## Abstract

In this paper, a curved hologram generation method with suppressed speckle noise is proposed. In the process of generating the curved hologram, the angle spectrum method is used to calculate the 3D object in layers. By analyzing the loss function relationship between the diffraction image of the curved hologram and the target light field, the loss function is calculated. The phase of the hologram is updated based on the stochastic gradient descent algorithm, thereby obtaining the optimal phase distribution of the curved hologram. When the curved hologram is reproduced at different bending center angles, the reconstructed image of the corresponding angle can be seen. The proposed method has obvious advantages compared with the error diffusion method and Fresnel diffraction algorithm. The feasibility of the proposed method is verified by the experiments.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Holographic display is an ideal 3D display technology because it can reconstruct the amplitude and phase information of the three-dimensional (3D) target by means of coded fringes [1–3]. Compared with the traditional optical holography, computer-generated holography uses a computer to calculate the wavefront information and realizes optical holographic reconstruction with a spatial light modulator (SLM) [4–5]. Therefore, the process wavefront recording is not limited by optical conditions, and dynamic holographic 3D display can be realized. In recent years, the authenticity of holographic reconstruction has attracted much attention. However, the existing holographic 3D display is difficult to meet people’s viewing needs. For example, the calculation speed of a hologram is relatively slow, which will take a long time to calculate a 3D object [6–7]. Besides, the viewing angle of the holographic display is small. The diffraction angle of the existing commercial SLM is only a few degrees, which is far from the large-viewing-angle display [8–10]. Moreover, due to the high coherence of the laser, the speckle noise exists in the reconstructed holographic image, which affects the viewing effect [11–15].

In order to obtain the holographic 3D display effect with large viewing angle, some researchers propose a method of splicing multiple SLMs, but the cost of this method is high and the required system is complex [16–17]. In addition, the method to achieve viewing angle expansion by fast refreshing a single SLM is also proposed [18–19]. This time multiplexing method has a high requirement for SLM refreshing rate. Nevertheless the abovementioned time and space multiplexing methods can only expand the viewing angle of the holographic display to a certain extent, so recently researchers have encoded curved holograms to further expand the viewing angle. In 2014, a spherical hologram generation method of real-existing object was proposed [20]. Since the cylindrical hologram can achieve 360° holographic display effect, many calculation methods based on the fast Fourier transform method and planar angular spectrum method are developed to generate the cylindrical hologram [21–24]. However, it takes a long time to generate the cylindrical hologram, and the cylindrical hologram cannot be combined with the existing curved screen. Therefore, it is still in the basic research stage. In 2019, a curved hologram generation method was proposed to expand the viewing angle of the holographic display [25]. In 2020, the acceleration algorithm based on curved holograms was also proposed [26]. As a part of cylindrical hologram, the curved hologram is easier to be combined with curved screens and flexible materials to reproduce the holographic 3D image with large viewing angle. Therefore, it has gradually become one of the hot research topics. However, due to the interference of the speckle noise, the current display quality of curved holograms urgently needs to be improved.

In previous work, we proposed a method to reduce speckle noise in curved hologram by using error diffusion [27]. However, when 3D objects with complex details are used for reproduction, the quality of the reproduced image is greatly affected by the speckle noise. Since the calculation process of curved holograms is different from that of planar holograms, there are few methods for generating the curved hologram with low speckle noise.

To solve this problem, a curved hologram generation method with suppressed speckle noise is proposed in this paper. Different from the traditional method, the key of the proposed method is to compare the target complex amplitude distribution of the 3D object with the complex amplitude reconstructed by the curved hologram, and calculate the loss function. Firstly, the angle spectrum method is used to calculate the target complex amplitude of the 3D object. Secondly, the phase distribution of the curved hologram is set to a random phase initially and the complex amplitude distribution of the planar hologram is obtained by adding the approximate phase compensation. In this way, complex amplitude reconstructed by the curved hologram can be calculated and the loss function can be obtained. Then the phase of the hologram is updated based on the stochastic gradient descent (SGD) algorithm, thereby obtaining the optimal phase distribution of the curved hologram. When the curved hologram is reproduced at the corresponding bending center angle, the reconstructed image can be seen accordingly.

The proposed method can be used not only for the curved hologram calculation of a single object, but also for the composed curved hologram of multiple objects. In the calculation process of the composed curved hologram, we do not calculate and compose the curved hologram of a single object separately. Instead, the phase distribution of the initial composed curved hologram is set to a random phase, and a method for optimizing the composed curved hologram based on the total loss function of all objects is further proposed. Finally, the composed curved hologram is optimized based on the SGD algorithm. Compared with our previous work [27], the reconstructed image by using the proposed method has higher quality, especially for complex objects, and the speckle noise is significantly reduced. Compared with the Fresnel diffraction method [25], the proposed method also has obvious advantages in the suppression of speckle noise. Through comparison, we verified the advantages and feasibility of the proposed method. The proposed method is expected to contribute to the development of holographic display.

## 2. Principle of the method

#### 2.1 SGD method for a curved hologram

In the previous studies, the loss function is usually obtained by comparing the light field of the reconstructed object obtained by hologram diffraction with the target light field [2,28]. However, it is difficult to directly calculate the light field of the reconstructed object based on the complex amplitude distribution of the curved hologram. In order to make better use of the current fast algorithm when calculating the light field of the reconstructed image, it is essentially necessary to transform the complex amplitude distribution of the curved hologram. Therefore, our idea for a single curved hologram is to convert the curved hologram of the object into a planar complex amplitude light field by using the phase approximation compensation method, so as to quickly calculate the light field of the reconstructed image through the angular spectrum method. The light field distribution of the object is compared with the target light field distribution to obtain the loss function and the complex amplitude distribution of the curved hologram is optimized. For the curved hologram of a single object, the flowchart of the proposed method is shown in Fig. 1.

Firstly, the recorded object is divided in different layers (*L*_{1}, *L*_{2}…*L _{l}*) and the complex amplitude distribution of the light field of the object is calculated by using the angular spectrum method. The phase distribution of the curved hologram is set to a random phase initially. Then, the complex amplitude distribution of the planar hologram is obtained by adding the approximate phase compensation. The light field distribution of the reconstructed image can be calculated by using the angular spectrum algorithm. After that, the total value of the loss function can be obtained by comparing the amplitude and phase of the reconstructed light field and the target light field, respectively. The SGD optimization algorithm is used to iteratively optimize the phase distribution of the curved hologram. Finally, the phase-optimized curved hologram is obtained when the loop meets the output conditions.

In the first step, we use the angular spectrum algorithm to simulate the diffraction and calculate the complex amplitude distribution of the target light field, that is, the complex amplitude distribution of the recorded object on the target plane. The recorded 3D object can be regarded as a superposition of *L* layers of different depth planes, where *L* = 1, 2, 3…, the expressions of the angular spectrum algorithm are shown as follows:

*U*

_{target}(

*x*

_{1},

*y*

_{1}) and

*U*

_{o}(

*x*,

*y*) represent the light field of the target plane and the light field of the recorded object, respectively.

*H*(

_{f}*f*,

_{x}*f*) represents the conversion equation in the angular spectrum method.

_{y}*FFT*and

*IFFT*represent Fast Fourier Transform and Fast Inverse Fourier Transform.

*k*,

*λ*and (

*f*,

_{x}*f*) represent the wave number, wavelength and spatial frequency, respectively.

_{y}*z*represents the diffraction distance. In addition, since the recorded object is treated as a superposition of

*L*layers of different depth planes, the distances from different depth layers to the target plane are also different. In the target complex amplitude calculation process, a compensate phase is used to multiply with the object at each depth plane. In order to eliminate the phase shift from different depth planes to the target plane, the phase compensation is added during the recording, as shown in Eq. (3), where

*z*represents the distance from the depth of the

_{l}*l*th layer to the target plane. Otherwise, there will exist an interference among different planes in the edge area since the phase distribution of the different depth planes on the hologram plane is different [29–30]. In this way, the reconstructed phase by the proposed method can exhibit a relatively smooth variation. However, the smooth variation of the phase implies the limited angular spectrum range of the reconstructed field.

After calculating the complex amplitude distribution of the target plane, we set up a mask to divide the light field of the target plane into the signal area and the background area, and optimize them with the SGD algorithm. The expression of the mask can be expressed as follows:

*m*and

*n*represent the sampling points number in

*x*and

*y*dimensions, respectively. Through the setting of different thresholds, the proportion of the signal area and the background area in the SGD optimization process can be controlled. Generally speaking, in order to suppress the effect of speckle noise in the background, the proportion of the signal area should be larger. In this paper, the threshold is set to 0.05.

After calculating the complex amplitude distribution of the target plane, we set the initial curved hologram with a certain bending center angle to a random phase. The curved hologram is superimposed with a phase conversion factor to convert it into a plane hologram through the phase approximation compensation method. The expression of the phase conversion factor is as follows:

Among them, *T*(*x*, *y*; *β*) represents the phase conversion factor corresponding to the curved hologram with a central angle of *β*. *z*(*x*, *y*; *β*) represents the distance between the corresponding pixel points on the curved hologram and the planar hologram. *z*_{c} represents the maximum distance of the corresponding pixel points. *U _{p}*

_{_β}and

*U*represent the complex amplitude distribution on the planar hologram and the curved hologram, respectively.

_{c}After the planar complex amplitude distribution is obtained, the angular spectrum method is used to calculate the corresponding light field distribution of the reconstructed image. Next, by comparing the complex amplitude of the reconstructed image with that of the target plane, the loss function is obtained. In the proposed method, the combination of real loss and imaginary loss is treated as the complex amplitude loss function. The expression of the loss function is expressed as follows:

*Loss*is the loss function.

*R*and

_{r}*R*represent the real part of the reconstructed image and the target plane, respectively.

_{t}*I*and

_{r}*I*represent the imaginary part of the reconstructed image and the target plane, respectively.

_{t}*A*and

_{rb}*A*represent the amplitude of the background area for the reconstructed image and target plane, respectively. The parameter

_{tb}*γ*is used to adjust the ratio between background loss and signal loss. Then, a more flexible form of the loss function can be obtained to balance the optimization direction of the SGD method. Here we need to choose a balance value, because a small value means the optimization process tends to optimize the signal, and a large value means the optimization process tends to optimize the background. Usually,

*γ*can be set from 0.1 to 1. Here, it is set to 0.1.

Finally, based on the loss function, the SGD algorithm is used to optimize the initial curved hologram in a loop. During the optimization process, the Adam optimizer can be used as an update rule. Adam is an adaptive learning rate method which only requires the first-order gradient. The first and second moments of the gradient are used to adapt to the learning rate. The first moment is the mean and the second moment is the centerless variance. After SGD optimization, by setting the output threshold, the phase distribution of the curved hologram can approach the global optimum. Since we optimize the complex amplitude distribution of the reconstructed image in this process, the reconstruction result matches the complex amplitude reconstruction result, which guarantees the quality of the reconstructed image.

#### 2.2 SGD method for a composed curved hologram

For composed curved holograms, its biggest feature is to record different 3D objects into holograms with different bending angles and composite them together. When the composed curved hologram is bent into different angles for reconstruction, the different reconstructed 3D objects can be seen, which can increase the information capacity of the hologram. For composed curved holograms that composite multiple objects, if the curved hologram of each object is calculated one by one according to the above process and then superimposed and composited, the optimization time will be increased linearly as the number of curved holograms increases. In order to solve this problem, we use a composed loss function to replace the loss function of a single curved hologram for SGD optimization. The specific process is shown in Fig. 2.

Firstly, similar to the first step of a single curved hologram, the tomography algorithm and angular spectrum method are used to calculate the composed complex amplitude distributions of target light fields of multiple 3D objects. Next, the phase distribution of the initial composed curved hologram is set to a random phase, and the composed curved hologram is bent to curved holograms with different central angles. The curved holograms with different central angles are approximately compensated for the phase to obtain the corresponding planar complex amplitude distributions. Then, the angular spectrum method is used to obtain the corresponding light field distribution of the reconstructed object. A loss function is acquired by comparing the amplitude and phase of the light field of the reconstructed object and the corresponding target light field. Each central angle of the curved hologram corresponds to a loss function, and these loss functions are superimposed to obtain a total loss function. Finally, we use the optimization loop idea of SGD to iteratively optimize the phase distribution of the composed curved hologram. The composed hologram with the phase optimization can be obtained when the output conditions of the loop are satisfied. The phase information of the composed curved hologram is extracted and loaded on the SLM. When the composed curved hologram is bent into different angles, 3D objects at different angles can be seen with reduced speckle noise.

## 3. Simulation, experiments and results

Based on the principle of the proposed method, a proof-of-principle system is built based on an SLM, as shown in Fig. 3, to verify the feasibility of the proposed method. In addition to the SLM, the proof-of-principle system also includes a laser, lens, beam splitter (BS), 4f filter and CCD camera. The wavelength of the laser is 532nm. The SLM has 1920×1080 pixels, with a pixel pitch of 6.4µm, which means the active area is 12.28mm×6.91mm. The 4f filter is composed of two lenses and a filter. When the laser passes through the lens, the collimated light is generated and illuminates the SLM after passing through the BS. The diffracted image can be received on the CCD camera after being filtered by the 4f system. The PyTorch 1.7.0 and Python 3.6.9 are used to achieve the SGD optimization. The number of iterations is 100. Since there is no curved SLM in the market now, we have to perform phase compensation on the planar SLM for curved experimental verification.

Firstly, we use a single 3D object for verification. Here, the four letters ‘A’, ‘B’, ‘C’, ‘D’ at different depth planes are used as a 3D object for experiment. The resolution of the 3D object is 1920×1080. The diffraction distances of the ‘A’, ‘B’, ‘C’, and ‘D’ are 250mm, 290mm, 330mm, and 370mm, respectively. The distance between the hologram plane and the target plane is set to 250mm. The proposed method is used to generate the curved hologram of the 3D object, and then the curved hologram is loaded onto the SLM. The resolution of the curved hologram is 1920×1080. The center angle of the curved hologram is 5°. Figures 4(a)-(d) are the reconstructed images of the 3D object when ‘A’, ‘B’, ‘C’, ‘D’ are focused respectively. To facilitate observation, the focus plane is marked with a red frame. At the same time, the error diffusion method and the Fresnel diffraction algorithm are used to generate the curved holograms for comparison. The results of the error diffusion method are shown in Figs. 4(e)-(h), and the results of the Fresnel diffraction algorithm are shown in Figs. 4(i)-(l). It can be seen from the results that the proposed method can reproduce the image of 3D objects very well. For 3D objects with simple details, the reproduced image quality of the proposed method does not seem to be much different from the error diffusion method, but compared with the Fresnel diffraction algorithm, the speckle noise is greatly suppressed.

In order to prove the advantage of the proposed method, the 3D object with complex details is used for experimental verification. Here, the ‘dragon’ and ‘train’ located in two different planes are regarded as the 3D object, and the resolution of the 3D object is 1920×1080. The diffraction distances of the ‘dragon’ and ‘train’ are 250mm and 350mm, respectively. Figures 5(a)-(b) are the reconstructed image of the 3D object when ‘dragon’ and ‘train’ are focused respectively by using the proposed method. To facilitate observation, the partial reproduced image of the object is enlarged, as shown in the red box in Fig. 5. The results of the error diffusion method and the Fresnel diffraction algorithm are also given, as shown in Figs. 5(c)-(f).

Compared with the error diffusion method, the speckle noise of the proposed method is obviously suppressed. It can be clearly seen that the reproduced image quality of the proposed method is better than those of the other two methods. In order to quantitatively analyze the quality of the reconstructed image, the peak signal-to-noise ratio (PSNR) of the reconstructed image is calculated. Here we calculate the PSNR value of the reproduced image in the red box of Fig. 5. The PSNR value of Figs. 5(a)-(f) is 7.6656, 7.3912, 7.4500, 7.3480, 6.3755 and 6.4791, respectively. Figure 6 is the simulation result of Fig. 5. The intensity distribution of the red box in Fig. 5 is shown in Fig. 7. It can also be seen from the intensity distribution that, compared with the other two methods, the intensity distribution of the reproduced image is more uniform by using the proposed method.

Next, the composed curved hologram of the two 3D objects is calculated. Among them, ‘cube’ and ‘cone’ in two different planes are regarded as 3D object 1, while ‘dragon’ and ‘train’ in two different planes are regarded as 3D object 2. The diffraction distances of the ‘dragon’ and ‘train’ are 250mm and 350mm, respectively. And the diffraction distances of the ‘cube’ and ‘cone’ are 250mm and 350mm, respectively. The resolutions of the 3D object and curved hologram are both 1920×1080. The angular spectrum algorithm is used to calculate the target complex amplitude information of 3D object 1 and 3D object 2, and then the target complex amplitude information is calculated. The phase distribution of the initial composed curved hologram is preset to a random phase, and the composed curved hologram is bent to form the curved hologram with different central angle. Among them, the bending center angles of 3D object 1 and 3D object 2 are 5° and 10°, respectively. The corresponding complex amplitude distribution of the reconstructed image can be obtained based on the phase approximate compensation. The loss functions of 3D object 1 and 3D object 2 are superimposed to obtain the total loss function. Then the SGD algorithm is used to optimize the phase distribution of the composed curved hologram, so as to obtain the optimized composed curved hologram. When the composed curved hologram is bent into 5°, the reconstructed image of 3D object 1 can be seen, as shown in Figs. 8(a)-(b), where Fig. 8(a) is the result when ‘cube’ is focused and Fig. 8(b) is the result when ‘cone’ is focused. From the results we can see that the 3D objects can be displayed correctly. When the composed curved hologram is bent into 10°, the reconstructed image of 3D object 2 can be seen. Figure 8(c) and Fig. 8(d) are the results when ‘train’ and ‘dragon’ are focused respectively.

Moreover, the results of the error diffusion method and the Fresnel diffraction algorithm have also been captured. Figures 8(e)-(h) are the results by using the error diffusion method and Figs. 8(i)-(l) are the results by using the Fresnel diffraction algorithm. When the composed curved hologram is bent into 5°, although ‘cube’ and ‘cone’ can be seen in Fig. 8(e) and Fig. 8(d) respectively, some information is missing. When the composed curved hologram is bent into 10°, ‘train’ and ‘dragon’ are focused respectively, as shown in Figs. 8(g)-(h). It is obvious that the reproduced information of ‘dragon’ and ‘train’ is lost. When the Fresnel diffraction algorithm is used for the reproduction, there is a lot of speckle noise in the reconstructed image. So, compared with the other two methods, it can be seen that when the bending angle is changed, the proposed method can correctly reproduce the object with the corresponding angle. The proposed method has obvious advantages in the calculation of holograms for complex object and composed curved holograms.

In the previous methods, the Fresnel algorithm propagates to the SLM once and use a random phase. Besides, due to the use of phase-type SLM, only the phase information of the hologram is extracted during encoding. Compared with the Fresnel algorithm, the error diffusion method optimizes the phase distribution by reducing the error between adjacent pixels, thereby suppressing the speckle noise in the reconstructed image. However, it is still difficult to meet the high-quality display requirements using error diffusion algorithms in curved holographic display, especially in composed curved holographic display. The proposed method calculates the complex amplitude information of the object, which includes not only the amplitude but also the phase. Then, the diffraction field is encoded into a pure phase hologram. As the complex amplitude coding is used and the loss function is calculated, the quality of the reconstructed image can be improved greatly.

Previously, we used the SGD algorithm in the calculation of planar holograms to achieve accelerated processing of holograms [29]. In this paper, the proposed method is to suppress the speckle noise of the curved hologram. The hologram types the process of the two methods is different, and the purpose is different accordingly. Specifically, compared with our previous work, the proposed method has the following differences.

- 1) Since the calculation process of curved holograms is different from that of planar holograms, there are few methods for generating the curved hologram with low speckle noise. It is understood that this is the first time that the SGD algorithm is used to curved holograms for speckle noise suppression. The key of the proposed method is to compare the target complex amplitude distribution of the 3D object with the complex amplitude reconstructed by the curved hologram, and calculate the loss function. So, when calculating the light field of the reconstructed image, we need to transform the complex amplitude distribution of the curved hologram by using the phase approximation compensation method, which is different from the planar hologram.
- 2) In the calculation process of the composed curved hologram, the phase distribution of the initial composed curved hologram is set to a random phase, and a method for optimizing the composed curved hologram based on the total loss function of all objects is further proposed. For composed curved holograms, its biggest feature is to record different 3D objects into holograms with different bending angles and composite them together. When the composed curved hologram is bent into different angles for reconstruction, the different reconstructed 3D objects can be seen, which can increase the information capacity of the hologram. This is completely different from the principle of planar hologram, which cannot achieve multi-angle multiplexing. The planar hologram does not contain any composed objects at all, and the generation process of the planar hologram is completely different from the calculation of the composed curved hologram in the proposed method. By calculating the total loss function of the composed curved hologram, the speckle noise of the composed objects can be suppressed, which has not been studied in our previous work.

In previous studies, the loss function is usually obtained by comparing the light field of the reconstructed object obtained by hologram diffraction with the target light field. For the composed curved hologram algorithm that composites multiple objects, optimizing the time becomes a problem. The key issue in the SGD optimization algorithm for curved hologram is how to obtain and calculate the loss function of the curved hologram. Before proposing this method, we initially considered calculating the planar hologram of the object using the SGD algorithm, and then converting the planar hologram into a curved hologram through the phase approximation compensation method, that is, the curved hologram does not participate in the loop process (Here, we call it the traditional SGD algorithm). However, when using the angle multiplexing method to calculate the curved holograms of multiple objects, the previous operation needs to be repeated, and each different object is looped according to the SGD algorithm. It also means that the optimization time will be increased linearly with the number of composite objects. In addition, since the curved hologram is obtained by approximate compensation based on the planar hologram, it is not directly involved in the loop of the SGD algorithm. Then the optimization effect of SGD on the phase does not directly act on the curved hologram, resulting in a decrease in the quality of the reconstruction of the final curved hologram.

The proposed method can improve the calculation speed while ensuring the quality of the reproduced image. The calculation time of the curved hologram is given by using different methods, as shown in Table 1. The calculation configuration used is NVIDIA GeForce GTX 1050 Ti with CUDA version 10.1. The number of iterations for the proposed method and traditional SGD algorithm is set to 100. It can be seen that when the number of the composite surface is 1, there is not much difference in calculation speed. When the number of the composite surfaces increases, the calculation speed of the proposed method is significantly faster than that of the traditional SGD algorithm. Since the algorithm used in the proposed method contains an iterative part, its calculation speed is not fast compared to the non-iterative algorithm. So, compared with the error diffusion method, the calculation time of the proposed method is longer. However, the speckle noise is greatly reduced and the display quality is improved. In the reproduction of curved hologram, the eye-box is limited. However, the filed of view of the curved hologram is effectively increased. Besides, we can encode multiple scenes into a hologram and reconstruct each of them with corresponding reference waves easily. In theory, we can encode objects at different angles onto a hologram. However, there is an upper limit on the number of scenes that can be encoded into the composed curved hologram. Due to the crosstalk between different scenes, too many scenes will affect the quality of the reconstructed image. In the next work, we will try to synthesize the composed curved holograms for more scenes, and discuss the relationship between the number of composites and the quality of the reconstructed image. We believe that with the development of technology, holographic display can show good results as soon as possible and embark on application.

## 4. Conclusion

In this paper, a curved hologram generation method with suppressed speckle noise is proposed. The angle spectrum method is used to calculate 3D object in layers. By analyzing the loss function relationship between the diffraction image of the curved hologram and the target light field, the SGD algorithm is used to obtain the optimal phase distribution of the curved hologram, thereby reduce the speckle noise. When the curved hologram is reproduced at different bending center angles, the corresponding reconstructed image can be seen, and the speckle noise is well suppressed. The proposed method has advantages in the calculation of holograms for complex object and composed curved holograms. The experiments verify the feasibility of the proposed method. We believe that the proposed method can promote the holographic display application.

## Funding

National Natural Science Foundation of China (62011540406, 62020106010).

## Acknowledgement

We would like to thank Dr. Chen Chun of Seoul National University for the experimental discussion and Nanofabrication facility in Beihang Nano for technique consultation.

## Disclosures

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

## 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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