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Abstract
To realize the large-scale and high-precision co-phasing adjustment of synthetic-aperture telescopes, we propose a multichannel left-subtract-right feature vector piston error detection method based on a convolutional neural network, which inherits the high precision and strong noise resistance of the DFA-LSR method while achieving a detection range of (−139λ, 139λ) (λ = 720 nm). In addition, a scheme to build large training datasets was proposed to solve the difficulty in collecting datasets using traditional neural network methods. Finally, simulations verified that this method can guarantee at least 94.96% accuracy with large samples, obtaining a root mean square error of 10.2 nm when the signal-to-noise ratio is 15.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the gradual increase in the telescope aperture required for astronomical observations, single-aperture telescopes have been unable to meet the demand owing to technological limitations, and the development of synthetic-aperture telescopes has become increasingly important. At present, there are many synthetic-aperture telescopes in the world, such as the Keck series telescopes in Hawaii [1], HET in Texas, USA [2,3], SALT in Cape Town, South Africa [4], LAMOST in Hebei, China [5], and other ground-based telescopes, as well as the on-orbit JWST, which is expected to be launched in 2021 [6]. Synthetic-aperture telescope technology bodes well for the development of large-aperture telescopes.
However, a synthetic-aperture telescope will have residual tilt and piston errors between adjacent submirrors after unfolding, which significantly reduces the imaging quality of the system. The former can be well corrected by methods such as the Hartmann sensor [7]; however, the large range and high precision of many piston detection methods cannot be obtained at the same time. In 1996, Koechlin et al. proposed a dispersed fringe sensor (DFS) for coarse co-phasing [8]. Since 2004, Shi et al. have made significant contributions to improving the performance and algorithm of DFSs, including the use of the least-squares fitting method to fit the intensity distribution of the dispersion direction of the dispersed fringe [9], the simulation study of the dispersed fringe pattern and the performance of DFS [10], and the modification of the fitting formula to improve the robustness of detection [11]. Because the least-squares fitting intensity method in the dispersion direction in DFS cannot detect pistons within one wavelength, in subsequent developments, DFS has often been combined with other fine co-phasing methods [12].
In 2016, Jiang et al. detected a piston based on the relationship between the MTF sidelobe intensity and the piston, and obtained the detection range of the coherence length of the incident light and 0.026λ root mean square (RMS) [13]. Then, according to this method, detection under the influence of tilt error and research on multi-aperture nonredundant distribution were performed [14,15]. Since then, researchers have focused on finding a piston detection method that integrates coarse and fine co-phasing to achieve large-scale and high-precision detection of pistons. In 2019, Zhang et al. investigated the piston extraction method on DFS and proposed a piston detection method based on the slope of the dispersed fringe [16], a merit-function-based active scanning algorithm [17], and a digital dispersion method [18]. These methods can achieve coarse and fine co-phasing simultaneously; however, their detection range is below 100 µm and the detection accuracy must be further improved.
In 2017, Li et al. proposed a piston estimation method called the dispersed-fringe-accumulation-based left-subtract-right method (DFA-LSR) [19]. This method is based on the DFS device, accumulating the obtained dispersed fringe images along the dispersion direction, and uses the linear relationship between the value of the left peak minus the right peak of the signal obtained by the accumulation and piston error to achieve accurate detection of the piston error. Owing to its accumulation, this method has strong resistance to noise and high detection accuracy. However, because of the accumulation of the entire dispersed fringe, the coherence length of the incident light is so short that it can only be used for fine-phase measurements.
In recent years, there has been an increasing body of research on neural networks used for co-phasing detection. In 2019, Li et al. combined the PD method and a neural network to first correct the piston to the detection range of the PD method and then perform a fine co-phasing correction [20]. However, these neural network-based detection methods often require large datasets to train the network, resulting in the methods having a detection range of approximately 10λ. In 2020, Guerra-Ramos used reinforcement learning methods and recurrent neural networks to detect pistons [21,22]. The former uses a neural network to predict the adjustment commands of piston actuators to achieve fine co-phasing, whereas the latter achieves a large measurement range of [−21λ,21λ].
A multichannel left-subtract-right feature vector piston detection method based on a convolutional neural network, CNN-Multi-LSR, is proposed in this study. On the basis of the DFA-LSR method, this method introduces multiple monochromatic light channels to illuminate a double-aperture diaphragm for diffraction imaging, instead of using a broadband light and DFS device to form a dispersed fringe image as in the original method, and the DFA-LSR method is then used to generate the LSR value of the image distribution of each channel to construct the LSR feature vector. Finally, the trained neural network is used to distinguish the LSR feature vector to obtain the piston value. Because the DFA-LSR method itself has high accuracy and strong noise resistance, the method proposed in this work inherits these characteristics. In addition, to address the difficulty of obtaining datasets for training faced by many current neural methods, a new method is proposed in this study to use data within one wavelength to create a large dataset. Moreover, both high-precision and wide-range measurements are achieved in this work.
The remainder of this article is organized as follows. The principle of the imaging process of the CNN-Multi-LSR method, the construction process of the LSR feature vector, and the theoretical piston error detection range of the CNN-Multi-LSR method are discussed in Section 2. The construction of a large training dataset and the neural network structure are shown in Section 3. Section 4 presents the results of the simulation experiment and its advantages compared with other co-phasing methods. Finally, the article is concluded in Section 5.
2. Principle of LSR feature vector piston error detection method based on a convolutional neural network
2.1 Imaging model of optical system in a single wavelength channel
The CNN-Multi-LSR method uses multiple monochromatic lights from multiple wavelength channels to perform interference imaging, processes the image results to obtain the LSR feature vector, and inputs it into the trained network to obtain the corresponding piston value. The imaging model of each monochromatic light channel and interference area selection stop is shown in Fig. 1.
Fig. 1. (a) Structure model of monochromatic light double-aperture interference imaging, (b) Structure model of the interference area selection stop.
In the optical path shown in Fig. 1(a), monochromatic light of wavelength λ passes through a stop with the structure shown in Fig. 1(b). The stop has two rectangular apertures of length a in the x direction, width b in the y direction, and center spacing d, where x and y are spatial coordinates. Subsequently, the piston error is applied to the light passing through the two apertures, denoted as $p = {\delta _2}- {\delta _1}$, where ${\delta _1}$ and are ${\delta _2}$ the optical path difference of the two beams with respect to the co-phased state, respectively. The light field distribution of the two apertures is shown in (1):
Here, rect is a rectangular function, which can be written as
Then, the two light beams with a phase difference are imaged on the target surface of the CCD camera by a lens with a focal length f. This process can be expressed as (3)–(5):
Table 1. Simulation parameters
The imaging model adopted in this study is a double-aperture diffraction model, which belongs to the front part of the DFS. In this model, the parameters a and b determine the diffraction effect in the y and x directions, respectively, and the center spacing d of the rectangular hole determines the interference effect. The combined effect of the three parameters is manifested in the density and contrast of the image fringes. The smaller a and b, the weaker the diffraction effect and the stronger the interference effect, which are manifested as an increase in the number of peaks in the image and a lower image contrast. In the DFS, the dispersed fringe becomes denser and the fringe contrast decreases. The smaller d is, the stronger the diffraction effect, the fewer peaks in the image, and the more blurred the image. In the DFS, the dispersed fringe is sparse, and the contrast decreases. The focal length f of the lens determines the broadening of the imaging result for each monochromatic wave. The larger the value of f, the lower is the contrast of the fringe. This result is the same for the DFS. The values of these four parameters together ensure that the imaging area in the simulation has three obvious peaks and a high fringe contrast.
The method proposed in this paper does not have a strict limit on the pixel size of the detector used. When a pixel size of 4.6 μm is used, the resolution of the imaging area is approximately 10×10, but the pixel size must ensure that the imaging area has sufficient resolution to allow the extraction of the superimposed signal peak. Therefore, the pixel size here is a typical value of 1.67 μm, and the resolution of the imaging area is adjusted to 40×40. To make the simulation more realistic, the monochromatic light in this study is a quasimonochromatic light with a full width at half maximum (FWHM) of 2.5 nm and a bandwidth of 4 nm. Figure 2 shows the intensity distribution of 660 nm monochromatic light with a 50 μm piston on the camera target surface.
Fig. 2. Image intensity when 660 nm monochromatic light (FWHM: 2.5 nm, bandwidth: 4 nm) with 50 μm piston is incident.
2.2 Multichannel LSR feature vector
In 2017, Li et al. proposed a piston estimation method called the dispersed-fringe-accumulation-based left-subtract-right method (DFA-LSR) [19], which is applied to the dispersed fringe image. After the dispersed fringe image is accumulated along the dispersion direction, the accumulated result is processed according to (6):
where ${L_p}$, ${R_p}$, ${M_p}$, and ${I_{\min }}$ are the left peak, right peak, main peak, and background intensity of the accumulated signal, respectively. Owing to its accumulation, this method has good noise immunity.In this study, to further improve the resistance to noise, we replace formula (6) with (7), where ${L_{p - aver}}$ is the arithmetic mean value of three adjacent positions centered on the left peak, and ${R_{p - aver}}$ and ${M_{p - aver}}$ have the same form as ${L_{p - aver}}$.
The LSR value is linearly distributed within the piston error range of (−λmin/2, λmin/2), where λmin is the minimum wavelength in the incident bandwidth light. Figure 3 shows the variation in the LSR value with the piston error in a wavelength range (λmin = 660 nm).
However, the DFA-LSR method can only be used for fine co-phasing detection, owing to the limited detection range. To improve the detection range, we introduce multiple wavelength channels and accumulate the image plane intensity distribution of each monochromatic light with a piston, instead of accumulating the entire dispersed fringe pattern as in the DFA-LSR method. Then, we obtain the LSR value corresponding to each wavelength channel.
Figure 4 shows the normalized image plane intensity distributions of the five wavelength channels and the accumulation process along the dispersion direction when the piston is 50 μm. Taking the 720 nm wavelength channel as an example, a signal containing three peaks (left, main, and right peaks) is obtained by accumulating the intensity distribution of the image plane, and then the LSR value of this channel can be calculated by formula (7). The other channels perform the same operation, then the feature vector composed of the LSR values of the five channels is [−0.3173, 0.6372,0.5918, −0.4433,0.1366].
Fig. 4. Accumulation process and results of the image plane intensity distribution of the 660, 690, 720, 750, and 780 nm wavelength channels when the piston is 50 μm.
2.3 Piston error detection range of CNN-Multi-LSR method
To illustrate how the LSR feature vector is used for large-scale measurements, we calculate the corresponding LSR value in the 660 nm channel according to the piston interval of 30 nm. The results are presented in Fig. 5.
Fig. 5. LSR value of the image intensity distribution in the 660 nm wavelength channel varies with the piston.
From the information of the six points marked in Fig. 5, it can be seen that the LSR value in this wavelength channel cycles according to two sets of values in the positive and negative piston ranges. These two sets of LSR values differ only in the highest and lowest points, with the highest and lowest points in the positive piston range being 0.6881 and −0.6345, respectively, and those in the negative piston range being 0.6345 and −0.6881, respectively, and their piston cycle period is 660 nm. Considering only the positive or negative piston error range, the number of data points in one cycle period is 22 (660 nm/30 nm). Similarly, the number of data and the LSR values of the other wavelength channels are also different, i.e., 690 nm: 23, 720 nm: 24, 750 nm: 25, 780 nm: 26; then, theoretically, for the five channels, we can construct $22 \times 23 \cdots \times 26 = 7893600$ groups of feature vectors. The piston interval between adjacent groups of these feature vectors is 30 nm, and thus a piston with a range of (−118404 μm, 118404 μm) can be specifically calibrated.
However, the actual detection range of the piston is determined by the optical coherence length of the minimum wavelength channel. When the piston exceeds the optical coherence length, the LSR value no longer cycles with the piston with the wavelength as the period, which is also the reason why the LSR value generated by the DFA-LSR method cannot be used to construct the feature vector. Therefore, we derive the theoretical maximum detection range based on the coherence of light.
The fringe contrast K is a quantitative description of the quality of the coherent light in physical optics. For nonmonochromatic light or quasimonochromatic light, it can be expressed by the classical Eq. (8):
where $\Delta k$ is the wave number and D is the coherence length of the light. When K = 1, the fringes are the clearest, and when K = 0, the fringes cannot be distinguished. When K reaches the first zero-point, Eq. (9) needs to be satisfied. Then, the maximum coherence length Dmax of light can be expressed by (10), where W is the effective bandwidth of each quasimonochromatic light channel that we define, which is different from the bandwidth $\Delta \lambda$ of each channel. This is because the weight of each wavelength in the bandwidth of the quasimonochromatic light emitted by the monochromator is different. ${\lambda _{\min }}$ is the wavelength of the smallest wavelength channel. W satisfies the relationship represented by (11). where ${\Delta _{1/2}}\lambda $ represents the FWHM of the quasimonochromatic light. By combining (10) and (11), we obtain (12).In theory, two beams of light whose optical path difference does not exceed the maximum coherence length can interfere, and thus we use $pisto{n_{\max - detected}}$ instead of ${D_{\max }}$. Then, we can obtain the relationship between the detection range of the piston of the CNN-Multi-LSR method and the coherence length of the incident light, which is given by (13):
In this study, ${\lambda _{\min }}$ is 660 nm, ${\Delta _{1/2}}\lambda $ and $\Delta \lambda $ are 2.5 and 4 nm, respectively, such that the range of the maximum piston error that can be detected is (108.90 μm, 174.24 μm). Figure 6 shows the normalized image plane intensity distribution of the 660 nm channel when the piston is 108.90 μm and 174.24 μm. It can be seen from Fig. 6 that when the piston is 108.90 μm, the image plane intensity can produce the LSR value; however, when the piston increases to 174.24 μm, there will be no left or right peaks in the accumulated signal, such that the piston at this time has exceeded the maximum detection range of the CNN-Multi-LSR method.
Fig. 6. (a) Normalized image plane intensity distribution of 660 nm wavelength channel when piston is 108.90 μm, (b) Normalized image plane intensity distribution of 660 nm wavelength channel when piston is 174.24 μm.
3. Multichannel LSR feature vector convolutional neural network
3.1 Dataset establishment and network structure
In subsection 2.2, we specifically calibrated the piston through the LSR feature vector and explained the theoretical maximum detection range of the CNN-Multi-LSR method. At present, many studies using neural networks to detect pistons are faced with the problem that a large-scale training dataset is not easy to obtain, resulting in a detection range often not exceeding 10λ. The LSR value of each channel will cycle with the piston, and this feature is used to artificially construct datasets for convolutional neural network training and verification. As shown in Fig. 5, the LSR of each channel continuously circulates through two sets of values (the two sets of values are only different at the highest and lowest points), and the period is one wavelength, such that we only need to obtain the LSR data within the piston range of one wavelength in each channel to construct the entire training set. Figure 7 shows the process of creating the dataset corresponding to the positive piston range, and the method of obtaining the dataset of the negative piston range is equivalent.
The method of obtaining the validation set is similar to that of the training set, except that the location when collecting data within one wavelength is different. After obtaining the training and validation sets within the negative translation error range in the same way, we can start training the network, and the piston range of the training set is (−108.9 μm, 108.9 μm). However, the training of neural networks becomes difficult as the number of output nodes increases. If the interval between adjacent output nodes in a network is set to 30 nm, then according to the above piston range, the structure and training difficulty of the entire network will become highly complex. To achieve high-precision, large-scale detection, we plan to train two networks to detect the piston. Table 2 shows the training data input format, detection range, and interval between the adjacent output nodes of the two neural networks.
Table 2. Input format, number of feature vectors in datasets, detection range of piston, and output node interval of two nets
The role of Net1 is to fit a linear relationship, and thus its training process can also be called a fitting process. Here, Net1 uses the simplest neural network, which only contains an input layer, a hidden layer, and an output layer, but its detection accuracy is high. The training and validation sets are a collection of 48 sets of LSR720 values in the 720 nm channel, and based on the 24 sets of LSR720 in the original data, we included an additional 24 sets to increase the training accuracy); the labels are the corresponding piston values.
The neural network Net2 adopted in this study was chosen after testing. Many networks were tested to select the network structure, including SqueezeNet and GoogleNet, but none of them realized the function of distinguishing feature vectors. Finally, we determined that Resnet18 can achieve this function. Next, to achieve lightweight training and detection, we selected three small difference networks of the Resnet18 network as Net2, and changed the input format and network parameters to match the feature vector. The input of Net2 is a feature vector composed of the LSR values of five channels, such as [LSR660, LSR690, LSR720, LSR750, and LSR780]. The actual piston interval between adjacent output nodes of Net2 is ${\lambda _{720}}$, which is precisely the detection range of Net1, and the number of output nodes in Net2 is 304. Net2 contains an input layer, eight convolutional layers, three addition layers, an average pooling layer, and a fully connected layer. The network structures of Net1 and Net2 are shown in Fig. 8.
Using the trained neural network to detect the piston, we obtain the final detection result using (14), which stems from the functions of the two networks. The function of Net2 is to locate the piston error in the range of [O2λ720, (O2+1)λ720] by distinguishing feature vectors. O2 is the output of Net2. λ720 is the length of each subrange, and it is also the actual piston interval between the adjacent output nodes of Net2, with a value of 0.72 μm. The function of Net1 is to provide a precise position within this small range; its output is O1.
In one detection, the two beams light with optical path difference piston0 is first imaged, and then the image plane intensity distribution in each channel is accumulated to obtain the feature vector L0 corresponding to piston0. L0 is used as the input of Net2 to obtain output O2, and then LSR720 in L0 is used as the input of Net1 to obtain output O1. Then, the detected piston can be calculated using (14). For example, when a feature vector with a piston of −30μm is input, O2 and O1 are −42 and 0.2342, respectively. Using (14), the piston detected is −30.0058 μm and the detection error is 5.8 nm. At this point, we have introduced the entire piston detection process, and we will show the performance of detecting a piston using a multichannel LSR feature vector convolutional neural network in the next section.
3.2 Training network
In subsection 3.1, we introduced two networks, Net1 (NN) and Net2 (CNN), which are used to detect pistons. In this subsection, we describe the training of these two networks. According to Table 2, there are 24 sets of data in the training and validation sets of Net1, and the training process can also be regarded as a fitting process. The training and validation sets of Net2 have 7260 and 4357 sets of data, respectively, which are evenly distributed among 304 nodes. The training results for the two networks are presented in Fig. 9.
Figure 9(a) shows the fitting results of the trained Net1. Because the training of Net1 is similar to the fitting process of a linear function, the training time is short (within 5 s), and the accuracy of the training results is high. The optimal performance was reached at epoch 19. The training process of Net2 is shown in (b); after 800 epochs, the verification accuracy reaches 97.04%, and the entire training process takes approximately 166 min. Next, we use these two networks to detect the piston in Section 4.
4. Performance demonstration based on numerical simulation
4.1 Detection results at the boundary of the detection range
To verify the performance of the trained network in piston detection, this study uses a trained network to detect random pistons in the range of [−100 μm, 100 μm]. The piston detection process is divided into four steps:
- (1) Perform imaging using five monochromatic light channels.
- (2) The image intensity distribution is accumulated, and Eq. (7) is used to process the accumulation result to obtain the corresponding LSR feature vector.
- (3) Input the feature vector into the two networks according to the input format of Net1 and Net2 to obtain O1 and O2.
- (4) Use Eq. (14) to calculate the piston value.
Here, we set 21 piston values uniformly distributed in [−100 μm, 100μm], and detected them under the condition of no noise and Gaussian noise of different intensities. The detection results and errors when there was no noise are shown in Fig. 10.
Fig. 10. (a) Detection results of 21 sets of piston values when there is no noise, (b) Detection error distribution of 21 sets of results in (a).
In Fig. 10, it can be seen that the 21 sets of detection results obtained by the CNN-Multi-LSR method are consistent with the actual piston, which indicates that the detected piston does not show a large deviation. The detailed error is shown in Fig. 10(b). The absolute value of all detection errors is less than 15 nm, and the RMS value was 6.7 nm. These results show that the CNN-Multi-LSR method can accurately measure the piston within [−100 μm, 100 μm]. Moreover, in order to verify the resistance of the CNN-Multi-LSR method to noise, we detected 21 sets of pistons under Gaussian noise with different signal-to-noise ratios, and the detection results and errors are shown in Fig. 11. Figure 11 (a) shows that the 21 sets of detection results have no large deviations compared to the actual piston, and from the detailed detection errors in Fig. 11(b), the errors of the 21 sets of detection results under three levels of noise are all less than 20 nm. Similarly, we have calculated the RMS value of the detection error at each signal-to-noise ratio to illustrate the detection level under noise: SNR = 15, RMS = 5 nm; SNR = 20, RMS = 7.3 nm; SNR = 25, RMS = 6.6 nm. Combining the detection results in Fig. 10 and Fig. 11, it is not difficult to find that the CNN-Multi-LSR method proposed in this study not only achieves large-scale and high-precision measurements of the piston, but also has strong resistance to noise. Figure 12 shows the image intensity distribution of the 660 nm channel corresponding to the 50 μm piston in the presence of Gaussian noise with signal-to-noise ratios of 15, 20, and 25.
Fig. 11. (a) Detection results of 21 sets of piston values under three different intensities of Gaussian noise: SNR=15, 20, 25, (b) Detection error distribution of 21 sets of results in (a).
Fig. 12. Image intensity distribution with 50 μm piston of 660 nm channel with Gaussian noise of SNR = 15, 20, 25.
4.2 Performance verification of CNN-Multi-LSR method at a large sample level
In this study, 5000 sets of random simulation experiments were conducted to verify the performance maintenance of the CNN-Multi-LSR method at a large sample level. Because the piston exceeding 100 μm is close to the upper limit of the detection range, it may cause 5000 sets of continuous detection to be interrupted, and thus the detection range here is selected as (−100 μm, 100 μm). The detection results for the noise-free scenario and different signal-to-noise ratios are shown in Fig. 13.
Fig. 13. (a) 5000 random detection results with no noise, (b) 5000 random detection results when SNR is 15, 20, and 25.
In Fig. 13(a), the 5000 sets of detection results have only eight sets of large deviations. Overall, it can be considered that the detection value of the piston is consistent with the actual value. For the results that seem to deviate significantly from the actual value, the main reason is that the training level of the CNN is not sufficient, which can be solved by further training. To further verify the performance of the CNN-Multi-LSR method under noise, we also performed 5000 sets of detections under the three signal-to-noise ratios of Gaussian noise, and the detection results are shown in Fig. 13(b). Compared with the situation when there is no noise, the abnormal point of the detection result increased significantly. The smaller the SNR, the more abnormal are the points.
To determine the error distribution of the detection results, the abnormal points must first be eliminated. Here, we regard results with a detection error of less than 100 nm as correct. Then, there are 4872 groups (97.44%, no noise), 4748 groups (94.96%, SNR=15), 4804 groups (96.08%, SNR=20), and 4845 groups (96.9%, SNR=25) with correct detection results. Among them, the detection accuracy under the condition of no noise reached 97.44%, which exceeds 97.04% of the final training accuracy of Net2 in Section 3. The error distributions of the results are shown in Fig. 14.
Fig. 14. (a) Histogram of detection error with no noise, (b) Histogram of detection error under Gaussian noise with SNR of 15, (c) Histogram of detection error under Gaussian noise with SNR of 20, (d) Histogram of detection error under Gaussian noise with SNR of 25.
Figure 14(a)–(d) show the error distributions for four noise intensities. The detection error distribution is the best when there is no noise. In this case, there are 4860 sets of results with an absolute error value of less than 20 nm, achieving detection accuracy of 99.72%, and the RMS error is 6.3 nm (0.0095λ). When the SNR is 15, the detection error distribution is the worst. There are 4587 sets of results with the absolute error value of less than 20 nm, achieving detection accuracy of 96.61% under this noise, and the RMS value of the error is 10.2 nm (0.0155λ). When the SNR is 20 and 25, the RMS value of the error is 7.8 nm (0.0118λ) and 8.1 nm (0.0123λ), respectively.
From the above results, it can be seen that this method can still maintain a high detection accuracy under large samples regardless of the noise level, and these accuracy values are close to or even exceed the final training accuracy of Net2. In the result of correct detection, the detection accuracy of RMS = 10.2 nm can be achieved when SNR = 15. This shows that the method proposed in this paper can not only be carried out with high precision in a large piston range, but also has strong noise resistance.
4.3 Comparison of CNN-Multi-LSR method and other co-phasing technology indicators
To illustrate the advantages of the method proposed in this paper compared with other common phase methods, we have listed the detection ranges and accuracies of several existing co-phasing methods. The comparison results are listed in Table 3. The method based on the slope of the dispersed fringe expands the detection range ${\pm} 107\lambda $ by increasing the contrast of the fringe, but as the piston is larger, the detection accuracy of this method is lower [16]. The broadband and narrowband methods used on the Keck telescope must be used in combination to achieve piston detection with an accuracy of 6 nm in the range of ${\pm} 41\lambda $ [23,24]. For some fine co-phasing algorithms, the detection accuracy is usually within 10 nm, but their detection range is very limited. For example, the detection range of the pyramid sensor is only 100 nm, and the detection range of the DFA-LSR method is less than ${\pm} \lambda /2$ [19,25]. The piston extraction method combining the PD method and neural network increases the detection range to 10λ, but it is difficult to meet the training dataset requirements to further increase the detection range [20].
Table 3. Comparison of detection range and detection accuracy of co-phasing technology
It can be seen from the piston detection methods shown in Table 3 that only the piston detection method proposed in this paper can simultaneously achieve large-scale and high-precision detection.
5. Conclusion
To realize the integration of coarse and fine co-phasing of synthetic-aperture telescopes, this paper proposes a multichannel LSR feature vector piston detection method based on a convolutional neural network (CNN-Multi-LSR method). This method uses multiple monochromatic light channels for imaging, and the imaging results are processed by the DFA-LSR method to obtain the LSR feature vector. Then, it is combined with the neural network to distinguish the LSR feature vector to realize the detection of the piston.
There are two main innovations in this method. First, in the algorithm, the DFA-LSR method is improved. It is no longer limited to the overall accumulation of the entire dispersed fringe pattern along the dispersion direction, but introduces multiwavelength channel imaging to form the LSR feature vector, which can also increase the coherence length of the incident light to increase the detection range. Second, in the acquisition of neural network training data, this article proposes a new solution to use data within one wavelength to create a large dataset, which can solve the problem of a lack of training data faced by traditional neural network methods.
Simulation studies of detecting pistons using this method have been performed. In a large number of random detections, the CNN-Multi-LSR method can always guarantee at least 94.96% accuracy, and even when the signal-to-noise ratio is 15, the RMS error is guaranteed to be 10.2 nm. As the signal-to-noise ratio increases, the RMS error decreases, and the detection accuracy gradually increases. When there is no noise, it has a detection accuracy of 97.44% and an RMS error of 6.3 nm.
In addition, the CNN-Multi-LSR method proposed in this study does not have strict requirements for the device. It can be used in the imaging model mentioned herein or integrated on the DFS device, which is also a major advantage of this method.
The following tasks remain to be completed in future work: First, we will establish a co-phase experimental platform based on the double-rectangular-aperture diffraction theory for a segmented mirror and start the experimental verification of the CNN-Multi-LSR method. Second, we will complete the verification of the CNN-Multi-LSR method on the DFS device. Third, to enhance the robustness of this method, tilt error, actual environmental noise, and other aberrations will be studied in further research.
Funding
Major Project of High-resolution Earth Observation System of China (GFZX04014307).
Disclosures
The author declares 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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