results show that the AWGS method also works well for the binary images, which are shown in Figs. 5(d) and 5(f).

To better compare the reconstructed image quality of the two methods, the iteration number is 20, and $n$ is 200. The simulation reconstructed images are shown in Fig. 6. Compared with the traditional method, the PSNR of the reconstructed image with the AWGS method is improved, and the proposed AWGS method can improve the quality of reconstructed images with 4.8 dB higher in PSNR on average. The reconstructed images by our method have better quality, and the speckle noise is greatly suppressed. Although the suppression of artifacts cannot be seen in the simulation since the artifacts only appear in optical experiments, the effectiveness of artifact suppression by the proposed method will be verified in the optical experiment.

We also perform simulations in various $n$ condition, as shown in Fig. 7. $n$ is set to 100, 200, 400, and 800, respectively. From the curve of Fig. 7, when $n$ is 200 and the iteration number is greater than 16, the PSNR is highest. As $n$ further increases, the PSNR would decrease. Because the $\varphi (x,y)$ turns to the quadratic phase to some extent when $n$ increases, as shown in Figs. 3(e) and 3(f). Moreover, it might lead to the appearance of artifacts since the effect of suppression for the power of the quadratic phase is reduced. Therefore, the selection of parameter $n$ is very important.

## 4. Experiment and discussion

In this section, we perform the optical experiments to verify our proposed method. The optical reconstructed system is depicted in Fig. 8. The experimental conditions are as follows. A He-Ne laser with a wavelength of 671 $nm$ is used as the light source. The laser is expanded and collimated by the beam expander and collimating lens to illuminate the SLM. The reflective phase-only 8-bit SLM is used, where sampling interval and resolution are 8 $\mu {m}$ $\times$ 8 $\mu {m}$ and 1920 $\times$ 1080, respectively. The focal length of the lens 1 is 30 $cm$, and the target image size is 600 $\times$ 600 and enlarged to 1024 $\times$ 1024 by zero-padding. A 4-f system composed of the lens 2 (focal of length: 15 $cm$) and the lens 3 (focal of length: 30 $cm$) with a spatial filter is used to eliminate the effect of zero-order light caused by the SLM. To better filter the zero-order light, a blazed grating is introduced to superimposed on the POH. The reconstructed image is combined with the real world through a beam splitter. We use a camera (Nikon D810) with a lens (105 $mm$) to capture the reconstructed image.

The results of the reconstructed image by the traditional GS method and AWGS method are shown in Fig. 9. The reconstructed image is captured in the back focal plane of the lens 1 by a camera without the lens. In the AWGS method, $n$ is set to 200 and 600, respectively. The iteration numbers are set to 20 times in all methods. We first observe that the optical results by the traditional method in Figs. 9(a), 9(b), and 9(c). The reconstructed images are contaminated by the artifacts which have been marked with green dashed box in the background. These artifacts overlap the primary image so that it increases the background noise. By comparison, the reconstructed image by the AWGS method is not influenced by the artifacts in Figs. 9(d) to 9(f), when $n$ is set to 200, because the proposed phase weakens the power of light convergence for the quadratic phase. Nevertheless, in Figs. 9(g) to 9(i), when $n$ is set to 600, the suppression effect of the artifacts is imperfect because the larger $n$ reduces the suppression effect.

In addition, the result of the proposed method for the speckle noise reduction is shown in Fig. 10. The bottom image is an enlarged part marked by the green dashed box in Fig. 10. Compared Fig. 10(a) with Fig. 10(b), we can see that the speckle noise is reduced in Figs. 10(b). While in Fig. 10(a), the reconstructed image is contaminated by the speckle noise. However, in Fig. 10(c), the speckle noise cannot be reduced since the use of the larger $n$, which is consistent with the simulation result in Fig. 7. In the meantime, we use other standard test images to perform the optical experiments. The experiment results are shown in Fig. 11. The Figs. 11(a) and (b) show that the artifacts do not appear when the random phase is used. However, the reconstructed images are filled with the speckle noise, which can be seen in the enlarged part. The speckle noise can be reduced when the quadratic phase is used, which are shown in Figs. 11(c) and (d). However, it results in the artifacts, which are marked with the green dashed box. Compared with the random phase and quadratic phase, the proposed method can reconstruct high-quality images without speckle noise and artifacts as shown in Figs. 11(e) and (f).

To illustrate the effectiveness of the proposed method in the field of AR, we reconstruct the images using the device that is shown in Fig. 8, and the reconstructed results are shown in Fig. 12. In Fig. 12(a), although a 4-$f$ filter system is adopted, the artifacts can still be seen since the 4-$f$ system does not work well for the traditional method. In contrast, in Fig. 12(b), there are no artifacts in the reconstructed image background. The proposed method can provide a better view of enjoyment. However, the quality of the reconstructed image is slightly degraded since the spherical aberrations caused by the use of multiple lenses compared with Figs. 9(b) and 9(e).

To reconstruct a multi-plane object using our method, we combine the method by adding the phase information of spherical wave phase $\phi _{s}$ to the precalculated phase distribution of the hologram plane [37], by which the reconstructed image can be shift from the focal plane of lens 1. When different $\phi _{s}$ is adopted, the target image can be reconstructed at different depths. The shift distance $\Delta {Z}$ is $f_{lens}^2 / (r-d+f_{lens})$, where $r$, $d$, $f_{lens}$ are the radius of spherical wave phase $\phi _{s}$, the distance from the SLM to lens 1, and the focal length of the lens 1, respectively. The multi-plane object is composed of the school badge and four Chinese characters, where $r$ and $d$ are set to 90 $cm$ and 30 $cm$, respectively. The experimental results are shown in Fig. 13. In Fig. 13(a), when the camera focus on the dice, the Chinese characters are clearly visible, and the school badge is blur since it is defocused. When focus the rear of a car, the school badge is in focus and visible in Fig. 13(b). The distance between the two reconstruction depths is 10 $cm$, which is in conformity to the theoretical calculation. The dynamic focusing on these two objects is shown in supplementary materials. However, for the AR device, the size of the system is an issue that cannot be ignored. In our designed prototype of the AR device, the 4-$f$ filter system is imposed, which increases the complexity of the device. For this matter, the lens 2 and the lens 3 can be replaced by the short-focus lenses to reduce the bulk. In addition, we also can combine some lensless method to suppress the zero-order light. By these means, this problem can be solved.

## 5. Discussion

#### 5.1 Comparison of calculation time

The extensive computation is also a common problem in our method since it is an iteration algorithm. In our experiment, the calculation platform includes Python3.7, Windows 10 operation system, and Core I5-4590 (3.3GHz). The detailed time composition in 1 iteration is shown in Table 1. As can be seen from Table 1, the adaptive weight feedback is not time-consuming, and most of the time is taken by the FFT. For a better comparison of calculation time, the total calculation time in different iterations is shown in Table 2. When the iterations are set to 20 times, the AWGS method takes 12.19s, which is 1.3 seconds slower than the traditional method under the same iterations. The reason is the accumulation of time differences in weight calculation under multiple iterations. Fortunately, we can accelerate the iteration process using the Graphics Processing Unit to solve this defect, or combine the proposed method with the optimized phase method [26].

#### 5.2 Analysis of zero-padding

The zero-padding is adopted in our proposed method. To analyze the influence of zero-padding, the simulation results by the traditional GS method and the AWGS method with and without zero-padding are shown in Fig. 14. The original image size is 800 $\times$ 800, and the iterations are set to 20. In the zero-padding method, the image is enlarged to 1024 $\times$ 1024 by zero-padding. From the results in Fig. 14, the reconstructed image quality with zero-padding is higher than without zero-padding in the traditional GS method, and the AWGS method also shows similar property. Furthermore, the proposed method yields better quality reconstructed images than traditional methods whether with zero-padding or without zero-padding.

Moreover, we also perform the optical experiments to verify it, as shown in Fig. 15. The first and second rows of Fig. 15 are the reconstructed images by the traditional GS method and AWGS method without and with zero-padding, respectively. In Fourier holography, the size of the reconstructed image is determined by $\lambda {f_{focal}}/{\Delta }_{SLM}$, ($\lambda$, $f_{focal}$, ${\Delta }_{SLM}$ are the wavelength, focal length of Fourier lens, and sampling interval of the SLM, respectively). Therefore, the reconstructed images without zero-padding will be larger compared to the results with zero-padding, which are shown in Figs. 15(a) to (d). However, the larger image size results in some issues. The brightness of the reconstructed image would be dimmed since the energy needs to be distributed over a larger area. Another issue is the uneven brightness distribution areas of the image because of the off-axis reconstruction and larger image size, as shown in Figs. 15(a), (b), (e), and (f). In the results of the zero-padding method, the above issues can be solved. Although it decreases the image size, it can reconstruct a high-quality image with uniform brightness.

## 6. Conclusion

In this study, we have firstly analyzed firstly the reason for the instability of convergence in the conventional WGS method. To solve the problem, we propose a phase-only hologram generation method based on the adaptive weighted GS iteration method. Compared with the conventional method, our proposed method can ensure the stability of the convergence and improve the quality of reconstructed images with 4.8 dB higher on average. Moreover, we propose an approximate phase that is used to replace the random phase and the quadratic phase as the initial value for iteration, which can suppress the appearance of the artifacts in optical reconstruction. Experiment results have demonstrated that our method can decrease the background noise and speckle noise efficiently. Meanwhile, the proposed method can reconstruct the images with a better view experience in our designed the prototype of the AR system. Therefore, the proposed method has potential applications in the field of AR.

## 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.

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