Min Fan | Jin Yang | Chengtao Du* | Jie Fang | Haibo Wang | Chenlu Li
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
Reconstruction of target images from phase-only hologram (POH) has the advantages of high diffraction efficiency and no conjugate terms. The Gerchberg-Saxton (GS) algorithm is a classical algorithm applied to recover the phase, but it most likely stagnantes after a few iterations. This paper proposes a hybrid iterative algorithm of Amplitude Weighting and Phase Gradient Descent (AW-PGD) to generate a higher-quality POH. Firstly, the quadratic phase is used as the initial phase, zero-pads the periphery of the target image, and then multiplies the two to form the complex amplitude as the iterative initial value. During iteration, the amplitude of the reconstructed image is constrained by an adaptive dynamic exponential term in the signal region to improve the reconstruction accuracy, the constraint in the non-signal region is relaxed to reduce the computational effort at the same time; and the phase gradient descent technology is used to increase the iteration step and speed up the convergence. Finally, the target image amplitude is reconstructed based on the generated POH. The numerical simulation results show that the algorithm does not have a significant increase in time cost with better reconstruction quality than the GS, Weighted GS (WGS) and Adaptive Weighted GS (AWGS) algorithm.
computer-generated hologram, phase-only hologram, phase gradient descent, quadratic phase
Since the birth of holography more than 60 years ago [1], it has evolved from traditional laser holography to today's Computer-Generated Holograms (CGHs) [2, 3]. CGHs don’t require special holographic recording materials, but instead use Spatial Light Modulators (SLMs) to load CGHs. SLMs are highly flexible and are widely used in beam shaping [4], holographic displays [5] and atom trapping [6]. Since the current mainstream commercial SLMs can only modulate either the amplitude or the phase of the light wave, but not both directly, only phase-only or amplitude-only hologram can be loaded on the SLM [7]. Phase-Only Hologram (POH) has a higher diffraction efficiency than Amplitude-Only Hologram (AOH) and doesn’t produce conjugate images [8], making POH an effective method for reconstructing high quality holograms.
The POH is a non-convex problem derived from complex amplitude images and there is no unique exact solution. Only an optimal solution under certain conditions can be obtained, and the quality of the POH will constrain the quality of the reconstructed image [9, 10]. Currently, there are two typies algorithms for generate POH, one is the iterative algorithms and the other is non-iterative algorithms. Non-iterative algorithms include phase mask-like methods [11-14], where the target image is multiplied with a phase mask on the image plane and then the phase of the complex amplitude on hologram plane is taken to reconstruct the target image, where the quadratic mask method mentioned in the references [15, 16] can obtain a higher signal-to-noise ratio. Other non-iterative algorithms, such as the Sampled Phase-Only Hologram algorithm [17] and the Complementary Phase-Only Hologram algorithm [18] are available for ensuring reconstruction quality still, they are computationally intensive and require a high-performance SLM. These non-iterative algorithms can obtain the desired POH in one step, but the quality of the reconstructed image is not as high.
Iterative algorithms propagate light wave over the hologram plane and the image plane by the Fourier Transform (FT) and Inversion Fourier Transform (IFT), optimizing the calculation results based on the measurable amplitude distribution in the two planes and the constraints imposed on the two planes to obtain the phase information of the light field in the hologram plane. The Gerchberg-Saxton (GS) [19] algorithm, proposed in 1972, is a classical iterative algorithm in the field of phase extraction and can achieve high-quality reconstructed images. But it suffers from the problem of falling into optimal local solution and slow convergence, i.e., increasing the number of iterations won’t lead to better reconstruction quality. In addition, using a random phase as the initial phase causes unexpected interference between adjacent sampling points in the reconstructed image based on POH, resulting in uneven line and edge levels and speckle noise in the reconstructed image. Thus, iterative algorithms need to be optimized in suppressing speckle noise and improving reconstruction quality. In terms of suppressing speckle noise, improvements can be made in the initial phase setting, with non-random phases such as the quadratic phase and the conical phase used as the initial value of the phase iteration in some studies [20, 21]. To avoid the speckle noise caused by the random initial phase, in 2021 Chen et al. proposed an initial phase with bandwidth constraint, which ensures the bandwidth constraint of the reproduced optical field and distributes the whole signal energy as evenly as possible to the hologram region [22]. Pang et al. proposed to use the quadratic phase as the initial phase, and was able to determine the relevant parameters of the quadratic phase according to the imaging and geometric relationships of the lens, thus significantly improving the scattering noise, but there are optical artifacts in the reconstructed image [23]. In the same year, Wu Y. proposed an approximate phase, using a set of plane waves with discrete orientations to represent the approximate quadratic phase, which can effectively suppress the appearance of speckle noise and artefacts [24]. In terms of improving the reconstruction quality and convergence speed, constraints need to be continuously optimised. For example, Fienup proposed to improve the convergence speed by introducing a difference between the target image and the computed image in 1982, but the noise suppression parameters need to be manually controlled [25]. To better tackle this problem, the Weighting GS (WGS) algorithm for kinoform implemented with POH was proposed by Alexander Kuzmenko in 2011, where the computed amplitude is multiplied by the weighting coefficients in each iteration to improve convergence [26]. But this method suffers from feedback instability. To solve this problem, reference [24] proposed the Adaptive Weighted GS (AWGS) algorithm with a flexible setting of amplitude constraints, which optimizes the weighting coefficients from the structure of proportional terms to the exponential terms to avoid feedback instability. In contrast to the previous approaches which is imposing amplitude constraints on the whole target image, some scholars have adopted the approach of dividing the target image into signal and non-signal regions, where different amplitude constraint strategies are adopted in the different regions in order to improve the reconstruction quality [27-28]. In addition to the above amplitude constraint-based algorithms, scholars have also proposed methods to constrain the phase as well. For example, in 2017, Wang combined gradient descent and weighting techniques with GS algorithm and applied them to Diffractive Optical Element (DOE) and CGH to demonstrate the effectiveness of this method [29]. Another example is Double-Constraint GS (DBGS) algorithm, which suppresses speckle noise [30]. This algorithm constrains the amplitude and phase in the signal region only. However, these algorithms still suffer from insufficient reconstruction accuracy. Therefore, the methods need to be optimized for higher reconstruction quality without increasing the complexity of the optical path system.
This paper proposes a hybrid constrained iterative algorithm for generating POH. In this algorithm, zero-pads the periphery of the target image, which is thus divided into signal and non-signal regions, and multiplied with the quadratic phase to form the initial complex amplitude. An exponential amplitude constraint is applied on the reconstructed image to improve the reconstruction accuracy, and a gradient descent technology is used on the phase of hologram to accelerate the convergence speed. Numerical experimental results show that the algorithm proposed in this paper can effectively accelerate the convergence and improve the quality of the reconstructed images.
This paper presents an optimization algorithm based on the GS algorithm, whose calculated POH can be loaded into the SLM in the Fourier hologram projection system to realize hologram projection. Here is an introduction to the principle of Fourier hologram projection and the GS algorithm.
2.1 Principle of Fourier hologram projection
The principle of Fourier hologram projection is shown in Figure 1. The hologram plane is placed on the front focal plane of the Fourier lens, and the image plane is on the back focal plane. The SLM of the hologram plane is loaded the POH firstly,which is carrying all the information of the target image. The plane light is incident vertically onto the hologram plane and modulated by the SLM. The light then passes through the Fourier lens for FT. The intensity distribution of the target image is displayed on the image plane of the back focal plane of the lens.
Figure 1. Schematic for Fourier hologram projection system
2.2 Principle of generating POH based on GS method
The GS algorithm is a classical iterative phase retrieval algorithm. Figure 2 presents its fundamental principle. The steps are as follows: 1) Assume $\mu_0$ as the initial phase value and form the initial complex amplitude $u_0=1 \cdot \exp \left(j \mu_0\right)$ into the iteration; 2) Propagate it through FT to the image plane to obtain $U_1$; 3) Replace its amplitude with the target image amplitude $\left|A_{t \text { arget }}\right|$ to obtain $U_1^{\prime}$; 4) Propagate $U_1^{\prime}$ to the hologram plane through IFT to obtain $u_1^{\prime}$; 5) Replace the amplitude of $u_1^{\prime}$ by 1 as the initial value for the next iteration. Repeat steps 2-5 until the quality of the reconstructed image meets the requirements, where uk is the complex amplitude of the hologram after k iterations, its phase is POH, and the amplitude of Uk is the reconstructed image amplitude.
Although the GS algorithm can generate higher quality POH, it still suffers from slow convergence, falling into local minima, insufficient reconstruction accuracy and feedback instability. Many scholars have studied many improved algorithms based on the GS algorithm, such as the WGS algorithm and the AWGS method. Figure 3 shows a comparison of the Root Mean Squared Error (RMSE) curves of the GS, WGS and AWGS algorithm, in which the horizontal coordinate is the number of iterations and the vertical coordinate is the value of RMSE. It can be seen that the WGS algorithm suffers from instability, and AWGS algorithm converges faster and has better reconstruction quality than GS algorithm.
The three methods are all based on amplitude constraints. There is still room to improve the reconstruction quality. If the generated phase is also constrained on the hologram plane, it will lead to better reconstruction quality. In this paper, we combine amplitude weighting strategy and gradient descent technology to propose an Amplitude Weighting and Phase Gradient Descent (AW-PGD) hybrid algorithm.
Figure 2. Principle of generating POH based on GS algorithm
Figure 3. Comparison of RMSE of GS, WGS and AWGS algorithm
The proposed AW-PGD hybrid algorithm consists of two parts. In the first part, zero-padding is applied around the target image to improve the reconstruction quality. The black frame formed by zero-padding is called the non-signal region, and the target image region inside the black frame is called the signal region. The image after zero-padding is called the target image, as shown in Figure 4. The purpose of zero-padding is to move the background noise from the signal region to the non-signal region for a high Signal-to-Noise Ratio (SNR) in the reconstructed image [31].
The second part is the hybrid constraint iteration. The hybrid constraint reflects two aspects. One is the feedback constraint of the exponential term is applied on the magnitude of the reconstructed image on the image plane to improve the reconstruction accuracy. The other is the phase of hologram is constrainted by gradient descent technology to increase the iteration step to speed up the convergence. Figure 5 presents the block diagram of the algorithm.
Figure 4. The image is divided into signal region and non-signal region by zero-padding
Figure 5. Scheme of amplitude weighting and phase gradient descent (AW-PGD) hybrid algorithm
$A_0$ is the initial complex amplitude of the image plane, which can be expressed as follows:
$A_0=\left|A_{\text {targ } e t}\right| \cdot \exp (j \phi)$ (1)
where, $\left|A_{\text {target }}\right|$ is the intensity of the target image containing the zero-padding region. Since multiplying the target image with a quadratic phase mask has the effect of removing some of the speckle noise, ϕ takes the quadratic phase, the expression of which is given as Eq. (2):
$\phi=a m^2+b n^2$ (2)
here, m and n are the coordinates of the signal region; a and b are the parameters between 0 and 1. According to reference [32], a and b can take the values:
$a \approx \frac{\pi}{M}$ (3)
$b \approx \frac{\pi}{N}$ (4)
M and N are the numbers of horizontal and vertical pixel points in the signal region respectively.
a0 is the complex amplitude of A0 propagated to the hologram plane after IFT. Only the phase of a0 is taken to obtain a1, starts the iterative process. k denotes the number of iterations, and k starts from k=1. The iterative process is as follows:
Step 1: ak represents the complex amplitude of hologram of k-th iteration, which phase distribution is $\varphi_k$. ak is propagated through FT to the image plane to obtain the complex amplitude Ak.
Step 2: Replace the amplitude of Ak with $\left|A_{o b}\right|$ to become $A_k^{\prime}$.
Step 3: $A_k^{\prime}$ is propagated back to the hologram plane via IFT to obtain the complex amplitude $a_k^{\prime}$, $\psi_k$ is the phase distribution of $a_k^{\prime}$, and $\psi_k$ becomes $\varphi_{k+1}$ after a gradient descent to the next iteration.
The amplitude of $A_k$ is the reconstructed image and the phase of $a_k$ is the desired POH. Repeat the above steps until the quality of the reconstructed image or the number of iterations meets the requirements.
In the above iterative process, the amplitude constraint of $A_k$ on the image plane is
$\left|A_{o b}\right|=\left\{\begin{array}{c}\left|A_{\text {target }}\right| \cdot e^{\left|A_{target}\right|-\left|A_k\right|} \mid, \quad \text { signal region } \\ \left|A_k\right|, \quad \text { non-signal region }\end{array}\right.$ (5)
This gives the complex amplitude of the image plane after the amplitude constraint $A_k^{\prime}$:
$A_k^{\prime}=\left|A_{o b}\right| \frac{A_k}{\left|A_k\right|}$ (6)
After IFT of $A_k^{\prime}$, we got $a_k^{\prime}=f_k * \exp \left(j \psi_k\right)$, where, $f_k$ is the amplitude of the hologram in the k-th iteration and $\psi_k$ is the phase of the hologram before gradient descent in the k-th iteration, using the gradient descent technology to act with the phase for faster convergence. The phase $\varphi_{k+1}$ after gradient descent is defined as Eq. (7):
$\varphi_{k+1}=\psi_k+\beta_k \gamma_k$ (7)
where, $\beta_k$ is the direction of the gradient, defined as the difference between $\psi_k$ and $\psi_{k-1}, \psi_{k-1}$ is the the phase of the hologram before gradient descent in previous iteration:
$\beta_k=\psi_k-\psi_{k-1}$ (8)
$\gamma_k$ is the acceleration factor:
$\gamma_k=\frac{\sum t_k t_{k-1}}{\sum t_{k-1}^2}$ (9)
where, the expression for $t_k$ is as Eq. (10):
$t_k=\psi_k-\varphi_k$ (10)
thus constitutes an iterative add-on $\beta_k \gamma_k$, increasing the gradient step size of the phase to speed up convergence.
To verify the effectiveness of the algorithm proposed in this paper, we carried out numerical simulations and compared with the quadratic-phase-based GS, WGS and AWGS algorithms. Here, we applied some evaluation functions to compare the experimental results quantitatively. The first evaluation function is the RMSE between the reconstructed image and the target image, which is defined as follows:
$R M S E=\sqrt{\frac{1}{M N} \sum_{m=1}^M \sum_{n=1}^N\left[A_k(m, n)-A_{tar\mathrm{~g} e t}(m, n)\right]^2}$ (11)
where, M and N are the numbers of horizontal and vertical pixel points in the signal region image, $A_k(m, n)$ is the intensity distribution of the reconstructed image of $k-t h$ iteration, and $A_{t a r g e t}(m, n)$ is the intensity distribution of the target image. The RMSE is a number greater than or equal to zero, with smaller values indicating better reconstruction quality.
The second evaluation function is Structural Similarity (SSIM), which is defined as:
$\operatorname{SSIM}=\frac{\left(2 \mu_{\text {targer }} \mu_k+c_1\right) \cdot\left(2 \sigma_{\text {targ } e t, k}+c_2\right)}{\left(\mu_{\text {target }}^2+\mu_k^2+c_1\right) \cdot\left(\sigma_{\text {target }}^2+\sigma_k^2+c_2\right)}$ (12)
where, $\mu_{t \text { arger }}$ and $\mu_k$ are the mean of the target image and the reconstructed image respectively; $\sigma_{t \operatorname{arget}}$ and $\sigma_k$ are the standard deviation of the two respectively; $\sigma_{t \text { arget }, k}$ is the covariance between the two; and c1 and c2 are tiny numbers to avoid the denominator being zero. SSIM takes values between 0 and 1, with larger values representing better reconstruction.
The third evaluation function is the Peak Signal-to-Noise Ratio (PSNR), which is defined for 8-bit greyscale images as:
$P S N R=20 \log \left[\frac{255}{R M S E}\right]$ (13）
The higher the value, the better the reconstruction.
In this simulation experiment, the target image selected is ‘cameraman’. The resolution of both the target image and the hologram plane after zero-padding is taken as 512×512 pixels, the size of the signal area is 300×300 pixels, and the pixel size is $12 \mu m$. See Figure 6(a). According to Eq. (3) and Eq. (4), the parameters of the quadratic phase a and b are both set to 0.01, as shown in Figure 6(b).
Figure 6. Target image ‘cameraman’ and initial quadratic phase
Figure 7 presents the POHs and reconstructed images generated by the GS, WGS, AWGS and AW-PGD GS algorithms at different numbers of iterations, as well as the values of the three evaluation parameters foreach reconstructed image. It shows that the reconstruction quality based on these four algorithms generally improves as the number of iterations increases. The RMSE of the GS algorithm is 0.073 at 50 iterations and drops to 0.0635 at 300 iterations, indicating that the reconstruction quality does not increase significantly with the number of iterations and falls into the trap of a locally optimal solution. The quality of the reconstructed images doesn’t become better when the number of iterations increases. Instead, the quality decreases. The AWGS algorithm is better than the GS and WGS algorithms. Although the reconstruction quality using the AW-PGD algorithm is average at a low number of iterations, the reconstruction quality is better than the other three algorithms as the number of iterations increases, with the RMSE value down to 0.0057 at 300 iterations. The SSIM and PSNR also indicate the superiority of the AW-PGD algorithm.
Figure 7. Reconstructed image and POHs under different numbers of iterations by different algorithm
Figure 8. Comparison of reconstructed quality parameter
To see more clearly the relationship between the number of iterations and reconstruction quality, Figure 8 compares the values of the three parameters under the four algorithms. Figure 8(a) shows the RMSE under different number of iterations. It shows that although the AW-PGD algorithm is not the best when the number of iterations is small, the RMSE of the AW-PGD algorithm starts to perform better than the remaining three algorithms after 11 iterations, with an RMSE value of 0.08423. At the 100th iteration, the RMSE drops to 0.01101, which is much better than the remaining three algorithms. Figure 8(b) shows the SSIM curve and Figure 8(c) shows the PSNR value curve. Both parameter curves show the superiority of the AW-PGD algorithm.
To verify the effectiveness of the method proposed in this paper for other images, we used 'Lena', 'Lifting body' and 'Baboon' as the target images for a simulation experiment. The reconstructed images based on these four algorithms are compared under the same parameters setting. In this experiment, the number of iterations was 100 and the rest of the parameters were set as above. In Figure 9, (a), (b), (c) and (d) present the reconstructed results and POHs obtained based on the GS, WGS, AWGS and AW-PGD algorithms respectively, with the values of RMSE, PSNR and SSIM compared to the target image marked below each reconstructed image. It can be seen that the AW-PGD algorithm also delivers better results than the other three algorithms when used for other target images.
We also compared the time cost of the AW-PGD algorithm
with the other three algorithms, as shown in Table 1. Since above we found that the algorithm converges at around ten iterations, we also wanted to investigate further whether the time cost increases when the number of iterations is increase compared to the other three algorithms. Therefore, the number of iterations was set to 20, 50 and 100, and the rest of the parameters were the same as above. It can be seen that at 20 iterations, the time cost of the AW-PGD algorithm is similar to that of the AWGS algorithm; at 100 iterations, the AW-PGD algorithm increases the time cost by less than 1s.
Table 1. Comparison of time cost of the four algorithms under different number of iterations
|
20 |
50 |
100 |
|
|
GS |
1.486s |
2.101s |
3.151s |
|
WGS |
1.560s |
2.370s |
3.830s |
|
AWGS |
1.814s |
2.596s |
3.693s |
|
AW-PGD |
1.816s |
2.700s |
4.490s |
In the AW-PGD method, the amplitude of the target image is divided into signal and non-signal regions, hence it is necessary to investigate the effect of the selection of the signal region size on the reconstruction effect and the time cost.
Figure 9. Reconstruction of different target amplitude from POH based on GS, WGS, AWGS and AW-PGD
Figure 10. The relationship between the size of the signal region and RMSE and time cost
Figure 10 illustrates the relationship between the signal region size and the RMSE value and time cost. Here, the size of the entire target image, including the non-signal region, is 512 × 512 pixels, where the size of the signal region is 100 × 100, 200 × 200, 300 × 300, 400 × 400, 500 × 500 and 512 × 512. The corresponding parameters of the quadratic phase are selected as 0.03, 0.01, 0.007, 0.006 and 0.0061, with 100 iterations. It can be seen that as the signal area increases, the RMSE values increase and the reconstruction quality decreases, with little effect on the time cost.
This paper proposes a hybrid constrained AW-PGD iterative algorithm to compute and generate higher quality POHs. Use the quadratic phase as the initial phase because the quadratic phase has the advantage of suppressing speckle noise. By the way of zero-padding around the target image to improve the reconstruction quality. In the process of iteration, an adaptive dynamic constraint strategy is used on the image plane to improve the reconstruction quality and relax the constraint on the non-signal region to reduce the computational effort; a gradient descent algorithm is used on the hologram plane to speed up the convergence. Numerical simulation results show that this method has higher reconstruction quality than the GS, WGS and AWGS algorithms, and does not significantly increase the time cost. In addition, this paper investigates the effect of the selection of the signal region size on the reconstruction effect and time cost. The algorithm has the advantages of low computational effort, high reconstruction quality and simple optical path. Accordingly, the algorithm has good prospects for applications in hologram projection and atom capture.
This work is supported by the National Science Foundation (Grant No.: 61302179); The Project of Excellent Talents Support Plan in Colleges of Anhui Province (Grant No.: gxyq2021206); The Universities Natural Science Research Foundation of Anhui province (Grant No.: KJ2019A0619); The Research Foundation for Advanced Talents of Hefei Normal University (Grant No.: 2020rcjj21); The Open Foundation from Anhui Province Key Laboratory of Simulation and Design for Electronic Information System (Grant No.: 2020ZDSYSYB01) and the Provincial Scientific Research Platform Special Program of Hefei Normal University (Grant No.: 2020PT11).
|
$u_0$ |
the initial complex amplitude of the hologram plane in GS |
|
$U_1$ |
the complex amplitude of FT of $u_0$ |
|
$\left|A_{t a r g e t}\right|$ |
amplitude of target image |
|
$U_1^{\prime}$ |
complex amplitude of $U_1$ after amplitude replacement |
|
$u_1^{\prime}$ |
Inverse Fourier Transform of $U_1^{\prime}$ |
|
$u_k$ |
complex amplitude of the hologram plane after k iterations in GS |
|
$U_k$ |
complex amplitude of the image plane after k iterations in GS |
|
$A_0$ |
the initial complex amplitude of the image plane in AW-PGD |
|
$a_0$ |
IFT of $A_0$ |
|
$a_1$ |
the phase of $a_0$ |
|
$a_k$ |
the complex amplitude of the k -th iteration of the hologram plane |
|
$\varphi_k$ |
the phase of $a_k$ |
|
$A_k$ |
the complex amplitude of the k -th iteration of the plane |
|
$\left|A_{o b}\right|$ |
the amplitude of $A_k^{\prime}$ |
|
$A_k^{\prime}$ |
complex amplitude after the amplitude of $A_k$ is replaced |
|
$a_k^{\prime}$ |
IFT of $A_k^{\prime}$ |
|
$f_k$ |
the amplitude of the hologram plane in the k -th iteration |
|
$t_k$ |
$\psi_k-\varphi_k$ |
|
Greek symbols |
|
|
$\mu_0$ |
initial phase in GS |
|
$\phi$ |
quadratic phase |
|
$\psi_k$ |
the phase distribution of $a_k^{\prime}$ |
|
$\varphi_{k+1}$ |
phase value of $\varphi_k$ after gradient descent |
|
$\beta_k$ |
the direction of the gradient in the k -th iteration |
|
$\gamma_k$ |
the acceleration factor in the k -th iteration |
|
Subscripts |
|
|
k |
number of iteration |
|
m |
horizontal coordinates of signal area |
|
n |
vertical coordinates of signal area |
|
a |
the parameters between 0 and 1 |
|
b |
the parameters between 0 and 1 |
|
M |
horizontal pixel points in the signal area |
|
N |
vertical pixel points in the signal area |
|
RMSE |
root mean squared error |
|
PNSR |
peak signal noise ratio |
|
SSIM |
structural similarity |
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