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Abstract
This paper proposes an unsupervised learning-based wavefront sensing method for SHWFS with insufficient sub-apertures. By modeling the light propagation of SHWFS in the neural network, the proposed method can train the model using unlabeled datasets. Therefore, it is convenient for the proposed method to be deployed in AO systems. The performance of the method is investigated through numerical simulations. Results show that the wavefront estimation accuracy of the proposed method is comparable to the existing methods based on supervised learning. This paper proposes a novel wavefront detection approach for SHWFS, the first application of unsupervised learning in wavefront detection.
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1. Introduction
The Shack-Hartmann wavefront sensors (SHWFS) [1,2] are now widely employed in applications requiring adaptive optics (AO) [3], such as astronomical observations [4], microscopic imaging [5], and free-space laser communications [6]. The SHWFS typically consists of a micro-lens array (MLA) and an imaging sensor located at the focal plane of the MLA. The sub-apertures of the MLA sample the corresponding parts of the distorted wavefront on the pupil plane and then focus the light onto the sensor, yielding a spot image. The averaged local slopes could be determined by calculating the centroid displacements of the spots. After that, the overall wavefront can be calculated from these slopes by a wavefront reconstruction algorithm, such as a modal or zonal approach [7,8]. Therefore, the wavefront sensing accuracy of SHWFS commonly relates to the number of sub-apertures; more sub-apertures can improve the resolution of the output array for the zonal approach or increase the number of Zernike modes used for the modal approach.
When aiming for detecting wavefront aberrations caused by atmospheric turbulence, the sub-aperture diameter d of the SHWFS needs to follow d ≤ r0, with r0 being the atmospheric coherence length [9]. If the guide star is faint at this point, the design of sub-aperture diameter d suffers from contradictions: on the one hand, the design of d needs to follow d ≤ r0 to obtain accurate wavefront estimations; on the other hand, it requires a large d to raise the total energy within the sub-aperture, preventing the spot from drowning in noise. When the atmospheric coherence length r0 is smaller than the sub-aperture diameter d, the sub-wavefront cannot be seen as a plane, and the higher modes in the sub-aperture can change the energy distribution of the focal spot, resulting in a wrong slope calculation. Hence, the detection error of SHWFS will increase significantly as the number of sub-apertures decreases. The above discussions reveal the current shortcomings of SHWFS in detecting wavefront aberrations of atmospheric turbulence.
Researchers made several attempts to improve the performance of SHWFS with insufficient sub-apertures. However, these approaches were not promoted because of the excessive structural [10,11] or complicated computations [12,13]. Recently, deep learning has gradually been applied in adaptive optics, providing novel solutions to improve the performance of SHWFS [14–16]. Xu et al. employed the extreme learning machine to fit the nonlinear corresponding relationship between the centroid displacement and the Zernike coefficients under sparse microlens [17]. Paine et al. used a convolutional neural network (CNN) operating point spread function(PSF) to determine an initial estimate of the wavefront [18]. Andersen et al. used focused and defocused images to measure atmospheric turbulence [19]. Hu et al. applied the above concept to the SHWFS; they directly took SHWFS images as input for a CNN, and the model can output wavefront aberrations [20,21]. We have previously demonstrated that CNN can improve the performance of SHWFS compared to the existing modal/zonal approach in the presence of insufficient sub-apertures [22]. We performed wavefront reconstruction using a modified ReaNet50 network with SHWFS images (d/r0≈2∼4) as input, and the wavefront reconstruction accuracy is comparable to the modal approach with sufficient sub-apertures. This approach reduced the number of sub-apertures required for SHWFS to detect atmospheric turbulence, providing a promising solution to wavefront detection using faint beacons.
Among the deep learning-based wavefront detection approaches described above, the neural networks are typically trained through supervised learning. A labeled dataset containing the SHWFS images and wavefront aberrations is required. However, obtaining the true values of turbulence aberrations in AO systems is challenging, as it typically requires cooperative beacons together with an extra high-resolution SHWFS. Therefore, it is hard to deploy wavefront detection based on supervised learning in AO systems.
Unsupervised learning is a branch of deep learning [23]. It can train models using unlabeled datasets, commonly applied in clustering or dimensionality reduction. Its feature of eliminating labeled datasets is well suited for application in learning-based wavefront detection, making it more convenient to implement in AO systems. In this paper, we propose a neural network based on unsupervised learning for SHWFS with insufficient sub-apertures and named it SH-Autoencoder. By modeling the light propagation of SHWFS in the neural network, the proposed method can train the model using an unlabeled dataset (SHWFS images only). The performance of the SH-Autoencoder is investigated and optimized through numerical simulations. Results show that the SH-Autoencoder can provide wavefront estimations via unlabeled datasets with almost the same accuracy as supervised learning.
The remaining parts of the article are arranged as follows: Section 2 describes the details of the proposed SH-Autoencoder. In section 3, the wavefront estimation performance of the SH-Autoencoder is investigated and optimized through numerical simulations. Section 4 discusses the simulation results. Finally, we conclude the paper in Section 5.
2. Method
The overall architecture of the SH-Autoencoder is shown in Fig. 1(a). It consists of an encoder and a decoder [24]. The encoder is a conventional CNN for predicting the wavefront aberrations from SHWFS images whose parameters are optimized during training. The decoder comprises our custom layer without learnable parameters, characterizing light propagation in SHWFS.
Fig. 1. Neural Network Structure of the proposed SH-Autoencoder. (a) The overall framework of the SH-Autoencoder. (b) The neural network structure of the encoder. Orange, blue, gray, and pink squares represent BottleNeck modules with different parameters. The number of each module is [3,3,3,3], respectively. (c) The neural network structure of the decoder. The decoder receives the Zernike polynomial coefficients, and the corresponding SHWFS image is computed by the Fresnel transfer method.
The network architecture diagram of the encoder is displayed in Fig. 1(b). We chose the simplified ResNet50 network to complete the wavefront prediction [25]. ResNet is currently a popular convolutional neural network model used in image-related problems, and it has achieved satisfactory fitting results in several types of tasks. The SHWFS image is input to extract the features, and the last Residual module outputs a 2048 × 1 feature vector. The feature vector is sent to the fully connected layer, which ultimately outputs the Zernike polynomial coefficients of the wavefront aberrations. For wavefront estimation, we found that a simpler network is sufficient for the Zernike coefficient regression, so the number of residual modules in the ResNet50 was simplified from [3 4 6 3] to [3 3 3 3]. The fewer parameters shortened the training and predicting process, which is a good compromise for high-speed wavefront sensing between simplicity and performance.
The output of the encoder continues to be input to the decoder, whose architecture diagram is shown in Fig. 1(c). Using the Fresnel transform method, the decoder calculates the SHWFS image from the known Zernike coefficients. The primary process of calculation involves the following [26].
Calculating the wavefront $\varphi ({x,y} )$ from the Zernike coefficients output from the encoder using Eq. (1).
Then, the complex amplitude of the optical field on the front surface of the SHWFS can be denoted as Eq. (2).
where P(x,y) is the pupil function, typically, the pupil shape is circular, with P(x,y) being 1 inside the circular and 0 outside the circular. Therefore, the light field at the rear surface of the MLA can be expressed as:In Eq. (3), t(x,y) is the phase screen of the MLA, which can be obtained by expanding the phase screen of the single-lens in space. Lastly, the optical field received by the imaging detector can be calculated from the Fresnel diffraction integral as:
Equations (4) can be calculated using Eq. (5) with the angular spectrum method.
We implemented the procedures from Eq. (1) to Eq. (5) in the neural network, constructing each of these single operations, e.g., array addition, array multiplication, fast Fourier transform, and fast Fourier inverse transform, as a network layer. To realize the computation of complex numbers, the decoder's network layer is designed to contain two inputs and two outputs corresponding to the real and imaginary parts of the complex amplitude. Finally, the reconstructed image is the sum of the squares of the real and imaginary parts, and the pixel values are normalized to 0∼1.
These are the neural network structures of the SH-Auotoencoder. It consists of two parts with different functions. The encoder is designed to predict Zernike coefficients from SHWFS images. It contains learnable parameters whose values will be determined during training. The decoder, without learnable parameters, calculates the light propagation in the SHWFS represented by the Fennel transmission method. It should be mentioned that both the input and output of the SH-Autoencoder are SHWFS images.
Typically, the learnable parameters in a neural network are determined by minimizing the loss function. We choose the negative Pearson correlation coefficient (NPCC) shown in Eq. (6) as the loss function [27]. The NPCC loss treats images as vectors and evaluates the similarity between images through the cosine of the vectorial angle; this ensures linear amplification and bias-free reconstruction, which increases the convergence probability. In the problems researched in this paper, the NPCC ranges from -1 to 0. Minimizing the NPCC loss is practically equivalent to maximizing the similarity between the input and output images. During backpropagation, the optimization of the parameters will make the encoder output the Zernike coefficients corresponding to the SHWFS image. Thus, the learnable parameters in the encoder can be trained in an unsupervised approach through unlabeled datasets.
3. Numerical simulation
3.1 Simulation setup
To validate the wavefront sensing performance of the SH-Autoencoder, we carried out numerical simulation research using MATLAB. A dataset containing 30,000 SHWFS images was generated for training. Table 1 displays the key parameters of the simulation. The dataset was constructed with two steps. In the first step, the atmospheric turbulence phase screens were generated with specific parameters. In the second step, the corresponding SHWFS image was calculated using the generated phase screen, with the procedure consistent with that demonstrated in Eq. (2)–Eq. (5). In addition, an extra dataset containing 2000 SHWFS images and corresponding phase screens was constructed for testing the model. The samples in the training set and test set are different from each other.
Table 1. Key Parameters of the Simulation
After that, we constructed the SH-Autoencoder and fed the training dataset into the model to train the learnable parameters. The model was trained and tested on a desktop workstation (Intel Core i9-7920X CPU @ 2.90 GHz, NVIDIA RTX 2080 Super). The optimizer for SH-Net training is adaptive moment estimation (ADAM) with an initial learning rate of 10−4. It took about 20 hours for 50 epochs with a batch size 20 for the training process.
3.2 Simulation results
We set the Zernike modes number of the encoder's output to 63 (corresponding to the first 66 Zernike modes with piston, tip, and tilt removed). Training and test datasets were then constructed, with the phase screen being a linear combination of Zernike polynomials. Following the hyper-parameters described in section 3.1, we obtained the trained model. Figure 2 shows a set of wavefront prediction results for the model. Figure 2(a) and 2(b) list the input SHWFS image and the SH-Auoencoder's output. The output image is analogous to the input, with their NPCC loss being about -0.9986. The simulated wavefront aberration and the encoder's output are listed in Fig. 2(c) and 2(d). The residual wavefront after wavefront compensation is shown in Fig. 2(e). The root mean square error (RMSE) of the residual wavefront is about 0.0167λ, and the apparent residuals can barely be observed, suggesting that the wavefront aberration is accurately estimated. The point spread function (PSF) before and after wavefront correction is shown in Fig. 2(f) and 2(g), and the corresponding center intensity profile is shown in Fig. 2(h). The morphology of the corrected PSF is close to the ideal Airy spot, with the Strehl Ratio (SR) being 0.9909, 14.1 times higher than that of the pre-corrected one, which indicates that the wavefront aberrations are well compensated.
Fig. 2. Wavefront estimation results on aberrations constructed from Zernike polynomials. (a) Input SHWSF images. (b) Output image of the SH-Autoencoder. (c) Wavefront aberration (d) Reconstructed wavefront via SH-Autoencoder. (e) The wavefront residual after compensation. (f) Focal spot before wavefront compensation. (g) Focal spot after wavefront compensation. (h) The center intensity profile of the focal spots in (f) and (g).
In the test set, the average RMSE of wavefront residuals for the 2000 samples is 0.0346λ, and the average SR of the corrected PSF is 0.9539. Wavefront aberrations in the test set are accurately estimated and well-compensated. The above results provide a preliminary indication of the validity of the proposed SH-Autoencoder. The encoder's learnable parameters of the SH-Autoencoder can be trained via unlabeled datasets. The trained model shows high precision in wavefront estimation and the PSF after wavefront compensation is close to the Airy spot.
To further validate the performance of SH-Autoencoder, we regenerated the dataset to repeat the above numerical experiments, but this dataset used phase screens constructed by the “power spectrum method”. Each phase screen was generated by taking the inverse Fourier transform of a function with an amplitude filtered according to the desired power spectrum. The phase screen generated by the power spectral method contains more high-frequency components closer to the natural atmospheric turbulence aberrations.
Figure 3 shows the wavefront estimation results of the trained model. In Fig. 3(a) and 3(b), the NPCC loss of the output image with respect to the input is -0.8334. The similarity of the images decreases as the high-frequency components in the wavefront aberrations increase. The RMSE of the residual wavefront shown in Fig. 3(e) is 0.2343λ, and the SR of the corrected PSF listed in Fig. 3(g) is about 0.156. Although the SR is improved by a factor of 3.8 compared to the uncorrected focal spot, there is still a significant discrepancy between the corrected focal spot and the ideal Airy spot in the morphology.
Fig. 3. Wavefront estimation results on aberrations generated through the power spectrum method. (a) Input SHWSF images. (b) Output image of the SH-Autoencoder. (c) Wavefront aberration (d) Reconstructed wavefront via SH-Autoencoder. (e) The wavefront residual after compensation. (f) Focal spot before wavefront compensation. (g) Focal spot after wavefront compensation. (h) The center intensity profile of the focal spots in (f) and (g).
Among the test dataset, the average RMSE of residual wavefront for all samples was 0.2546λ, and the average SR for the corrected PSF was 0.1130. Both showed a significant decrease compared to the previous numerical simulation results. All these results show that when the wavefront aberrations contain high-frequency components, the wavefront estimation performance of the SH-Autoencoder will be degraded, and the wavefront aberration will not be well compensated accordingly. However, actual atmospheric turbulence wavefronts typically contain some high-frequency components. Therefore, the SH-Autoencoder needs further improvement to increase its wavefront estimation precision.
3.3 Improvements of the SH-Autoencoder
Analyzing the results in Fig. 2 and Fig. 3, we suggest that the main reasons for the degradation in the wavefront estimation performance of SH-Autoencoder may be as follows.
The SH-Autoencoder is designed for wavefront estimation. However, the learnable parameters of SH-Autoencoder are optimized by minimizing the NPCC loss function. When the wavefront aberrations contain high-frequency components, the first 66 Zernike polynomials cannot represent all the details of the wavefront, which leads to a decrease in the similarity between the input and output images. In this case, minimizing the NPCC loss function may be unequal to minimizing the wavefront residuals. This may lead to a biased gradient of the learnable parameter, bringing the model to a local optimum and introducing wavefront estimation errors.
Besides, SHWFS images are usually considered sparse arrays: most pixels have zero values, and a few have large values. Pixels with large values contribute significantly to the NPCC loss, while the contributions of pixels with smaller values are often neglected. This could be another explanation for the bias in the gradient calculation during the optimization.
To improve the performance of the SH-Autoencoder, we modified the neural network and the form of the loss function. The number of Zernike modes in the encoder's output was increased to 207(corresponding to the first 210 Zernike modes with piston, tip, and tilt removed). More Zernike modes can accurately represent the detailed information of wavefront aberrations and improve the similarity between the input and output images. The form of the loss function is modified to the NPCC of the square root of the images. The exact form is shown in Eq. (7). The square root function can magnify small pixel values without changing the value domain, increasing its contribution to the loss function
After completing the above improvements, we retrained the model using the same dataset. Figure 4 lists the wavefront estimation results, and the wavefront aberration to be measured is the same as that in Fig. 3. As can be seen from Fig. 4(e), the wavefront residual decreases compared with that in Fig. 3(e). The RMSE of the residual wavefront is reduced from 0.2546λ to 0.117λ. Meanwhile, the corrected PSF listed in Fig. 4(g) can be observed as being better focused, with an improvement in SR from 0.1130 to 0.6757. In the test set, the averaged RMSE of residual wavefronts decreases to 0.1365λ among all samples, and the average SR of the corrected PSF improves to 0.5519. These results illustrate the improvement in the performance of the SH-Autoencoder. Increasing the Zernike modes output by the encoder and improving the contribution of all pixel values to the NPCC loss can contribute to the SH-Autoencoder better dealing with the high-frequency component in the wavefront aberrations.
Fig. 4. Wavefront estimation results of the improved SH-Autoencoder, the wavefront aberration to be measured is the same as that in Fig. 3. (a) Input SHWSF images. (b) Output image of the SH-Autoencoder. (c) Wavefront aberration (d) Reconstructed wavefront via SH-Autoencoder. (e) The wavefront residual after compensation. (f) Focal spot before wavefront compensation. (g) Focal spot after wavefront compensation. (h) The center intensity profile of the focal spots in (f) and (g).
3.4 Comparison with existing approaches
This subsection will compare the proposed SH-Autoencoder with the modal and supervised learning-based approaches. We simulated a 20 × 20 SHWFS to measure the first 63 Zernike coefficients of the wavefront aberrations in the test set. The key parameters of the SHWFS remain consistent with those shown in Table 1, and it follows the principle of d/r0 ≤ 1 during detection. Additionally, we also trained the convolutional neural network in the encoder using supervised learning, and the model outputs the first 63 Zernike coefficients of the wavefront aberration. Comparisons of two wavefront estimation results for the three methods are presented in Fig. 5. Figure 5(a) and 5(e) show the wavefront aberrations and the corresponding PSF. Results constructed by the SH-Auotencoder are presented in Fig. 5(b) and 5(f). For comparison, we intercepted the first 63 Zernike polynomial coefficients of the encoder's output to reconstruct the wavefront. The RMSE of the residual wavefront in the two results are 0.1438λ and 0.1640λ, and the SR of the PSFs after wavefront compensation are 0.488 and 0.3854, respectively. The wavefront estimation results of supervised learning are shown in Fig. 5(c) and 5(g). The corresponding RMSE and SR are 0.1436λ, 0.4952, and 0.1587λ, 0.4120, respectively. Figure 5(d) and 5(h) list the results of the modal approach, with RMSE and SR being 0.1367λ, 0.5366, and 0.1505λ, 0.4580, respectively. From the results in Fig. 5, it can be seen that all three methods can provide an accurate wavefront estimation, and the PSF is focused well after wavefront compensation.
Fig. 5. Wavefront estimation results for the SH-Autoencoder, supervised learning, and the modal approach. (a) and (e). Wavefront aberrations and the uncorrected PSFs. (b) and (d). Wavefront estimation results constructed by the SH-Autoencoder. (c) and (g). Wavefront estimation results of the CNN in the encoder. The parameters were trained via supervised learning. (f) and (g). Wavefront estimation results of the modal approach.
Figure 6 displays the wavefront estimation results for all samples in the test set. The RMSE of residual wavefronts for all samples is shown in Fig. 6(a), and the SR of the focal spot after wavefront compensation is listed in Fig. 6(b). Each point in the figure represents a wavefront estimation result, with each method distinguished by a color. The average values of the two metrics are presented in Table 2. Among the three methods, the average RMSE of the wavefront residuals was 0.1769λ(SH-Autoencoder), 0.1677λ(supervised learning), and 0.1599λ(modal), respectively, and the average SR of the corrected PSF was 0.3405(SH-Autoencoder), 0.3786(supervised learning), and 0.4157(modal). The results of Fig. 5 and Fig. 6 show that the performance of the three methods in wavefront estimation is approximately comparable. It indicates that the convolutional neural network in the encoder can accurately fit the mapping between the SHWFS image and the Zernike coefficients. In addition, the SH-Autoencoder shows virtually no degradation in accuracy compared to supervised learning methods.
Fig. 6. Wavefront estimation results for samples in the test set. Each point in the figure represents a wavefront estimation result, with each method distinguished by a color. (a) RMSEs of the residual wavefronts. (b) SR of the focal spots after wavefront compensation.
Table 2. Comparison of metrics of the three approaches
3.5 Simulation results of closed-loop control
To illustrate the effect of SH-Autoencoder in closed-loop control systems and further validate the generalization capability of SH-Autoencoder. We numerically simulated a deformable mirror containing 489 units as the wavefront corrector (the number of units in the radius direction is 25). The proportional-integral algorithm was employed to achieve closed-loop control by utilizing wavefront measurements by the SH-Autoencoder. Figure 7 shows a set of simulation results. Figure 7(a) and 7(b) represent the focal spots during the correction process and the SHWFS images before and after correction, respectively. Figure 7(c) compares the variations of some sub-aperture images before and after correction, with these sub-apertures marked with red lines in Fig. 7 (b). As we can see in Fig. 7(a), the focal spot is gradually centralized as the closed-loop control proceeds, with the sub-aperture spots exhibiting the same trend and the centroid offsets decreasing. After correction, the RMSE of the wavefront residual is 0.1249λ, and the SR of the focal spot is 0.5701. Figures 7(d) and 7(e) give the trends of wavefront residual RMSE and focal spot SR during 500 closed-loop corrections. The averaged value of the wavefront residual RMSE and the SR of the corrected PSF are 0.138λ and 0.5055, respectively. The above results demonstrate the generalization ability of the SH-ZerAutoencoder, which enables the correction of wavefront aberrations through the measurement of wavefront residuals.
Fig. 7. Numerical simulation results for closed-loop control. (a) Focal spots during the correction process. (b) The SHWFS images before and after correction. (c) The comparisons of some sub-aperture images before and after correction. (d) The average trend of wavefront residual RMSE over 500 simulations. (e) The average trend of SR in 500 simulations.
4. Discussion
In Section 3, we investigate the performance of the SH-Autoencoder and compare it with two existing methods. From the numerical simulation results, the properties of these methods can be summarized as follows:
The supervised learning-based approach can fit the mapping relationship between SHWFS images and wavefront aberrations, making wavefront estimation via fewer sub-apertures (d/r0 ≈ 3) feasible. However, labeled datasets are usually necessary for supervised learning. When applied in AO systems, it typically requires cooperative beacons and a high-resolution SHWFS, resulting in challenges in implementing the approach. In contrast, the SH-Autoencoder eliminates the necessity of labels, which makes it more feasible to be implemented in AO systems. In the numerical simulation results, extra degradation in accuracy is not observed either. The above features give SH-Autoencoder advantages over existing wavefront detection methods based on supervised learning.
5. Conclusion
In order to realize the wavefront estimation when the beacon is faint, this paper proposes an unsupervised learning-based approach named SH-Autoencoder. Compared with the conventional zonal or modal approaches, the SH-Autoencoder can realize wavefront estimation with fewer sub-apertures(d/r0 ≈ 3). Besides, the learnable parameters of SH-Autoencoder can be optimized with the unlabeled dataset, which makes it more convenient to implement in AO systems. The performance of the SH-Autoencoder is investigated and compared via numerical simulations. Results show that SH-Autoencoder can provide wavefront estimations with almost the same accuracy as supervised learning.
In summary, this paper realizes a novel wavefront detection for SHWFS. It employs neural networks to utilize more details in the SHWFS image, achieving wavefront estimation with fewer sub-apertures. In addition, the unsupervised learning-based SH-Autoencoder is convenient to implement in the system, offering feasibility advantages. These features make the SH-Autoencoder promising for applications in wavefront detection in atmospheric turbulence.
Acknowledgements
The author thanks Jialong Peng for their help in the manuscript revision.
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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