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Reducing crosstalk of a multi-plane holographic display by the time-multiplexing stochastic gradient descent

Zi Wang, Tao Chen, Qiyang Chen, Kefeng Tu, Qibin Feng, Guoqiang Lv, Anting Wang, and Hai Ming

Open Access Open Access

Abstract

Multi-plane reconstruction is essential for realizing a holographic three-dimensional (3D) display. One fundamental issue in conventional multi-plane Gerchberg-Saxton (GS) algorithm is the inter-plane crosstalk, mainly caused by the neglect of other planes’ interference in the process of amplitude replacement at each object plane. In this paper, we proposed the time-multiplexing stochastic gradient descent (TM-SGD) optimization algorithm to reduce the multi-plane reconstruction crosstalk. First, the global optimization feature of stochastic gradient descent (SGD) was utilized to reduce the inter-plane crosstalk. However, the crosstalk optimization effect would degrade as the number of object planes increases, due to the imbalance between input and output information. Thus, we further introduced the time-multiplexing strategy into both the iteration and reconstruction process of multi-plane SGD to increase input information. In TM-SGD, multiple sub-holograms are obtained through multi-loop iteration and then sequentially refreshed on spatial light modulator (SLM). The optimization condition between the holograms and the object planes converts from one-to-many to many-to-many, improving the optimization of inter-plane crosstalk. During the persistence of vision, multiple sub-hologram jointly reconstruct the crosstalk-free multi-plane images. Through simulation and experiment, we confirmed that TM-SGD could effectively reduce the inter-plane crosstalk and improve image quality.The proposed TM-SGD-based holographic display has wide applications in tomographic 3D visualization for biology, medical science, and engineering design, which need to reconstruct multiple independent tomographic images without inter-plane crosstalk.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Computer-generated holography (CGH) is a technique that can reconstruct the whole optical wave field of a 3D scene and provide all the depth cues that the human eye can perceive [1]. It can apply to various fields that need accurate light distribution control, including optical trapping [2] and neural photo-stimulation [3–6]. Compared to the traditional 3D display based on binocular parallax, the 3D display based on CGH can provide all monocular and binocular depth cues without vergence-accommodation conflict. Thus, it is also considered as a promising solution for 3D display [7–9].

In 3D display, multiple images need to be reconstructed simultaneously. There have been several multi-plane algorithms like superposition algorithms [10] and Global Gerchberg-Saxton (GGS) [11–13]. However, the inter-plane interference causes severe crosstalk in reconstructed images, which remains a challenge to overcome [14,15]. Several researches have been reported to address this issue. Makey et al. preshaped the wavefronts of object planes to locally reduce Fresnel diffraction to Fourier holography, then broken the crosstalk limits to dynamic holography using orthogonality of high-dimensional random vectors [16]. Li et al. proposed a dynamic compensatory Gerchberg-Saxton (GS) algorithm to reduce the crosstalk in the object planes near the SLM, thus solving the images quality imbalance [17]. Zhang et al. proposed a new non-convex optimization algorithm that computed holograms by minimizing a custom cost function, reducing the crosstalk to some extent [18]. Pang et al. proposed a method based on an analytical formula with no iteration to generate phase-only holograms for low crosstalk full-color multi-plane holographic projection [19]. However, the quality improvement remains a challenge due to the imbalance between input and output information. As the number of object planes increases, this imbalance becomes more severe and degrades the quality of multi-plane reconstruction. Thus, the inter-plane crosstalk cannot be easily relieved without increasing input information.

In this paper, we proposed the TM-SGD optimization algorithm to reduce the inter-plane crosstalk of multi-plane holography display. First, the crosstalk of multi-plane GS was demonstrated and analyzed. The main reason was the neglect of other planes’ interference when enforcing amplitude replacement at one object plane. Secondly, the SGD was used to reduce inter-plane crosstalk and improve image quality. Although SGD has been used to calculate the multi-plane hologram [20–26], it is not used to solve the inter-plane crosstalk problem. Here, we verified that SGD could reduce the crosstalk among object planes to a certain extent when the objects are unrelated discrete planes. However, as the number of object planes increased, its optimization effect would degrade due to the imbalance between input and output information. Finally, we further proposed TM-SGD to reduce the inter-plane crosstalk of multi-plane holographic display. In this method, multiple sub-holograms were obtained through multi-loop iteration and then sequentially refreshed on SLM. During the persistence of vision [27], the amplitude information of multi-plane images is reconstructed by those sub-holograms together. Through simulation and experiment, we confirmed that TM-SGD could significantly reduce the crosstalk among multiple planes and improve images quality.

2. Comparison between multi-plane GS and multi-plane SGD

2.1 Multi-plane GS algorithm

Fig. 1 shows the workflow of a commonly used multi-plane GS algorithm, also called the GGS [11–13]. Here, the number of object planes and iterations is set to N and k, respectively, and the target amplitude in n-th object plane is $A_{n}^{target}$ (n = 1…N). H1 is the initial hologram of GGS, with a random phase φ0 and a uniform amplitude. The process of its iteration loop is expressed as the followings:

$${H_k} = \left\{ \begin{array}{l} \exp (i{\phi_0}),k = 1\\ H_{k\textrm{ - }1}^{^{\prime\prime}} = \exp (i\phi_{k\textrm{ - }1}^H),k > 1 \end{array} \right.$$
$${U_{(k\textrm{,n)}}} = ASM({H_k},{z_n}) = {A_{(k,n)}} \cdot \exp (i{\phi _{(k,n)}})$$
$$U_{(k\textrm{,n)}}^{constraint} = A_n^{target} \cdot \exp (i{\phi _{(k,n)}})$$
$$H_k^{\prime} = \frac{1}{N}\sum\limits_n {ASM(U_{(k,n)}^{constraint}, - {z_n})} = A_k^H \cdot \exp (i\phi _k^H)$$
$$H_k^{^{\prime\prime}} = \exp (i\phi _k^H)$$
where ASM represents the angular spectrum method [28]. In k-th loop, the hologram Hk propagates to object planes and gets complex amplitude U(k,n) as shown in Eq. (2), in which znis the distance between n-th object plane and SLM. Eq. (3) denotes the amplitude constraint enforcement in n-th object plane, where A(k,n) (the amplitude of U(k,n)) is replaced by target amplitude $A_{n}^{target}$. After amplitude constraint, the complex amplitude $U_{(k,n)}^{constraint}$ is inverse transformed back to the hologram plane. In the last step, the amplitude of hologram H’ k is set back to a constant to ensure pure phase modulation. Then the output hologram H’’ k will serve as the starting point for the (k + 1)-th loop.

 figure: Fig. 1.

Fig. 1. Scheme of a commonly used multi-plane GS algorithm.

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Fig. 2(a) shows the simulation and experimental results of two-plane GS, and the distances from object planes to SLM were 25 cm and 35 cm, respectively. In multi-plane display, the relevance between adjacent planes influences the crosstalk. When two object planes are identical, there is little crosstalk between them. And the crosstalk increases as the relevance between them decreases. The crosstalk among object planes is embodied in the defocused images. We can see that the reconstructed images were obviously influenced by the defocused images of other planes when using multi-plane GS. The main reason was that the inter-plane influence was not considered when enforcing amplitude replacement on each object plane, which lacked overall consideration and was easy to fall into the local optimal solution. The inter-plane influence is related to the relevance between adjacent planes. In addition, the amplitude replacement on SLM resulted in the loss of image information. We can see that the inter-plane crosstalk severely degraded the visual effect of the multi-plane display.

 figure: Fig. 2.

Fig. 2. Simulation and experimental results of (a) multi-plane GS and (b) multi-plane SGD.

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2.2 Multi-plane SGD algorithm

To solve the problem that multi-plane GS independently optimizes each object plane [11–13], we applied SGD to optimize the process as shown in Fig. 3. The multi-plane SGD algorithm begins with an initial hologram H1, which has a random phase φ0 and a uniform amplitude. By defining a loss function in the optimization process, the gradient of the loss function can be used to update the initial phase. The initial phase is then propagated to the target plane, and the loss value can be obtained again. The process of its iteration loop is expressed as the followings:

$${H_k} = \left\{ \begin{array}{l} \exp (i{\phi_0}),k = 1\\ \exp (i{\phi_k}),k > 1 \end{array} \right.$$
$${U_{(k\textrm{,n)}}} = ASM({H_k},{z_n}) = {A_{(k,n)}} \cdot \exp (i{\phi _{(k,n)}})$$
$$Los{s_k} = \sum\limits_n {loss(A_{(k,n)}^{},A_n^{target})} = \sum\limits_n {MSE(A_{(k,n)}^{},A_n^{target})} = \sum\limits_n {[\frac{1}{{XY}}\sum\limits_{x,y} {{{(A_{(k,n,x,y)}^{} - A_{(n,x,y)}^{target})}^2}} } ]$$
$${\phi _{k + 1}} \leftarrow {\phi _k}\textrm{ - }\alpha {(\frac{{\partial Los{s_k}}}{{\partial \phi }})^T}Los{s_k}$$
where the learning rate α is set as 1, which determines the iteration speed. In k-th loop, the hologram Hk propagates to object planes and gets the complex amplitude U(k,n) as shown in Eq. (7). The amplitude A(k,n) is extracted and compared with the target amplitude $A_{n}^{target}$ to calculate the loss function of each object plane. The overall loss Lossk is obtained by adding the loss of different depth planes. In this process, the mean square error (MSE) function is used to calculate the loss function between the reconstructed image and the object image. X is the number of vertical pixels in the images, and Y is the number of horizontal pixels in the images. In reconstructed image, A (k,n,x,y) is the amplitude of the pixel in row x and column y. Then the gradient of the loss function is calculated using automatic differentiation, and the hologram is optimized according to it, as shown in Eq. (9).

 figure: Fig. 3.

Fig. 3. Scheme of the multi-plane SGD algorithm.

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In multi-plane GS, the inter-plane crosstalk is mainly caused by the defocused information of object planes and related to the relevance between adjacent planes. Multi-plane SGD takes that interference into the calculation process of the loss function and takes the relevance into account. Thus, the inter-plane crosstalk is reduced through the global optimization, as shown in Fig. 2(b). By contrast, multi-plane GS performed amplitude replacement simply and independently, without considering the inter-plane interference and leading to the crosstalk in reconstructed images, as shown in Fig. 2(a). At the same time, the peak signal-to-noise ratio (PSNR) of multi-plane SGD was improved by 2.48 dB and 1.973 dB compared to multi-plane GS, and the structural similarity index measurement (SSIM) was increased by 0.21 and 0.198, respectively.

Next, we studied the relationship between image quality and the output plane number. Figure 4(a) shows 6 independent target images with 10 cm intervals. Fig. 4(b) shows that as the output plane number increases, the average PSNR decreases. Figs. 4(c)–4(d) are the simulation results of three object planes and six object planes, respectively. We can see the inter-plane crosstalk becomes more apparent as the object plane number increases. That is because only one hologram corresponds to the multiple object planes in the iteration process of multi-plane SGD, thus leading to the imbalance between input and output information. The increase of object planes’ number would worsen this imbalance, reducing the crosstalk optimization effect.

 figure: Fig. 4.

Fig. 4. (a) Diagram of multi-plane holographic display, where six independent images were apart from 10 cm interval; (b) The reconstructed results’ average PSNR changed with the increase of object planes’ number; (c) Simulation results of the three object planes; (d) Simulation results of the six object planes.

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3. TM-SGD algorithm

To solve the problem of insufficient input information mentioned above, we proposed the TM-SGD algorithm with the time-multiplexing strategy applied to both iteration and reconstruction of multi-plane SGD.

Figure 5(a) shows the workflow of this algorithm. Its iteration process has multiple loops, and each loop optimizes one sub-hologram, propagating separately to object planes. When the number of sub-holograms and iterations are m and k, its process can be described below.

$$H_k^m = \left\{ \begin{array}{l} \exp (i\phi_0^m),k = 1\\ \exp (i\phi_k^m),k > 1 \end{array} \right.$$
$$U_{(k,n)}^m = ASM(H_k^m,{z_n}) = A_{(k,n)}^m \cdot \exp (i\phi _{(k,n)}^m)$$
$${A_{(k,n)}} = \sqrt {\frac{1}{M}\sum\limits_m^M {{{\left| {A_{(k,n)}^m} \right|}^2}} }$$
$$Los{s_k} = \sum\limits_n {loss(A_{(k,n)}^{},A_n^{target})} = \sum\limits_n {MSE(A_{(k,n)}^{},A_n^{target})} = \sum\limits_n {[\frac{1}{{XY}}\sum\limits_{x,y} {{{(A_{(k,n,x,y)}^{} - A_{(n,x,y)}^{target})}^2}} } ]$$
$$\phi _{k + 1}^m \leftarrow \phi _\textrm{k}^m\textrm{ - }\alpha {(\frac{{\partial Los{s_k}}}{{\partial {\phi ^m}}})^T}Los{s_k}$$
where $\varphi_{m}^{0}$ is the initial phase of the m-th sub-hologram. In k-th loop, the m-th sub-hologram $H_{k}^{m}$ propagates to object planes and gets the complex amplitude $U_{(k,n)}^{m}$ as shown in Eq. (11). In the n-th object plane, the amplitude of all sub-holograms’ reconstructed images are synthesized into the synthetic amplitude A(k,n) as shown in Eq. (12). The loss function of each object plane is calculated through the synthetic amplitude and target amplitude. The overall loss Lossk is obtained by adding the loss of different depth planes. Then the hologram is optimized according to it, as shown in Eq. (14). Compared to SGD, TM-SGD applies the time-multiplexing strategy in the process of iteration and reconstruction to increase the degree of freedom in the time dimension. In TM-SGD, each iteration process has multiple loops, and each loop optimizes one sub-hologram. Thus the optimization condition between the holograms and the object planes converts from one-to-many to many-to-many. Then those sub-holograms are sequentially refreshed on SLM in the reconstruction process, and their wavefronts are incoherently superimposed in the object planes to increase the input information. With a high-refresh-rate SLM, multiple sub-holograms’ diffraction amplitudes can be synthesized into a total amplitude distribution during the persistence of vision [27]. As shown in Fig. 5(b), TM-SGD could reduce the inter-plane crosstalk and improve the reconstructed image quality compared to multi-plane SGD. Two sub-holograms were generated for the reconstruction. In addition, the PSNR of the reconstructed images was improved by 7.467 dB and 7.946 dB, and the SSIM was increased by 0.085 and 0.129, respectively.

 figure: Fig. 5.

Fig. 5. (a) Scheme of the TM-SGD algorithm; (b) Simulation and experimental results of TM-SGD.

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Figure 6 shows the reconstruction images optimized by multi-plane GS, multi-plane SGD, and TM-SGD after different iterations number: 100, 500, 1000, and 2000. When the number of iterations is insufficient, the reconstruction quality of multi-plane GS is better than that of multi-plane SGD and TM-SGD. However, the reconstruction quality of multi-plane GS will quickly converge to a relatively low value. By contrast, the reconstruction quality of multi-plane SGD and TM-SGD are much better than that of multi-plane GS after enough iterations. In addition, although multi-plane SGD can improve the quality of reconstructed images, the inter-plane crosstalk is still apparent. TM-SGD can significantly reduce the inter-plane crosstalk and improve the quality of reconstructed images, which is much better than that of multi-plane SGD, with the PSNR increasing by about 7.467 dB and 7.946 dB.

 figure: Fig. 6.

Fig. 6. Reconstruction images optimized by multi-plane GS (line 1), multi-plane SGD (line 2) and TM-SGD (line 3) after iterations of (a) 100, (b) 500, (c) 1,000, and (d) 2,000.

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In addition to the above binarization images, we also did experiments of grayscale images whose propagation distances were set as 25 cm and 40 cm, respectively, and the resolution of object planes and holograms was 3000 × 2160. The output holograms were written into SLM illuminated with a collimated laser beam (λ = 532 nm). And the SLM was pure-phase-modulated, with 256 grey levels and a pixel size of 3.6 µm × 3.6 µm. In addition, the refresh rate of this SLM is 120 Hz.

We can see that our method is still valid for reducing crosstalk of grayscale images. As shown in Fig. 7, the PSNR of multi-plane SGD’s reconstructed images was increased by 4.499 dB and 2.635 dB compared to multi-plane GS, and the SSIM was increased by 0.208 and 0.190, respectively. In addition, the PSNR of TM-SGD was improved by 5.675 dB and 6.490 dB compared to multi-plane SGD, and the SSIM was increased by 0.043 and 0.060, respectively.

 figure: Fig. 7.

Fig. 7. Simulation and experimental results of two object planes.

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To better demonstrate the crosstalk elimination, a four-plane reconstruction was shown in Fig. 8. Figure 8(a) shows the target object images, and Fig. 8(b)–8(d) are the simulation and experimental results of the three algorithms. The propagation distances were set as 20 cm, 30 cm, 40 cm, and 50 cm, respectively. In conventional multi-plane GS, the reconstruction results almost loses their image features due to the defocused crosstalk of other planes. For example, the letter A on plane 1 is barely visible, and the background has obvious defocus information, as shown in Fig. 8(b). By contrast, the reconstruction images of multi-plane SGD has better visibility, with the PSNR averagely increased by 4.334 dB. But those results still has a certain amount of inter-plane crosstalk, worsening the visual effects. We can see that the letter A on plane 1 has a more distinct outline, but the background is still affected by the defocus crosstalk of other planes, as shown in Fig. 8(c). In TM-SGD, the SGD is combined with the time-multiplexing strategy. Multiple sub-holograms are sequentially refreshed on SLM to reconstruct the object images together. We can see that the inter-plane crosstalk is significantly reduced, with PSNR averagely increased by 6.781 dB compared to multi-plane SGD. As shown in Fig. 8(d), the letter A on plane 1 is clear, and the background is uniform with little crosstalk.

 figure: Fig. 8.

Fig. 8. (a) Object images; Simulation and experimental results of (b) multi-plane GS, (c) multi-plane SGD and (d) TM-SGD.

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Figure 9 shows the simulation results of three algorithms in full color, and the distances from object planes to SLM are 20 cm and 23 cm, respectively. The resolution of the images are 1920 × 1080. We can see that our method is still valid for reducing the inter-plane crosstalk of color images, especially the part in the red box. The reconstructed images were obviously influenced by the defocused images of another plane when using multi-plane GS. And the inter-plane crosstalk was reduced to a certain extent through SGD. In contrast, TM-SGD could reduce the inter-plane crosstalk obviously and improve the reconstructed image quality compared to multi-plane SGD. At the same time, the PSNR of multi-plane SGD’s reconstructed images was increased by 2.932 dB and 3.704 dB compared to multi-plane GS, and the SSIM was increased by 0.232 and 0.297, respectively. In addition, the PSNR of TM-SGD was improved by 4.509 dB and 4.456 dB compared to multi-plane SGD, and the SSIM was increased by 0.094 and 0.072, respectively.

 figure: Fig. 9.

Fig. 9. Simulation results of three algorithms in full color.

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One limitation of TM-SGD is the refresh rate, which needs to be fast enough to refresh the sub-holograms. Fortunately, the high refresh rate SLM will be commercially available in the near future with the development of ferroelectric liquid crystal [29–31].

Note that our work is independently developed from the recent works [21,32]. Although the previous works also used the TM technique, the purpose, working condition and technical effect are different. First, The purpose of the TM technique in [32] was to eliminate the speckle noise of random phase. It is known that each frame has an independent speckle pattern, and multiple frames will have an average effect and reduce the speckle noise. The purpose of the TM technique in [21] was to extend AI-driven CGH algorithms to operate with emerging fast but heavily quantized phase SLMs. In comparison, the TM technique in our paper was proposed for inter-plane crosstalk optimization by increasing input information, not for speckle noise and special hardwares. Secondly, the time sequence number in this paper is much less than previous works, due to its different purpose. Thirdly, In previous works [32], the TM technique was used to a binary hologram. Compared with a conventional 8-bit hologram, the gray scale was decreases a lot. Thus, the input information of such a TM technique is not certainly more than conventional 8-bit hologram. However, we used the TM technique to an 8-bit hologram, thus, the input information is certainly increased. The core contributions of this paper can be summarized to two aspects. The first is that the cause of inter-plane crosstalk in conventional multi-plane GS is explained theoretically and we consider the global optimization of SGD as the effective solution. The second is that we reveal the crosstalk optimization effect is limited by the imbalance between input and output information. And we propose that the increase of input information can be the effective solution for improving crosstalk optimization effect. In our paper, why and how to use TM-SGD for eliminating crosstalk are studied and demonstrated clearly. However, in [32], the crosstalk was reduced by existing random phase method [16], and the TM technique was used for reducing speckle noise.

It is noted that various previous works focused on the hologram generation of 3D focal stack [33–38]. In this kind of hologram generation, the target images in each focal plane have little overlap in the same lateral position, because the back planes will be occluded by the front planes in holographic recording procedure. Thus, the defocused images of planes will not affect the reconstruction quality. On the contrary, the research object of our paper is the optimization of multiple independent images. Different plane images will overlap in the same lateral position, and the reconstruction quality of one plane will be affected by the defocused images of other planes. Then severe inter-plane crosstalk will appear and affect the quality of multi-plane reconstruction.

This kind of holograms has wide applications in biology, medical science and engineering design. For example, to achieve the tomographic 3D visualization of a biological tissue [39], multiple independent tomographic images needs to be reconstructed at different depths with no occlusion. In order to achieve a high-quality tomographic visualization, it is necessary to reconstruct multiple independent plane images and reduce the inter-plane crosstalk.

4. Conclusion

In conclusion, we proposed the TM-SGD algorithm to reduce the inter-plane crosstalk of multi-plane holography display. First, the global optimization feature of SGD was utilized to reduce the inter-plane crosstalk. However, only one hologram corresponds to the multiple object planes in the iteration process of multi-plane SGD, leading to the imbalance between input and output information. In addition, the increase in object planes’ number would worsen this imbalance, reducing the crosstalk optimization effect. Thus, we further introduced the time-multiplexing strategy into the iteration and reconstruction process of multi-plane SGD to increase the optimization degree of freedom. In TM-SGD, each iteration process has multiple loops, and each loop optimizes one sub-hologram. The crosstalk optimization condition between the holograms and the object planes converts from one-to-many to many-to-many, increasing the input information of the system. The amplitude information of the object images is reconstructed by those sub-holograms together during the persistence of vision. Through simulation and experiment, we confirmed that TM-SGD could reduce the inter-plane crosstalk, and the reduced impact of defocus images essentially enhanced the image quality. The proposed TM-SGD-based holographic display has wide applications in tomographic 3D visualization for biology, medical science, and engineering design.

Funding

National Natural Science Foundation of China (61805065, 62275071); Major Science and Technology Projects in Anhui Province (202203a05020005); Fundamental Research Funds for the Central Universities (JZ2021HGTB0077).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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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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Figures (9)
Fig. 1.
Fig. 1. Scheme of a commonly used multi-plane GS algorithm.

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Fig. 2.
Fig. 2. Simulation and experimental results of (a) multi-plane GS and (b) multi-plane SGD.

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Fig. 3.
Fig. 3. Scheme of the multi-plane SGD algorithm.

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Fig. 4.
Fig. 4. (a) Diagram of multi-plane holographic display, where six independent images were apart from 10 cm interval; (b) The reconstructed results’ average PSNR changed with the increase of object planes’ number; (c) Simulation results of the three object planes; (d) Simulation results of the six object planes.

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Fig. 5.
Fig. 5. (a) Scheme of the TM-SGD algorithm; (b) Simulation and experimental results of TM-SGD.

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Fig. 6.
Fig. 6. Reconstruction images optimized by multi-plane GS (line 1), multi-plane SGD (line 2) and TM-SGD (line 3) after iterations of (a) 100, (b) 500, (c) 1,000, and (d) 2,000.

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Fig. 7.
Fig. 7. Simulation and experimental results of two object planes.

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Fig. 8.
Fig. 8. (a) Object images; Simulation and experimental results of (b) multi-plane GS, (c) multi-plane SGD and (d) TM-SGD.

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Fig. 9.
Fig. 9. Simulation results of three algorithms in full color.

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Equations (14)

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

H k = { exp ⁡ ( i ϕ 0 ) , k = 1 H k  -  1 ′ ′ = exp ⁡ ( i ϕ k  -  1 H ) , k > 1
U ( k ,n) = A S M ( H k , z n ) = A ( k , n ) ⋅ exp ⁡ ( i ϕ ( k , n ) )
U ( k ,n) c o n s t r a i n t = A n t a r g e t ⋅ exp ⁡ ( i ϕ ( k , n ) )
H k ′ = 1 N ∑ n A S M ( U ( k , n ) c o n s t r a i n t , − z n ) = A k H ⋅ exp ⁡ ( i ϕ k H )
H k ′ ′ = exp ⁡ ( i ϕ k H )
H k = { exp ⁡ ( i ϕ 0 ) , k = 1 exp ⁡ ( i ϕ k ) , k > 1
U ( k ,n) = A S M ( H k , z n ) = A ( k , n ) ⋅ exp ⁡ ( i ϕ ( k , n ) )
L o s s k = ∑ n l o s s ( A ( k , n ) , A n t a r g e t ) = ∑ n M S E ( A ( k , n ) , A n t a r g e t ) = ∑ n [ 1 X Y ∑ x , y ( A ( k , n , x , y ) − A ( n , x , y ) t a r g e t ) 2 ]
ϕ k + 1 ← ϕ k  -  α ( ∂ L o s s k ∂ ϕ ) T L o s s k
H k m = { exp ⁡ ( i ϕ 0 m ) , k = 1 exp ⁡ ( i ϕ k m ) , k > 1
U ( k , n ) m = A S M ( H k m , z n ) = A ( k , n ) m ⋅ exp ⁡ ( i ϕ ( k , n ) m )
A ( k , n ) = 1 M ∑ m M | A ( k , n ) m | 2
L o s s k = ∑ n l o s s ( A ( k , n ) , A n t a r g e t ) = ∑ n M S E ( A ( k , n ) , A n t a r g e t ) = ∑ n [ 1 X Y ∑ x , y ( A ( k , n , x , y ) − A ( n , x , y ) t a r g e t ) 2 ]
ϕ k + 1 m ← ϕ k m  -  α ( ∂ L o s s k ∂ ϕ m ) T L o s s k
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