Electromagnetic inverse scattering imaging method based on coring depth expansion network

Abstract

The invention relates to an electromagnetic inverse scattering imaging method based on a nucleation depth expansion network, and belongs to the technical field of electromagnetic inverse scattering imaging. A kernel convolution module and a random Fourier feature module are integrated at the bottleneck position of the U-Net network, the kernel convolution module effectively extracts local spatial features of dielectric attributes by integrating a nonlinear kernel function, and the random Fourier feature module realizes efficient capture of global spectrum features through cosine base projection. Therefore, the expression ability of the sub-wavelength structure and the wavelength scale interference mode is considered. Through recursive iteration and end-to-end training of a physical operator, a reconstruction result can be optimized step by step, and organic combination of data driving and physical prior is realized. According to the method, through recursive iteration and end-to-end training, the reconstruction precision, generalization ability and physical consistency are effectively improved, and the method is suitable for the electromagnetic wave imaging fields of medical imaging, nondestructive testing and the like.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic backscattering imaging technology, specifically relating to an electromagnetic backscattering imaging method based on a multi-physics-constrained nucleated depth unfolding network, which is applicable to medical microwave imaging, non-destructive testing, geophysical exploration and other application scenarios. Background Technology

[0002] Electromagnetic inverse scattering imaging (EISI) is an imaging technique that infers the distribution of internal dielectric properties by measuring the scattering response of a target object to electromagnetic waves. This technique has significant applications in medical imaging, non-destructive testing, and security inspection. However, EISI is inherently a highly nonlinear and ill-conditioned inverse problem. Traditional solution methods mainly rely on iterative optimization algorithms, such as the Born iteration method, contrast source inversion method, and subspace optimization method. Although these methods have a solid theoretical foundation, they typically suffer from high computational complexity, sensitivity to initial conditions, and susceptibility to local optima, making it difficult to meet the requirements of efficiency and robustness in practical applications.

[0003] In recent years, deep learning methods have been introduced into the EISI field, leveraging the powerful feature representation capabilities of neural networks to achieve efficient end-to-end imaging. However, most existing deep learning methods neglect the physical constraints in the electromagnetic scattering process, resulting in insufficient physical consistency of the network output and a high dependence on the distribution of training samples, limiting generalization ability and causing the network output to deviate from the laws of electromagnetic physics, thus affecting the reliability and practicality of imaging. Some studies have attempted to use deep unfolded networks, mapping the iterative steps of traditional optimization algorithms to the layers of neural networks, achieving a combination of physical models and neural network structures, improving physical interpretability and generalization ability. However, existing deep unfolding methods still have limitations in nonlinear feature modeling and the expression of complex physical relationships, making it difficult to fully explore the multi-scale and high-order features in the electromagnetic scattering process.

[0004] In summary, existing technologies in the field of electromagnetic backscattering imaging still face challenges such as low computational efficiency, poor physical consistency, and limited nonlinear feature extraction capabilities. Therefore, there is an urgent need for a novel, highly efficient imaging method that can organically integrate physical operators with deep networks and explicitly guarantee physical consistency, thereby improving imaging accuracy, robustness, and practical application value. Summary of the Invention

[0005] This invention overcomes the following technical problems existing in current electromagnetic backscattering imaging technology:

[0006] (1) Limitations of traditional optimization methods: Traditional methods such as Born iteration method and contrast source inversion method have high computational complexity, are sensitive to initial values, and are prone to getting trapped in local optima, making it difficult to meet the efficiency and robustness requirements of practical applications;

[0007] (2) Insufficient physical consistency of deep learning methods: Existing deep learning methods lack physical interpretability and ignore physical constraints in the electromagnetic scattering process, resulting in poor physical consistency of reconstruction results and limited generalization ability;

[0008] (3) Limited ability to model nonlinear features: Existing deep unfolding networks are insufficient in extracting high-order nonlinear features and fusing multi-scale information, which affects the reconstruction accuracy of complex electromagnetic scattering problems.

[0009] To achieve the above objectives, this invention provides an electromagnetic backscattering imaging method based on a nucleated deep unfolded network. This method transforms the iterative solution process of the subspace optimization method into a forward propagation process of a deep neural network, achieving gradual optimization and reconstruction of the induced current distribution through a multi-cascaded network structure. Therefore, the technical solution of this invention is: an electromagnetic backscattering imaging method based on a nucleated deep unfolded network, comprising two or more cascaded neural networks. Each layer of the neural network includes: a multi-physical quantity fusion module, a nucleated U-Net module, and a physical operator module. The input to the entire neural network is the determined induced current distribution and the dielectric constant distribution. The output of each layer includes: the predicted induced current distribution, the predicted scattered field, and the predicted dielectric constant distribution. The multi-physical quantity fusion module concatenates the predicted induced current distribution, the predicted scattered field, and the predicted dielectric constant distribution output from the previous layer, and then inputs this concatenation to the nucleated U-Net module. The nucleated U-Net module outputs the predicted induced current distribution. The predicted induced current distribution and the determined induced current distribution serve as inputs to the physical operator module, which outputs the predicted scattered field and the predicted dielectric constant distribution.

[0010] Furthermore, the iterative update formula for each k-layer neural network is transformed into:

[0011] ;

[0012] in This represents the induced current of the k-th layer of the neural network. For deterministic induced current, This represents the function mapping relationship for the k-th level network module. These are the learnable parameters of the neural network.

[0013] Furthermore, the kernelized U-Net module is based on the U-Net encoder-decoder architecture, with a kernelized bottleneck structure at the bottleneck layer. The kernelized bottleneck structure consists of a first kernelized convolutional module, a second kernelized convolutional module, a random Fourier feature module, and a max pooling module connected in series. The input of the max pooling module also includes the output of the second kernelized convolutional module.

[0014] The first and second kernelized convolutional modules have the same structure, and the output of the kernelized convolutional modules is the same. The calculation formula is:

[0015] ;

[0016] in, This is the output of a standard convolution operation. Indicates the input feature map, This represents the convolution operation. Represents the convolution kernel. Indicates the output position, and the exp(·) term is the response of the radial basis function kernel. It is the bandwidth of the RBF core.

[0017] D frequencies are randomly sampled from a Gaussian distribution. D phase shifts are sampled from a uniform distribution [0, 2π]. The output of the random Fourier feature module is obtained. The calculation formula is:

[0018] .

[0019] Furthermore, the calculation method of the physical operator module is as follows:

[0020] ;

[0021] ;

[0022] in, Represents the total electric field. Indicates the incident electric field. This represents the Green's operator within the domain, used to calculate the contribution of the scattered field generated by the induced current J at any point within the region of interest. This represents the observed Green's operator, used to calculate the scattered field from the induced current J in the region of interest to the location of the external receiver. ;

[0023] The dielectric constant distribution εᵣ is then calculated using the following formula;

[0024] ;

[0025] in, Indicates wave number, This represents a position vector, specifically an arbitrary point in space.

[0026] Furthermore, determine the loss function for the entire network. for:

[0027] The loss function comprises four main components: induced current loss Scattering field loss Structural similarity loss and mean square error loss :

[0028] ;

[0029] ;

[0030] ;

[0031] ;

[0032] in, , and These represent the predicted induced current, predicted scattered field, and predicted dielectric constant distribution, respectively, with corresponding reference values ​​as follows: , and , Indicates the number of transmitting antennas. Indicates the number of receiving antennas. Indicates the structural similarity between the reconstructed image and the original image. Denotes the 2-norm of a matrix;

[0033] ;

[0034] Where λ1 and λ2 are hyperparameters that balance the contributions of different loss terms.

[0035] Compared with the prior art, the present invention has the following significant advantages:

[0036] 1. Significantly improved physical consistency: Through the organic integration of physical operators and deep networks, the physical consistency of the imaging process is explicitly guaranteed, and the reconstruction results conform to the physical constraints of the Maxwell equations and the Lippmann-Schwinger equations.

[0037] 2. Enhanced nonlinear modeling capability: The kernelized U-Net structure utilizes the infinite-dimensional mapping characteristics of the RBF kernel to achieve accurate modeling of high-order nonlinear relationships in electromagnetic scattering, and has a stronger feature representation capability compared to traditional convolutional networks.

[0038] 3. Multi-scale feature extraction optimization: The synergistic effect of random Fourier feature modules and kernelized convolution captures both local subwavelength structures and global wavelength-scale interference patterns, achieving effective fusion of multi-scale information.

[0039] 4. Significantly improved reconstruction accuracy: Experimental results show that compared with SOM, KSOM and SOM-Net methods, the present invention improves the SSIM index by an average of 14.84%-75.82% and the PSNR index by an average of 17.95%-39.11%.

[0040] 5. Significantly enhanced generalization ability: The end-to-end optimization strategy with multiple physical constraints effectively suppresses overfitting and improves the network's generalization performance in different scenarios, making it suitable for multiple fields such as medical microwave imaging, non-destructive testing, and geophysical exploration.

[0041] 6. Optimized computational efficiency: The deep unfolding architecture significantly reduces computation time compared to traditional iterative methods, and the linear complexity of the RFF module further improves computational efficiency.

[0042] This invention provides a novel solution for electromagnetic backscattering imaging technology that combines physical and data-driven approaches. It achieves high-precision and high-efficiency imaging reconstruction while ensuring physical interpretability, and has significant theoretical implications and broad application prospects. Attached Figure Description

[0043] Figure 1 The overall architecture for deep neural networks.

[0044] Figure 2 The network architecture for kernel-based U-Net.

[0045] Figure 3 The results are from the inverse scattering imaging experiment using the MNIST handwritten digit dataset.

[0046] Figure 4 The improvement achieved by the proposed method compared to the subspace optimization algorithm (SOM) and the kernel subspace optimization algorithm (KSOM) is shown.

[0047] Figure 5 The induced current (including real and imaginary parts) and the corresponding reference value are calculated using the proposed method.

[0048] Figure 6 The scattered field calculated by the proposed method and the corresponding reference value are shown.

[0049] Figure 7 This is the experimental setup for actual measurement.

[0050] Figure 8 The results are from the backscattering imaging experiment.

[0051] Figure 9 The improvement rate of the proposed method compared to SOM and KSOM is given.

[0052] Figure 10The proposed method is used to calculate the induced current (including real and imaginary parts) and the corresponding reference value based on measured data.

[0053] Figure 11 The proposed method is used to calculate the scattering field and the corresponding reference value based on measured data. Detailed Implementation

[0054] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0055] Physical constraints for the iterative reconstruction process: In the inference phase, the input scattered field data is first used to obtain initial induced current estimates and dielectric constant guesses through singular value decomposition (SVD) and backpropagation (BP). Subsequently, the data is passed sequentially through multiple cascaded kernelized U-Net modules, each module outputting an updated induced current distribution. In each cascade, the Green's function operator G... D and observation operator G obs The corresponding physical quantities are calculated, including the total electric field, the scattered field, and the dielectric constant distribution. These calculations strictly adhere to electromagnetic scattering theory, ensuring that the reconstruction results satisfy the constraints of Maxwell's equations. The final cascaded output is the reconstructed dielectric constant distribution, which not only boasts high reconstruction accuracy but also guarantees good physical interpretability.

[0056] Example 1: Numerical Simulation Experiment Based on MNIST Dataset

[0057] This embodiment provides a numerical simulation experiment of electromagnetic backscattering imaging based on the MNIST handwritten digit dataset to verify the effectiveness of multi-physics-constrained kernelized deep unfolding networks.

[0058] System hardware configuration: This embodiment uses a computing platform equipped with dual NVIDIA GeForce RTX 3090 GPUs (24GB VRAM), an Intel(R) Xeon(R) Gold 6248R @ 3.00GHz CPU, and 256GB DDR4 memory. The software environment includes Python 3.8, PyTorch 1.12.0, and CUDA 11.6.

[0059] Dataset Construction and Preprocessing: This embodiment constructs a training dataset containing 9000 samples with different dielectric constant distributions. The imaging region of each sample is set to 64×64 pixels, corresponding to a physical size of 6λ×6λ (where λ is the operating wavelength). The real part of the dielectric constant ranges from 1.0 to 4.0, and the imaginary part ranges from 0.0 to 0.5, covering typical application scenarios such as biological tissues and dielectric materials. The scattered field data is generated through simulation using the method of moments (MOM). There are 16 transmitting antennas and 32 receiving antennas, uniformly distributed on a circular array around the imaging region with a radius of 4λ. To simulate actual measurement conditions, Gaussian white noise with a signal-to-noise ratio of 30dB is added to the simulation data.

[0060] Dataset Construction and Preprocessing: This embodiment constructs electromagnetic scattering simulation data based on the MNIST handwritten digit dataset. 10,000 samples are selected from the MNIST dataset, and each 28×28 pixel handwritten digit image is resized to 64×64 pixels, corresponding to an imaging region of physical size 6λ×6λ. The dielectric constant is set according to the following rules: the dielectric constant of the background region (pixels with a digit of 0) is 1.0; the real part of the dielectric constant of the foreground region (pixels with non-zero digits) is set to a random value between 1.0 and 3.0, and the imaginary part is set to a random value between 0.1 and 0.5. To increase the complexity of the inversion task, a small circular scatterer is randomly placed in each simulation sample, with its position, size, and dielectric constant all varying randomly.

[0061] Multi-station radar configuration: This embodiment uses a multi-station radar configuration for scattering field data acquisition. Number of transmitting antennas N t =16, number of receiving antennas N r =32 antennas, evenly distributed on a circular array with a radius of 4λ around the imaging area. The operating frequency is set to 2.45GHz, corresponding to a free space wavelength λ=12.2cm. Each transmitting antenna illuminates the target area sequentially, and all receiving antennas simultaneously receive the scattered field data, forming 16×32=512 scattered field measurements.

[0062] Noise model settings: To simulate actual measurement conditions, Gaussian white noise with a signal-to-noise ratio of approximately 30 dB is added to the simulated scattering field data. The noise model is calculated according to the following formula:

[0063] (12)

[0064] Among them G r and G i The noise consists of real and imaginary parts, respectively, both of which follow a standard normal distribution. , This represents the Frobenius norm.

[0065] Kernelized Deep Unfolding Network Architecture Design: The network in this embodiment comprises four cascaded modules, each corresponding to one iteration of the traditional SOM algorithm. The network input includes: an initial induced current estimate obtained through Singular Value Decomposition (SVD) and Backpropagation (BP) methods. The dimensions are 64×64×1; the initial dielectric constant is guessed. The dimensions are 64×64×1. The input of each cascaded module is the concatenation of the output of the previous stage and the physical quantity calculated by the physical operator. The number of input channels is 3 (real part of induced current, imaginary part, and dielectric constant).

[0066] Detailed parameters of the kernelized U-Net architecture: The encoder path contains four convolutional blocks, each containing two 3×3 convolutional layers with 64, 128, 256, and 512 channels respectively. Each convolutional layer is followed by batch normalization and a ReLU activation function. Downsampling is achieved through 2×2 max pooling. The decoder path adopts a symmetrical structure, achieving upsampling through 2×2 transposed convolutions and skipping connections with the corresponding layers in the encoder.

[0067] Kernel convolution module implementation: A kernel convolution module is integrated into the bottleneck layer (512 channels). The formula for calculating kernel convolution is:

[0068] (13)

[0069] in The bandwidth parameter for the RBF kernel is initially set to 1.0 and is adaptively adjusted during training. The kernelized convolution module fuses features with the output of a standard convolutional layer through a 1×1 convolutional layer.

[0070] Random Fourier Feature Module Configuration: The RFF module is integrated in the bottleneck layer, with a random frequency count D=512. The frequencies ω follow a Gaussian distribution. The sampling is performed in the middle, where σ = 1.0; the phase offset b is sampled from a uniform distribution [0, 2π]. The feature mapping function is:

[0071] (14)

[0072] The output dimension is 1024. The RFF module fuses the output of a fully connected layer with the kernelized convolutional output, and the fusion weights are in the range of [0,1] and are automatically learned during training.

[0073] Numerical Implementation of the Physical Operator: The physical operator in this embodiment is implemented based on the numerical calculation of the Green's function. For the two-dimensional TM wave scattering problem, the Green's function is:

[0074] (15)

[0075] in It is a zeroth-order Hankel function of the first kind. is the free space wavenumber.

[0076] In the numerical implementation, computational efficiency is improved by pre-computing and storing the Green's function matrix. Observation operator G obs The dimension is 256×4096 (corresponding to 16×16 receiving antennas and 64×64 imaging pixels), and the domain operator G D The dimension is 4096×4096. To avoid the numerical instability of matrix inversion, singular value decomposition (SVD) is used for regularization, with a truncation exponent L=30.

[0077] Weighting and optimization strategy of the loss function: In the comprehensive loss function of this embodiment, the weights of each item are set as follows: induced current loss weight is 1.0, scattered field loss weight is 1.0, SSIM loss weight λ1=0.5, and MSE loss weight λ2=0.1. These weights are determined by a grid search method to ensure the coordinated optimization of each physical quantity. The training process uses the Adam optimizer with an initial learning rate of 1×10⁻. 4 β1=0.9, β2=0.999, and the weight decay coefficient is 1×10⁻ 5 The learning rate scheduling strategy is as follows: maintain the initial learning rate for the first 20 rounds, and linearly decay to 5×10⁻ ... 5 The linear decay to 1×10⁻ in rounds 41-60. 5 .

[0078] The training process was implemented as follows: The training dataset was randomly divided into a training set (5000 samples), a validation set (2000 samples), and a test set (1000 samples). The batch size was set to 16, and each training round contained 60 batches. To improve training stability, gradient pruning was used, with a pruning threshold set to 1.0. The entire training process took approximately 6 hours to complete.

[0079] Detailed inference process: In the inference stage, the input scattered field data is first normalized, and then the initial induced current estimate is obtained through backpropagation. Specifically, the initial estimate calculation formula is as follows:

[0080] (16)

[0081] in The pseudo-inverse of the observation operator is obtained through SVD decomposition.

[0082] Subsequently, the data is processed sequentially through four cascaded kernelized U-Net modules, each with a computation time of approximately 50 ms. Within each cascade, the corresponding physical quantity, the total electric field, is calculated using physical operators. Scattered field and dielectric constant Their expressions are as follows (17) to (19).

[0083] (17)

[0084] (18)

[0085] (19)

[0086] Where k is the current sequence number of the cascaded U-Net.

[0087] Experimental results and evaluation metrics: Evaluation metrics used were metric 1 (Structural Similarity SSIM) and metric 2 (Peak Signal-to-Noise Ratio PSNR). The calculation methods for the evaluation metrics are as follows:

[0088] (20)

[0089] Where μ is the mean of the image, σ is the variance of the image, and σ xy c1 and c2 are the covariances of the images x and y, respectively, and constants added to avoid division by zero.

[0090] (twenty one)

[0091] Where m and n are the dimensions of the image, I(i, j) and K(i, j) are the pixel values ​​at pixel position (i, j) in the original image and the reconstructed image, and MAX is the maximum pixel value in the image.

[0092] Four samples were randomly selected from the test set and recorded as samples (a), (b), (c), and (d) to verify the effectiveness of the proposed method. The inversion results of each method are attached. Figure 3 As shown, the specific experimental data results are shown in Appendix Tables 1 and 2, and the performance improvement is shown in Appendix Tables 1 and 2. Figure 4 (The red solid and dashed lines represent the improvement rates of the proposed method compared to SOM in evaluation metrics 1 and 2, respectively; the blue solid and dashed lines represent the improvement rates of the proposed method compared to KSOM in evaluation metrics 1 and 2, respectively; and the black solid and dashed lines represent the improvement rates of the proposed method compared to SOM-Net in evaluation metrics 1 and 2, respectively.) Furthermore, the induced current and scattered field calculated from the inversion results are shown in the appendix. Figures 4 to 5 (The red solid line represents the predicted value, and the black solid line represents the actual value).

[0093] According to the appendix Figure 3Based on the results and the quantitative data in Tables 1 and 3, this invention demonstrates superior performance in all test scenarios, achieving an evaluation index (1) exceeding 0.94 and an evaluation index (2) exceeding 31 dB, significantly outperforming existing methods. Specifically, the percentage improvement in metrics compared to SOM, KSOM, and KSOM-Net is shown in Tables 2 and 4.

[0094] Considering average improvement, specifically:

[0095] Compared with SOM, the method of the present invention improved the evaluation index 1 by an average of 0.2415 (an improvement of 34.13%) and the evaluation index 2 by an average of 6.8082 dB (an improvement of 27.03%).

[0096] Compared with KSOM, the method of the present invention improved the evaluation index 1 by an average of 0.1709 (an improvement of 21.95%) and the evaluation index 2 by an average of 6.0988 dB (an improvement of 23.48%).

[0097] Compared with SOM-Net, the method of the present invention improves the average evaluation index 1 by 0.1214 (an improvement of 14.84%) and the average evaluation index 2 by 4.8569 dB (an improvement of 17.95%).

[0098] Physical consistency verification: Figure 5 The real and imaginary parts of the induced current, along with their corresponding reference values, are shown for comparison. Figure 6 The scattered field was reconstructed as a one-dimensional vector, from which 512 consecutive data points were randomly selected and compared with a reference value. These visualizations demonstrate that the induced current and scattered field reconstructed using the method of this invention highly match the reference values. For the induced current, the MSE value is 1.30 × 10⁻⁶. -4 1.53×10 -4 9.11×10 -5 and 1.51×10 -4 For the scattered field, the MSE values ​​are 2.60 × 10⁻⁶. -2 2.03×10 -2 1.44×10 -2 and 2.99×10 -2 The error data above demonstrate that the method of this invention can achieve high-precision reconstruction following electromagnetic scattering under multi-physics constraints, while providing strong physical interpretability.

[0099] Example 2: Field Verification of Millimeter-Wave Imaging System

[0100] This embodiment provides a field verification scheme based on a millimeter-wave imaging system to verify the technical effect of the multi-physics-constrained nucleated depth unfolding network method described in this invention in a real hardware system.

[0101] Millimeter-wave imaging system configuration: This embodiment uses a self-developed millimeter-wave imager, such as... Figure 7 As shown. The system includes: a vector network analyzer (VNA) as the signal source and receiver; a robotic arm to control the precise positioning of the antenna probe; and an image processing computer for data acquisition and processing. The operating frequency range is 8-12 GHz, with a center frequency of 10 GHz, corresponding to a free space wavelength of 3 cm. The remaining hardware is described in Example 1 and will not be repeated here.

[0102] Antenna array design: A single-antenna scanning method is adopted, with a robotic arm controlling the antenna probe to perform a circular scan around the target object. The scanning radius is 15cm, and the number of scanning points is 48, evenly distributed on the circumference. Each scanning position serves as both a transmitting and receiving antenna. The S11 scattering parameters are measured using a vector network analyzer to obtain the scattered field data of the target object.

[0103] Test target preparation: The test targets include two categories: (1) custom metal letters, such as the letters "U" and "S", with a thickness of 2 mm, made of aluminum alloy; (2) everyday items, including metal objects such as scissors and tweezers. All test targets are placed on foam supports with low dielectric constant to reduce background interference.

[0104] Data Acquisition and Preprocessing: For each test target, 48 scattered field measurements were acquired, corresponding to 48 transmit-receive position combinations. Data preprocessing included: background subtraction, using the scattered field measured when there was no target as the background; phase calibration, eliminating system phase errors; and amplitude normalization, normalizing the scattered field amplitude to the range of [-1, 1].

[0105] Network parameter adjustments: Based on the characteristics of the measured data, the network parameters were adjusted as follows: Due to the low signal-to-noise ratio (approximately 15dB) of the measured data, the network regularization strength was increased, and the weight decay coefficient was adjusted to 5×10⁻. 5 Adjust the kernel parameter σ of the kernelized convolution to 0.8; increase the number of training epochs to 80 epochs to fully learn the features of the measured data.

[0106] Experimental Results Analysis: Four samples were randomly selected from the test set and recorded as samples (e), (f), (g), and (h). The inversion results of each method are attached. Figure 8 As shown, specific experimental data results are shown in Appendix Tables 3 and 4, and performance improvements are shown in Appendix Tables 4 and 5. Figure 9(The red solid and dashed lines represent the improvement rates of the proposed method compared to SOM in evaluation metrics 1 and 2, respectively; the blue solid and dashed lines represent the improvement rates of the proposed method compared to KSOM in evaluation metrics 1 and 2, respectively; and the black solid and dashed lines represent the improvement rates of the proposed method compared to SOM-Net in evaluation metrics 1 and 2, respectively.) Furthermore, the induced current and scattered field calculated from the inversion results are shown in the appendix. Figures 10 to 11 (The red solid line represents the predicted value, and the black solid line represents the actual value).

[0107] According to the appendix Figure 8 Based on the results and the quantitative data in Tables 5 and 7, this invention demonstrates superior performance in all practical scenarios, fully achieving evaluation metrics exceeding 0.95 (1) and 32 dB (2), significantly outperforming existing methods. Specifically, the percentage improvement in metrics compared to SOM, KSOM, and KSOM-Net is shown in Tables 6 and 8.

[0108] Considering average improvement, specifically:

[0109] Compared with SOM, the method of the present invention improved the average evaluation index 1 by 0.3873 (an improvement of 34.13%) and the average evaluation index 2 by 9.36 dB (an improvement of 39.11%).

[0110] Compared with KSOM, the method of the present invention improved the evaluation index 1 by an average of 0.1709 (an improvement of 21.95%) and the evaluation index 2 by an average of 6.0988 dB (an improvement of 23.48%).

[0111] Compared with SOM-Net, the method of the present invention improved the evaluation index 1 by an average of 0.0813 (an improvement of 9.66%) and the evaluation index 2 by an average of 6.22 dB (an improvement of 20.81%).

[0112] Physical consistency verification: Figure 10 The real and imaginary parts of the induced current, along with their corresponding reference values, are shown for comparison. Figure 11 The scattered field was reconstructed into a one-dimensional vector, from which 512 consecutive data points were randomly selected and compared with a reference value. These visualizations demonstrate that the induced current and scattered field reconstructed using the method of this invention highly match the reference values. For the induced current, the MSE value is 3.09 × 10⁻⁶. -5 4.95×10 -5 2.82×10 -5 , and 2.38×10 -5 For the scattered field, the MSE values ​​are 8.04 × 10⁻⁶. −32.17×10 −3 1.73×10 −3 , and 3.81×10 −3 The error data above demonstrate that the method of this invention can achieve high-precision reconstruction following electromagnetic scattering under multi-physics constraints, while providing strong physical interpretability.

[0113] The above description of specific embodiments details the implementation process and application scenarios of an electromagnetic inverse scattering imaging method based on a multi-physics-constrained kernelized depth unfolding network, enabling technicians to implement kernel-based subspace optimization algorithms according to the present invention. The above embodiments are only used to illustrate the technical solution of the present invention and do not limit the scope of protection of the present invention. Those skilled in the art can make various modifications and variations based on the present invention; any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0114] Table 1. Comparison of Algorithms in the MNIST Handwritten Font Experiment (Evaluation Metric 1-SSIM)

[0115]

[0116] Table 2. Percentage improvement of each algorithm in the MNIST handwritten font experiment (Evaluation metric 1 - SSIM)

[0117]

[0118] Table 3 Comparison of Algorithms in the MNIST Handwritten Font Experiment (Evaluation Metric 2 - PSNR)

[0119]

[0120] Table 4. Percentage improvement of each algorithm in the MNIST handwritten font experiment (Evaluation metric 2 - PSNR)

[0121]

[0122] Table 5. Comparison of the effects of each algorithm in the actual test (Evaluation index 1 - SSIM)

[0123]

[0124] Table 6. Percentage improvement of each algorithm in the actual test (Evaluation index 1 - SSIM)

[0125]

[0126] Table 7 Comparison of the effects of each algorithm in the actual test (Evaluation index 2 - PSNR)

[0127]

[0128] Table 8. Percentage improvement of each algorithm in the actual experiment (Evaluation index 2 - PSNR)

[0129]

Claims

1. An electromagnetic inverse scattering imaging method based on a kernelized deep unfolded network, comprising two or more cascaded neural networks, each layer of which includes: The neural network consists of a multi-physical quantity fusion module, a kernelized U-Net module, and a physical operator module. The input to the entire neural network is the determined induced current distribution and dielectric constant distribution. The output of each layer includes the predicted induced current distribution, the predicted scattered field, and the predicted dielectric constant distribution. The multi-physical quantity fusion module concatenates the predicted induced current distribution, the predicted scattered field, and the predicted dielectric constant distribution from the previous layer, and then inputs them to the kernelized U-Net module. The kernelized U-Net module outputs the predicted induced current distribution. The predicted induced current distribution and the determined induced current distribution are used as inputs to the physical operator module, which outputs the predicted scattered field and the predicted dielectric constant distribution.

2. The electromagnetic backscattering imaging method based on a kernelized deep unfolded network as described in claim 1, characterized in that the iterative update formula for each k-layer neural network is converted to: ； in This represents the induced current of the k-th layer of the neural network. For deterministic induced current, This represents the function mapping relationship for the k-th level network module. These are the learnable parameters of the neural network.

3. The electromagnetic backscattering imaging method based on a nucleated depth unfolded network as described in claim 1, characterized in that, The kernelized U-Net module is based on the U-Net encoder-decoder architecture, with a kernelized bottleneck structure at the bottleneck layer. The kernelized bottleneck structure consists of a first kernelized convolutional module, a second kernelized convolutional module, a random Fourier feature module, and a max pooling module connected in series. The input of the max pooling module also includes the output of the second kernelized convolutional module. The first and second kernelized convolutional modules have the same structure, and the output of the kernelized convolutional modules is the same. The calculation formula is: ； in, This is the output of a standard convolution operation. Indicates the input feature map, This represents the convolution operation. Represents the convolution kernel. Indicates the output position, and the exp(·) term is the response of the radial basis function kernel. It is the bandwidth of the RBF core.

4. Randomly sample D frequencies from a Gaussian distribution. D phase shifts are sampled from a uniform distribution [0, 2π]. The output of the random Fourier feature module is obtained. The calculation formula is: 。 5. An electromagnetic backscattering imaging method based on a nucleated depth unfolded network as described in claim 1, characterized in that, The calculation method of the physical operator module is as follows: ； ； in, Represents the total electric field. Indicates the incident electric field. This represents the Green's operator within the domain, used to calculate the contribution of the scattered field generated by the induced current J at any point within the region of interest. This represents the observed Green's operator, used to calculate the scattered field from the induced current J in the region of interest to the location of the external receiver. ; The dielectric constant distribution εᵣ is then calculated using the following formula; ; in, Indicates wave number, This represents a position vector, specifically an arbitrary point in space.

6. The electromagnetic backscattering imaging method based on a nucleated depth unfolded network as described in claim 1, characterized in that, Determine the loss function for the entire network. for: The loss function comprises four main components: induced current loss Scattering field loss Structural similarity loss and mean square error loss : ； ； ； ； in, , and These represent the predicted induced current, predicted scattered field, and predicted dielectric constant distribution, respectively, with corresponding reference values ​​as follows: , and , Indicates the number of transmitting antennas. Indicates the number of receiving antennas. Indicates the structural similarity between the reconstructed image and the original image. Denotes the 2-norm of a matrix; ； Where λ1 and λ2 are hyperparameters that balance the contributions of different loss terms.

Images