A neural network wavefront reconstruction method based on second-order moments

Abstract

This invention discloses a neural network wavefront reconstruction method based on second-order moments, aiming to solve the problems of low reconstruction accuracy and poor robustness of traditional phase retrieval algorithms under strong turbulence conditions. This method constructs a Moment-U-Net neural network combining DenseBlock, SEBlock, and U-Net structures. It extracts spot shape information through second-order moments, calculates the offset and intensity information between the actual and reference centroids of the spot, and uses the spot shape, offset, and intensity information as input features to the Moment-U-Net neural network to train the mapping relationship between the input features and the atmospheric phase screen. This invention exhibits excellent robustness and millisecond-level computation speed under strong turbulence. This invention can achieve accurate wavefront reconstruction under strong turbulence, which is of great significance in the fields of image post-construction and optical system correction.

Description

Technical Field

[0001] This invention relates to a wavefront reconstruction method based on second-order moments, and more particularly to an improved wavefront reconstruction method based on deep learning. Background Technology

[0002] The Zernike mode method is one of the most common methods widely used in wavefront reconstruction. It effectively describes and reconstructs the wavefront shape by representing wavefront distortion as a combination of a series of Zernike polynomials, possessing a strong theoretical foundation and wide applicability. However, despite its significant achievements, the Zernike mode method still has some limitations. First, as the complexity of wavefront distortion increases, low-order Zernike modes may fail to adequately describe the complex wavefront morphology, leading to increased errors in higher-order modes. In cases of large field of view and high-order distortion, the mode method may not provide sufficiently accurate reconstruction results. Second, the Zernike mode method also faces the problem of modal cross-coupling, where Zernike modes of different orders may influence each other, leading to inaccurate wavefront slope calculations, which is particularly evident in higher-order reconstructions. To address this issue, Huang Shengyang et al. proposed a method combining Laplacian eigenfunctions, effectively reducing the influence of modal cross-coupling and thus improving the accuracy of higher-order wavefront reconstruction. Besides these issues, Zernike mode methods typically require significant computation when dealing with complex wavefront distortions, especially in high-order reconstructions, where computational complexity and efficiency become key factors limiting their application. The matrix multiplication simplification method proposed by Yong Feng Zhang et al. addresses this problem to some extent by simplifying the calculation of wavefront slopes, improving computational efficiency and accuracy, particularly in complex systems such as large telescopes. Finally, the sensitivity of Zernike mode methods to noise is another limitation. At low signal-to-noise ratios, the coefficients of the mode method may be affected by noise, leading to a decrease in wavefront reconstruction accuracy. Although some methods can mitigate the impact of noise, the inherent sensitivity of the mode method to noise means that noise-induced errors cannot be completely avoided, especially in complex environments. In summary, while Zernike mode methods offer certain advantages in wavefront reconstruction, particularly in simplifying computation and improving efficiency, they also face challenges such as modal cross-coupling, computational complexity, and noise effects. To improve the accuracy and efficiency of wavefront reconstruction, researchers are increasingly turning to new technologies such as deep learning to overcome the shortcomings of traditional mode methods.

[0003] In recent years, deep learning methods have made breakthroughs in wavefront reconstruction, especially in overcoming the limitations of traditional mode methods. Traditional mode methods, such as wavefront reconstruction based on Zernike polynomials, while possessing a sound theoretical foundation, often face challenges in handling complex wavefront distortions, noise interference, and high-order wavefront reconstruction. These methods require multiple iterations and highly complex computations, and are susceptible to modal cross-coupling and noise, leading to decreased accuracy.

[0004] To overcome these limitations, an increasing number of studies are turning to wavefront reconstruction methods based on neural networks. For example, Xin Liu et al. proposed using a U-shaped convolutional neural network (U-net) to handle wavefront reconstruction with irregular apertures, demonstrating the robustness and efficiency of deep learning methods in reconstructing irregularly shaped wavefronts. Similarly, Xin Tang et al. proposed a method based on a Bayesian convolutional neural network that can predict the absolute phase distribution from a single sheared phase map and provide uncertainty estimates, offering a new approach to improving reconstruction accuracy and reliability. Furthermore, Hong Guo explored noise issues in wavefront reconstruction using artificial neural networks (ANNs), showcasing the advantages of neural networks over traditional methods in noisy conditions.

[0005] These neural network-based methods not only effectively address the computational efficiency issues of traditional pattern decomposition methods, but also better handle complex wavefront distortion and noise problems through the nonlinear modeling capabilities of deep learning. More importantly, neural networks can automatically learn the complex features of wavefronts from large amounts of data, making wavefront reconstruction no longer dependent on precise pattern decomposition, thus reducing computational complexity and the need for prior knowledge.

[0006] However, despite the powerful potential these deep learning methods have shown in wavefront reconstruction, several challenges remain. Most methods rely on large amounts of training data, and computational efficiency remains an issue in some real-time demanding applications. To further improve efficiency and reduce reliance on large amounts of data, new wavefront reconstruction methods can be explored by combining the advantages of physical models and neural networks. Summary of the Invention

[0007] To address the limitations of existing wavefront reconstruction methods based on phase retrieval algorithms, this invention proposes a neural network-based wavefront reconstruction method based on second-order moments. This method optimizes traditional phase retrieval algorithms and overcomes the shortcomings of insufficient handling of atmospheric disturbances and nonlinear effects. By precisely adjusting the training dataset and the neural network loss function, this invention can effectively extract the shape information of the light spot through second-order moments and combine it with multiple features such as offset and intensity to achieve high-precision mapping with the atmospheric phase, thereby significantly improving the accuracy and robustness of wavefront reconstruction. Furthermore, this invention combines DenseBlock, SEBlock, and U-Net structures, fully leveraging their advantages in feature extraction and contextual information integration. DenseBlock and SEBlock enhance the network's expressive power through dense connections, while the U-Net structure retains feature information at different scales through skip connections, optimizing the training process of deep networks. Thanks to the powerful computational capabilities of deep learning, this invention's method can efficiently complete wavefront reconstruction within milliseconds, significantly improving processing speed and meeting the needs of real-time applications. This method demonstrates superior performance under challenging conditions such as complex wavefront distortion, strong turbulence, and atmospheric disturbances, and is particularly valuable for applications requiring high precision. Compared with traditional methods, this invention not only improves wavefront reconstruction accuracy but also enhances system stability and adaptability, exhibiting significant theoretical innovation and broad practical application prospects. It is especially suitable for fields such as astronomical observation, laser communication, and precision measurement, providing a novel, high-precision, and high-efficiency solution for real-time wavefront reconstruction.

[0008] The technical solution adopted in this invention is:

[0009] A second-order moment-based neural network wavefront reconstruction method is proposed. A Moment-U-Net neural network combining DenseBlock, SEBlock, and U-Net structures is constructed. The second-order moment is used to extract the shape information of the light spot, and the offset and intensity information of the actual centroid of the light spot from the reference centroid are calculated. The light spot shape information, light spot offset, and light spot intensity information are used as input features of the Moment-U-Net neural network to train the mapping relationship between the input features and the atmospheric phase screen.

[0010] The light spot shape information is extracted by calculating the second moment of the light spot intensity distribution using the following formula;

[0011]

[0012]

[0013] in and These represent the light spots at... and The second moment in the direction reflects the spread of the light spot. It is the mixed second moment of the light spot, representing the tilt of the light spot shape. It is the intensity distribution of the light spot. , These are the coordinates of the centroid of the light spot;

[0014] The spot offset is obtained by solving the change in the centroid position of the spot using the following formula;

[0015]

[0016]

[0017] in, and These represent the offsets of the light spot in the x and y directions, respectively. and For the actual centroid coordinates, Reference standard centroid coordinates;

[0018] The light spot intensity information is obtained by directly calculating the total light intensity of the light spot area using the following formula;

[0019]

[0020] in The total intensity of a single light spot The intensity value of each pixel within the light spot area. The coordinates of pixels within the spot area;

[0021] Finally, a loss function is used to optimize the backpropagation process. The network updates the gradient based on a smaller loss value and gradually learns the accurate mapping relationship from the input features to the atmospheric phase screen, thereby achieving high-precision wavefront reconstruction.

[0022] Furthermore, in the dataset of the Moment-U-Net neural network, a phase map describing atmospheric wavefront distortion is used as the output of the neural network. Each dataset contains information on the feature shape, offset, and intensity of the light spot, as well as its corresponding atmospheric phase information. Before inputting the data into the network, the input feature shape, offset, and intensity information of the light spot are normalized, and the data is divided into a training set, a validation set, and a test set. This allows the neural network to learn the mapping relationship from the feature shape, offset, and intensity information of the light spot to the atmospheric phase screen. After training, the accuracy of the network after training is tested using the test set.

[0023] Furthermore, the expression for the loss function is as follows:

[0024]

[0025] in, The phase screen value is calculated using a neural network. The standard output is the true phase screen corresponding to the spot feature information in the dataset. The loss function measures the error by calculating the root mean square error (RMSE) between the network output and the standard output. The formula for calculating RMSE is:

[0026]

[0027] in, The first prediction of the neural network Data point values, This represents the actual value of the corresponding data point.

[0028] Furthermore, the network structure adopts a multi-layer convolutional neural network architecture, with dense blocks (DenseBlock and SEBlock) serving as internal feature extraction modules, and U-Net as the overall network framework for multi-scale feature extraction and fusion. By combining dense blocks (DenseBlock and SEBlock) and downsampled blocks (DownsampleBlock), features are effectively extracted and atmospheric phase information is gradually recovered. The Adam optimizer is used during optimization, and L2 regularization is used to alleviate overfitting. At the same time, the network includes a group normalization layer (GroupNorm) to accelerate convergence and stabilize the training process, and the ReLU activation function is used to enhance nonlinear expressive power and solve the gradient vanishing problem.

[0029] During the training of the neural network, the Adam algorithm is used as the optimization method. Based on the backpropagation loss of each batch of data, the Adam algorithm calculates the gradient of the loss function with respect to each weight in the network, and iteratively updates the weights, causing the loss function value to gradually decrease and eventually converge to a lower level. During training, the network parameters are gradually optimized through the calculation of the loss function and the backpropagation process, thereby improving the performance of the model.

[0030] Furthermore, the backpropagation process calculates the gradient of each model parameter using the chain rule, with the following formula:

[0031]

[0032] in, It is a loss function. It is the first in the model One parameter, It is the predicted output of the neural network;

[0033] When using the RMSE loss function, for The gradient calculation formula is:

[0034]

[0035] Where N is the sample size. Standard output;

[0036] When the neural network performs backpropagation, it calculates the gradient of each parameter layer by layer according to the computational graph of the model, and uses these gradients to update the model parameters in the optimization step. By continuously adjusting the parameters, the network gradually reduces the loss function value, thereby improving the model performance and ultimately achieving high-precision wavefront reconstruction.

[0037] Furthermore, through dense connections and multi-layer convolution operations, multi-scale features of the input data are learned, and efficient information transfer is achieved; the output of each layer is connected to the outputs of all previous layers to avoid information loss.

[0038] Furthermore, feature compression is performed using a downsampling block (DownsampleBlock), while the resolution of the atmospheric phase screen is restored using an upsampling block (UpsampleBlock).

[0039] This invention proposes a neural network wavefront reconstruction method based on second-order moment information. By expanding the dataset and optimizing the loss function, it significantly improves traditional phase retrieval algorithms. The method employs a Moment-U-Net neural network, taking light spot intensity, offset, and second-order moment information as input, and outputting an atmospheric phase screen. By introducing dense blocks (DenseBlock, SEBlock) and a downsampling-upsampling mechanism, combined with the ReLU activation function and group normalization layers, the vanishing gradient problem is effectively solved, and the training stability and accuracy of the network are improved.

[0040] Compared with traditional phase retrieval algorithms, this invention not only optimizes the network architecture and enhances the network's ability to process complex data, but also extracts the spot shape information output by the wavefront sensor through a second-order moment mathematical model. This enables the network to extract effective features more comprehensively from the spot data, thereby achieving accurate wavefront reconstruction. By expanding the dataset and optimizing the loss function, this invention further improves the accuracy and robustness of phase retrieval, demonstrating strong application prospects. Attached Figure Description

[0041] Figure 1 This is the output diagram of the wavefront sensor.

[0042] Figure 2 The second moment is used to describe the shape of the light spot.

[0043] Figure 3Input features (spot intensity, offset, second moment) to the neural network.

[0044] Figure 4 This is a schematic diagram of the Moment-U-Net structure.

[0045] Figure 5 This is a schematic diagram of the structure of a dense block SEBlock in Moment-U-Net.

[0046] Figure 6 This is a schematic diagram of the structure of a dense block (DenseBlock) in Moment-U-Net.

[0047] Figure 7 This is a schematic diagram of the DownsampleBlock structure in Moment-U-Net.

[0048] Figure 8 This is a schematic diagram of the UpsampleBlock structure in Moment-U-Net.

[0049] Figure 9 This is a graph showing the prediction results on the test set. Detailed Implementation

[0050] The present invention will now be described in further detail with reference to the accompanying drawings.

[0051] This invention proposes a neural network wavefront reconstruction method based on second-order moments to achieve high-precision atmospheric phase reconstruction. This method first extracts the shape information of the light spot using second-order moments, including the position of the light spot within the wavefront. , and The directional extension characteristics, combined with the offset and intensity information of the light spot, are used as input features for the neural network. An atmospheric phase screen serves as the precise output target. To achieve efficient mapping between input features and atmospheric phase, this invention employs a U-Net as the main framework, combined with DenseBlock and SEBlock modules in a neural network, ensuring the network can accurately learn the complex nonlinear relationship between light spot features and atmospheric phase. During training, this invention uses RMSE as the loss function and optimizes the network through backpropagation, gradually approximating the precise mapping relationship between input and output. This method not only significantly improves the accuracy of wavefront reconstruction but also greatly enhances processing speed by utilizing the efficient computational architecture of deep learning, exhibiting superior reconstruction capabilities, especially in complex turbulent environments.

[0052] Compared to traditional pattern-based reconstruction methods, deep learning-based wavefront reconstruction techniques achieve more accurate and faster wavefront recovery. By extracting features such as the shape, intensity, and offset of the wavefront sensor output spot, a deep neural network can map these features to the atmospheric phase screen of the actual wavefront distortion, thus achieving wavefront reconstruction. This method can extract spot shape information through second-order moments, and the input inclusiveness of the neural network effectively solves the mathematical complexity of second-order moment calculation in traditional methods. Through the nonlinear modeling capabilities of deep learning, the neural network can extract key information from complex spot data and perform wavefront reconstruction directly without relying on traditional computational frameworks, greatly simplifying the wavefront recovery process. This advantage enables the method to significantly improve reconstruction accuracy and processing speed under complex environmental conditions such as strong turbulence, thereby overcoming the limitations of traditional methods.

[0053] The training dataset of this invention contains two parts of information in each group: spot feature information and the corresponding atmospheric phase screen. The spot feature information is extracted in the following way:

[0054] Light spot shape information: The shape information of the light spot is extracted by calculating the second moment of the light spot intensity distribution. The formula for calculating the second moment is:

[0055]

[0056] in, Let be the second moment of the light spot. Intensity distribution of the light spot As coordinates, and These are the orders of the moments. The moments are calculated using the second-order moments. The value can effectively characterize the geometry of the light spot.

[0057] Spot offset: The offset is extracted by calculating the change in the position of the spot centroid. The formula for calculating the offset is:

[0058]

[0059]

[0060] in, and These are the centroid coordinates of the current light spot. and The reference centroid coordinates.

[0061] Light spot intensity information: The intensity information of the light spot is extracted by calculating the total light intensity of the light spot area. The formula is as follows:

[0062]

[0063] in The total intensity of a single light spot The intensity value of each pixel within the light spot area. The coordinates of pixels within the spot area;

[0064] This invention combines U-Net as the main framework with DenseBlock and SEBlock as internal modules, employing a deep convolutional neural network to map light spot feature information to predict the corresponding atmospheric phase screen. This network extracts high-level features from the input light spot feature information through multi-layer convolution and pooling operations, and then performs wavefront reconstruction. The final mapping relationship can be described by the following formula:

[0065]

[0066] in, For the output atmospheric phase screen, The input is the feature information of the light spot. The mapping function representing the neural network. These are the weights and bias parameters of the network. The neural network processes the input data layer by layer, and finally generates a predicted phase screen through convolutional layers and fully connected layers. This includes the influence of atmospheric turbulence on the wavefront.

[0067] Data normalization: In the data processing of this invention, to optimize data quality and improve model training performance, the input data is normalized. The input data is a four-dimensional array, typically represented as (N, C, H, W), where N is the number of samples, C is the number of channels, and H and W are the width and height of the image, respectively. Each feature channel is normalized independently to ensure that the numerical range and variation characteristics of different features are handled reasonably, avoiding a decline in model training performance due to excessive feature differences. Considering the differences in the numerical distribution of different features in the input data, this invention employs outlier handling and two normalization methods. Specifically:

[0068] Outlier Handling: Since outliers in real-world data often appear at the edges of 16×16 regions and are frequently meaningless, this invention first applies a circular mask to remove invalid information from image edges. Subsequently, only the data within the valid region of the mask is normalized to its maximum and minimum values. This local normalization method effectively removes the influence of edge outliers, ensuring data validity and the accuracy of the normalization process.

[0069] Normalization of Spot Intensity and the First Two Eigenvalues ​​of the Second Moment: Spot intensity represents the sum of pixel values ​​within the spot region, while the first two eigenvalues ​​of the second moment describe the intensity of the spot and its spatial extent, and these features are always positive. To unify the numerical range of these features, this invention employs the Min-MaxNormalization method to compress their numerical range to [0, 1]. This method effectively reduces the numerical range of features, thereby improving the model's convergence speed and making the impact of different features on model training more balanced. The specific normalization formula is as follows:

[0070]

[0071] in, These are the original eigenvalues. and These are the minimum and maximum values ​​of the feature across all samples in the dataset, respectively. This normalization process maps the data to the range [0, 1], ensuring a balanced impact of different features on the loss function during model training and preventing negative effects on model training due to excessively large numerical ranges for certain features.

[0072] The slopes in the X and Y directions, as well as the last eigenvalue of the second moment, are normalized: the slope describes the rate of change of the light spot in the X and Y directions, i.e., the degree of tilt in each direction. The last eigenvalue of the second moment reflects the directionality of the light spot and can be positive or negative. To ensure that the values ​​of these features are adapted to a unified training standard, this invention employs normalization processing in the range of -1 to 1. This method effectively unifies the numerical range of these features, reduces the impact of differences in the distribution of different feature values, and thus improves the stability and convergence speed of model training, thereby ensuring that different features receive reasonable weight allocation during training.

[0073]

[0074] in These are the original feature values. and These are the minimum and maximum values ​​of this feature in the dataset, respectively. These are the normalized feature values. Through this normalization method, the data is compressed to the range of [−1,1], which can reduce the distribution differences between different feature values, thereby making the training of the model more stable and improving the convergence speed of the model.

[0075] loss function

[0076] This invention uses root mean square error (RMSE) as the loss function, specifically in the form of:

[0077]

[0078] Where N is the sample size. Let be the predicted value for the i-th sample. Let be the true value of the i-th sample. The RMSE loss function optimizes network parameters through backpropagation, minimizing prediction error and thus improving wavefront reconstruction accuracy.

[0079] The Moment-U-Net neural network takes light spot intensity, offset, and second-order moment information as input and outputs an atmospheric phase screen. This network employs a multi-layer convolutional neural network architecture, combining dense block (DenseBlock, SEBlock) and downsampled block (DownsampleBlock) modules to effectively extract features and gradually recover atmospheric phase information. During training, the ReLU activation function is used to address the vanishing gradient problem, and a group normalization layer (GroupNorm) is used to stabilize the training process, improving the network's training efficiency and stability.

[0080] In particular, Moment-U-Net's advantages are reflected in the following aspects:

[0081] Multi-layer dense blocks: Through dense connections and multi-layer convolutional operations, the network can effectively learn multi-scale features of the input data and achieve efficient information transfer. The output of each layer is connected to the outputs of all previous layers, avoiding information loss.

[0082] Combining downsampling and upsampling: The network compresses features using downsampled blocks while simultaneously restoring the resolution of the atmospheric phase screen using upsampled blocks. This strategy helps the network effectively integrate information between global and local features.

[0083] Solving the vanishing gradient problem: Using the ReLU activation function enables neural networks to better handle the vanishing gradient problem, especially in deep networks, avoiding the risk of the activation function becoming completely inactive in the negative half-axis region.

[0084] Stable training process: Group normalization layers (GroupNorm) are widely used after each convolutional operation to accelerate training, reduce memory consumption, and improve the model's generalization ability.

[0085] The network design fully considers the multidimensional characteristics of the data and can effectively reconstruct the atmospheric phase screen through deep convolution and dense connections.

[0086] The method described above uses the Adam algorithm as the network optimizer during the training of Moment-U-Net. Based on the backpropagation loss of each batch of data, the Adam algorithm calculates the gradient of the loss with respect to each weight in the network and updates the weights so that the loss can be reduced and eventually converge to a lower level.

[0087] This invention provides a wavefront reconstruction method based on a second-order moment neural network, comprising the following steps:

[0088] I. Acquisition of light spot images

[0089] Images of light spots are acquired from a Shack-Hartmann wavefront sensor, such as... Figure 1 As shown, each image contains multiple light spots, representing local wavefront information at different locations within the sensor array. By applying a second-order moment mathematical model, this invention can accurately extract the shape features of each light spot, while simultaneously acquiring the spot's offset and intensity information. This information is crucial for wavefront reconstruction. Finally, by combining the light spot data output by the sensor, the corresponding atmospheric phase map is collected and analyzed, providing accurate input data for subsequent wavefront reconstruction.

[0090] II. Feature Selection and Calculation

[0091] Extract the following features from each spot image (e.g.) Figure 3 (as shown)

[0092] Spot intensity: The intensity value of each spot is determined by the brightness of each pixel in the image, reflecting the intensity distribution of the spot. Intensity information provides important clues about the local features of the wavefront and plays a crucial role in the subsequent wavefront reconstruction process.

[0093] Displacement information: By calculating the offset between the actual centroid of the light spot and the reference centroid, the displacement of the light spot can be accurately obtained. direction and Displacement in direction. This information reveals the tilt and changes of the local wavefront, helping to accurately locate wavefront deformation and distortion.

[0094] Second-order moment information: The calculation of the second-order moment for each light spot includes... , and Components can effectively describe the geometric characteristics of a light spot, such as its shape, orientation, and ellipticity. Figure 2 As shown. This information provides a deeper understanding of the light spot distribution, further aiding in the analysis and reconstruction of the wavefront's structure and properties.

[0095] III. Feature Fusion

[0096] The extracted features (spot intensity, displacement, second moment, etc.) are integrated into a multi-channel input data. Each channel represents a feature, and these features are finally combined into a four-dimensional tensor. The dimensions of each sample are: [number of samples, number of channels, width, height].

[0097] IV. Data Normalization Processing

[0098] In the data processing of this invention, to optimize data quality and improve model training performance, the input data is normalized. The input data is a four-dimensional array, typically represented in the form (N, C, H, W), where N is the number of samples, C is the number of channels, and H and W are the width and height of the image, respectively. Each feature channel is normalized independently to ensure that the numerical range and variation characteristics of different features are handled reasonably, avoiding a decline in model training performance due to excessive feature differences. Considering the differences in the numerical distribution of different features in the input data, this invention employs outlier handling and two normalization methods. Specifically:

[0099] Outlier Handling: Since outliers in real-world data often appear at the edges of 16×16 regions and are frequently meaningless, this invention first applies a circular mask to remove invalid information from image edges. Subsequently, only the data within the valid region of the mask is normalized to its maximum and minimum values. This local normalization method effectively removes the influence of edge outliers, ensuring data validity and the accuracy of the normalization process.

[0100] Normalization of Spot Intensity and the First Two Eigenvalues ​​of the Second Moment: Spot intensity represents the sum of pixel values ​​within the spot region, while the first two eigenvalues ​​of the second moment describe the intensity of the spot and its spatial extent, and these features are always positive. To unify the numerical range of these features, this invention employs the Min-MaxNormalization method to compress their numerical range to [0, 1]. This method effectively reduces the numerical range of features, thereby improving the model's convergence speed and making the impact of different features on model training more balanced. The specific normalization formula is as follows:

[0101]

[0102] in, These are the original eigenvalues. and These are the minimum and maximum values ​​of the feature across all samples in the dataset, respectively. This normalization process maps the data to the range [0, 1], ensuring a balanced impact of different features on the loss function during model training and preventing negative effects on model training due to excessively large numerical ranges for certain features.

[0103] V. Mask Application

[0104] To improve the training accuracy and robustness of the wavefront reconstruction model, a masking technique was employed when processing the spot input features to remove the spot features from the edge regions of the wavefront sensor. These edge regions typically do not contain valid wavefront information and may therefore interfere with the training process. By setting a threshold, the mask retains only the core region of the spot, thus avoiding the impact of invalid information on model performance.

[0105] Furthermore, the output of the atmospheric phase screen was masked to ensure that only valid phase information was considered. This masking operation helps remove irrelevant edge effects in the atmospheric phase map, ensuring that the model only processes the core region of the wavefront during training, thereby improving data validity, training efficiency, and final reconstruction accuracy.

[0106] VI. Network Structure Design

[0107] like Figure 4-8 As shown, a deep neural network combining DenseNet and U-Net structures is used to improve the accuracy of image reconstruction and wavefront restoration. The DenseNet part effectively utilizes the feature information extracted from previous layers through densely connected DenseBlock and SEBlock modules, enhancing the network's expressive power and feature transfer efficiency. Each DenseBlock and SEBlock module fuses the output features of each layer with those of the preceding layers through dense connections, thereby optimizing the feature learning and wavefront restoration process. The U-Net structure progressively restores the feature map size through an upsampled module (UpsampleBlock) and combines high-level features with low-level details using skip connections, effectively preserving spatial information and improving the accuracy of image reconstruction. This network structure, combining the feature utilization of DenseNet and the skip connection advantages of U-Net, provides an efficient and accurate solution for wavefront reconstruction tasks.

[0108] VII. Forward Propagation

[0109] During the forward propagation process, the input data passes through multiple convolutional layers, pooling layers, and upsampling layers in the network, progressively extracting and fusing features. Convolutional layers capture local structural information in the spot image through local perceptual fields, while pooling layers reduce data dimensionality through downsampling, enhancing feature robustness. Upsampling layers progressively restore the spatial resolution of the image, ensuring that details are not lost in the reconstructed image. Combined with skip connections in the U-Net structure, the network effectively fuses high-level abstract features with low-level detailed information, further improving reconstruction accuracy. Through this multi-level feature extraction and information fusion, the network can accurately handle spatial structure, displacement, and intensity variations in spot images, thereby achieving high-precision wavefront reconstruction.

[0110] VIII. Loss Function Calculation

[0111] A loss function is calculated based on the difference between the network output and the actual atmospheric phase screen. The loss function uses the root mean square error (RMSE) to measure the error between the reconstructed result and the true value. By quantifying the reconstruction error, RMSE guides the network to adjust parameters during training, thereby gradually reducing the gap between prediction and reality and improving the network's reconstruction accuracy. The loss function plays a crucial role in the optimization process, ensuring that the network is trained in the direction of minimizing error, ultimately achieving high-precision wavefront reconstruction.

[0112] IX. Gradient Vanishing and Regularization

[0113] To avoid the vanishing gradient problem, an appropriate activation function (ReLU) was chosen. This activation function effectively mitigates the risk of gradient vanishing by introducing a small slope to preserve gradients in negative regions, thus promoting deeper learning by the network. Furthermore, GroupNorm is applied after each convolutional layer to normalize the input data, accelerating the network's convergence speed and helping to reduce overfitting, thereby improving the model's generalization ability. These design features help ensure that the network optimizes stably and efficiently during training, ultimately achieving better reconstruction results.

[0114] In terms of regularization, L2 regularization (weight decay) was adopted to further improve the model's generalization ability and reduce the risk of overfitting. L2 regularization penalizes network weights, preventing them from becoming too large, and prompting the network to focus more on important features during optimization, thus improving the model's stability.

[0115] 10. Gradient Descent Optimization

[0116] The Adam algorithm is used to optimize network parameters. During optimization, the gradient is calculated using backpropagation, and the network weights are updated based on the gradient to minimize the loss function. Through this process, the network continuously adjusts its parameters, gradually improving performance and optimizing the accuracy and precision of the output. The Adam optimizer combines adaptive learning rate and momentum methods, enabling the network to converge more efficiently and stably during training, ultimately achieving better wavefront reconstruction results.

[0117] XI. Skip Connections and Feature Fusion

[0118] Skip connections are introduced into the network design to effectively fuse features at different levels. Through skip connections, low-level detailed features and high-level abstract features are combined, thus preserving more spatial information, which is crucial for image reconstruction tasks. Low-level features provide rich spatial details, while high-level features capture deeper semantic information. Skip connections allow the network to better convey this information, improving reconstruction accuracy and ensuring a balance between detail and global structure. This design significantly enhances the network's performance in handling complex image reconstruction tasks, especially in high-precision tasks such as wavefront reconstruction, achieving more accurate results.

[0119] 12. Training Phase

[0120] During training, an early stopping strategy was employed, meaning training was stopped prematurely when the error on the validation set stopped decreasing. This strategy helps prevent overfitting during training, ensuring good generalization ability on unseen data. To further evaluate model performance, mean squared error (RMSE) was used as the evaluation metric to measure the difference between the predicted atmospheric phase screen and the actual phase screen. By continuously monitoring the validation error and using appropriate evaluation metrics, the model was ensured to converge towards the optimal direction throughout training, thereby achieving high-precision wavefront reconstruction.

[0121] XIII. Reconstruction of Atmospheric Phase Screen

[0122] A well-trained neural network can accurately reconstruct the atmospheric phase screen based on the input light spot intensity, displacement, and second-order moment information. The atmospheric phase screen reflects the phase changes experienced by light waves as they pass through the atmosphere and is a key component of the wavefront control system. This neural network model allows for the efficient extraction of crucial phase information from the light spot image, enabling precise compensation for atmospheric interference, improving the overall performance of the wavefront control system, and ensuring high-quality imaging and transmission.

[0123] XIV. Application Scenarios

[0124] This method can be applied to various optical wavefront control systems, such as telescopes, laser communications, and astronomical observation equipment. In these applications, real-time and high-precision reconstruction of the atmospheric phase screen helps improve imaging and communication quality and overcome the effects of atmospheric disturbances.

[0125] XV. Dataset and Training Evaluation

[0126] The network was trained using a large-scale labeled dataset, with the labels being actual atmospheric phase screens. The dataset contained various light spot features and their corresponding atmospheric phase screens. During training, the reconstruction accuracy of the model was gradually improved by continuously optimizing the network parameters.

[0127] This invention provides a wavefront reconstruction method based on a second-order moment neural network, aiming to solve the problems of low reconstruction accuracy and poor robustness of traditional phase retrieval algorithms under strong turbulence conditions. This method is based on the Moment-U-Net neural network structure, which combines DenseBlock, SEBlock, and U-Net. It extracts the shape of the light spot output from the wavefront sensor using a second-order moment model and correlates it with the wavefront's... , The actual centroid offset from the standard centroid and the spot intensity are used together as input features for training and reconstructing a neural network. Deep learning is used to optimize the wavefront reconstruction process, achieving accurate prediction of atmospheric phase screens. Figure 9 As shown. The improved dataset design enables the network to quickly and accurately reconstruct the wavefront, even in strong turbulence (Cn^2=2.1×10). -12 It exhibits excellent robustness and millisecond-level computation speed in environments with a density of m⁻² / ³. The improved loss function effectively solves the gradient vanishing problem, improving training stability and prediction accuracy. This invention effectively overcomes the limitations of traditional phase retrieval algorithms, enabling accurate wavefront reconstruction under strong turbulence, and has significant implications for image post-construction and optical system correction.

[0128] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any transformations or substitutions that can be conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of the present invention.

Claims

1. A wavefront reconstruction method based on second-order moments in a neural network, characterized in that: A Moment-U-Net neural network combining DenseBlock, SEBlock, and U-Net structures is constructed. The shape information of the light spot is extracted through the second moment, and the offset and intensity information of the actual centroid of the light spot from the reference centroid are calculated. The shape information, offset, and intensity information of the light spot are used as the input features of the Moment-U-Net neural network, and the mapping relationship between the input features and the atmospheric phase screen is trained. The light spot shape information is extracted by calculating the second moment of the light spot intensity distribution using the following formula; ； ； ； in and These represent the light spots at... and The second moment in the direction reflects the spread of the light spot; It is the mixed second moment of the light spot, representing the tilt of the light spot shape. It is the intensity distribution of the light spot; , These are the coordinates of the centroid of the light spot; The spot offset is obtained by solving the change in the centroid position of the spot using the following formula; ； ； in, and These represent the offsets of the light spot in the x and y directions, respectively. and For the actual centroid coordinates, Reference standard centroid coordinates; The light spot intensity information is obtained by directly calculating the total light intensity of the light spot area using the following formula; ； in The total intensity of a single light spot The intensity value of each pixel within the light spot area. The coordinates of pixels within the spot area; Finally, a loss function is used to optimize the backpropagation process. The network updates the gradient based on a smaller loss value and gradually learns the accurate mapping relationship from the input features to the atmospheric phase screen, thereby achieving high-precision wavefront reconstruction.

2. The neural network wavefront reconstruction method based on second-order moments according to claim 1, characterized in that: The Moment-U-Net neural network dataset uses a phase map describing atmospheric wavefront distortion as the neural network output. Each dataset contains information on the shape, offset, and intensity of light spots, as well as their corresponding atmospheric phase information. Before inputting the data into the network, the input light spot shape, offset, and intensity information are normalized, and the data is divided into a training set, a validation set, and a test set. This allows the neural network to learn the mapping relationship from the light spot shape, offset, and intensity information to the atmospheric phase screen. After training, the accuracy of the network is tested using the test set.

3. The neural network wavefront reconstruction method based on second-order moments according to claim 1, characterized in that: The loss function expression is as follows: ； in, The phase screen value is calculated using a neural network. The standard output is the true phase screen corresponding to the spot feature information in the dataset. The loss function measures the error by calculating the root mean square error (RMSE) between the network output and the standard output. The formula for calculating RMSE is: ； in, The first prediction of the neural network Phase value, Let N be the actual phase value in the dataset, and N be the total number of phase values.

4. The neural network wavefront reconstruction method based on second-order moments according to claim 1, characterized in that: The network structure adopts a multi-layer convolutional neural network architecture, with dense blocks (DenseBlock and SEBlock) serving as internal feature extraction modules, and U-Net as the overall network framework for multi-scale feature extraction and fusion. By combining dense blocks (DenseBlock and SEBlock) and downsampled blocks (DownsampleBlock), features are effectively extracted and atmospheric phase information is gradually recovered. The Adam optimizer is used during optimization, and L2 regularization is used to alleviate overfitting. At the same time, the network includes a group normalization layer (GroupNorm) to accelerate convergence and stabilize the training process, and the ReLU activation function is used to enhance nonlinear expressiveness and solve the gradient vanishing problem.

5. The neural network wavefront reconstruction method based on second-order moments according to claim 1, characterized in that: During the training of the neural network, the Adam algorithm is used as the optimizer. Based on the backpropagation loss of each batch of data, the Adam algorithm is used to calculate the gradient of the loss with respect to each weight in the network and update the weights so that the loss can be gradually reduced and eventually converge to a lower level. During the training process, the gradient of the loss function with respect to the network parameters is calculated by automatic differentiation technique and combined with the backpropagation algorithm to gradually optimize the model parameters.

6. The neural network wavefront reconstruction method based on second-order moments according to claim 5, characterized in that: Backpropagation involves calculating the gradient of each model parameter using the chain rule, with the following formula: ； in, It is a loss function. It is the first in the model One parameter, It is the predicted output of the neural network; When using the RMSE loss function, for The gradient calculation formula is: ； Where N is the sample size. Standard output; When a neural network performs backpropagation, it calculates the gradient of each parameter according to the computation graph and applies these gradients to the update steps of the model parameters. By continuously adjusting the parameters, the network can reduce the loss and gradually improve the performance of the model, ultimately achieving high-precision wavefront reconstruction.

7. The neural network wavefront reconstruction method based on second-order moments according to claim 1, characterized in that: By employing dense connections and multi-layer convolution operations, the system learns multi-scale features of the input data and achieves efficient information transfer. The output of each layer is connected to the outputs of all preceding layers to avoid information loss.

8. The neural network wavefront reconstruction method based on second-order moments according to claim 1, characterized in that: Feature compression is performed using a downsampling block (DownsampleBlock), while the resolution of the atmospheric phase screen is restored using an upsampling block (UpsampleBlock).

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