A kernel space optimization method for high-precision full-wave inverse scattering imaging

Abstract

In view of the problem that the optimization method based on the subspace is insufficient in precision in the field of full-wave inverse scattering imaging, a kernel subspace optimization method for full-wave inverse scattering high-precision imaging is disclosed, which relates to the fields of machine learning and full-wave inverse scattering. The method combines the robust framework of the traditional subspace optimization method and the adaptive learning advantage of the kernel method, and transfers the iteration of the original subspace optimization method to the reproducing kernel Hilbert space, so that the evolution in the learning iteration process is learned with the least mean square error as the learning criterion. The kernel-based subspace optimization method combines the mature and interpretable framework of the subspace optimization method with the adaptive learning ability of the kernel method, and the method provides a robust and general algorithm for full-wave inverse scattering in the fields of medical imaging, geophysical exploration and nondestructive testing, and can provide accurate reconstruction results.

Description

Technical Field

[0001] This invention relates to the fields of machine learning and full-wave inverse scattering. Background Technology

[0002] Full-wave inverse scattering plays a crucial role in fields such as medical imaging, geophysical exploration, and nondestructive testing. Accurately reconstructing the physical properties of a target from scattered electromagnetic fields has profound significance for both scientific research and practical applications. Among the algorithms developed for this purpose, subspace optimization methods have become widely used due to their strong data processing capabilities. However, in full-wave inverse scattering problems with highly nonlinear data and significant noise, traditional subspace optimization algorithms still face challenges. These limitations lead to decreased accuracy, slower convergence, and instability during the reconstruction process. In recent years, deep learning has become a powerful tool, with some techniques integrating deep neural networks enabling the reconstruction and inversion of complex scatterers in highly nonlinear data. Although some deep learning-based methods have achieved good results, the black-box nature of neural networks often raises concerns about interpretability. This lack of interpretability makes the algorithm's decision-making process difficult to trace, especially in the medical field, where diagnostic results require high reliability and transparency to ensure patient safety. This lack of transparency not only undermines doctors' trust in the method's output but may also make it difficult to verify the reliability of results in critical diagnostic scenarios. Therefore, how to improve reconstruction accuracy and efficiency while ensuring the interpretability of the method in highly nonlinear and significantly noisy data has become an urgent need.

[0003] Adaptive filtering, as a classic adaptive data processing method, boasts advantages such as lightweight, real-time performance, adaptability, robustness, and high accuracy, achieving great success in areas such as system identification, inverse modeling, prediction, and interference cancellation. Recently, a powerful adaptive filter, called the kernel adaptive filter, has been proposed, which incorporates kernel methods into the adaptive filter framework to solve highly nonlinear prediction problems.

[0004] This invention combines kernel adaptive filtering with subspace optimization. By mapping the data optimized by traditional subspace to the regenerated Hilbert space, it obtains high-dimensional information of the data and updates the weight coefficients using the similarity between the current sample information and historical information. It has successfully achieved excellent performance in the full-wave inverse scattering problem with highly nonlinear data. Summary of the Invention

[0005] Inspired by the research background described above, this invention proposes a kernel subspace optimization method with strong interpretability and generalization ability. This method introduces a kernel into the traditional subspace optimization method. The kernel can map data into the regenerated Hilbert kernel space and perform implicit solutions. It has high-precision, robust, and highly adaptive data processing capabilities under limited computing resources.

[0006] This invention is applied to the field of inverse scattering imaging. The nucleus space optimization algorithm significantly improves imaging accuracy and computational efficiency by effectively extracting key information from complex scattering data. This algorithm not only helps accurately reconstruct the internal structure of the target object but also possesses strong interpretability, helping researchers understand the contribution of different scattering characteristics to the imaging results. Therefore, it is suitable for applications requiring high-precision and high-reliability imaging, such as medical imaging, geophysical exploration, and non-destructive testing.

[0007] This method first divides the input data of inverse scattering imaging into a discrete grid of fixed size (e.g., a 64x64 grid) for grid-by-grid processing. By using a kernel space optimization algorithm, detailed information and background features in the grid can be efficiently extracted, and the features of the scattering object are dynamically updated using kernel space optimization. The technical solution of this invention is: a kernel space optimization method for high-precision full-wave inverse scattering imaging, comprising:

[0008] Step 1: Define the number of iterations based on the kernel subspace optimization method as T, and the contrast χ of the generated matrix sequence as... The filter order n will Divided into new sample sequences Using implicit mapping χ t Mapped to Hilbert space, it is represented as

[0009] Step 2: Initialize the weight vector Ω(0) = 0 in the regenerating kernel Hilbert space. Using stochastic gradient descent, the weight vector for the i-th iteration is calculated as follows:

[0010]

[0011] Where e(i) is the prediction error, Ω(i) is the weight vector in the Hilbert space of the regeneration kernel, i represents the time step, d(i) represents the expected value of the i-th iteration, and χ(i) represents the input contrast vector of the i-th iteration.

[0012] Step 3: Calculate the instantaneous gradient at the current time step i to update the weight vector. The weights at time step i are as follows:

[0013]

[0014] Where η is the step size;

[0015] During the iteration process, Ω(i) is further expressed as:

[0016]

[0017] Step 4: After training at time step i, the new weight vector Ω(i) represents the historical samples. A linear combination of the historical error e(j); for a new input sample The system output y(i) is:

[0018]

[0019] Step 5: y(i) is represented in Hilbert space as the inner product of the input samples; by the kernel least mean square method, y(i) is:

[0020]

[0021] Where κ(·) is the kernel function;

[0022] Step 6: Iterate through Steps 2 to 5 alternately to realize the learning and iterative evolution of the subspace optimization algorithm in the Hilbert space.

[0023] Furthermore, the kernel function is κ(χ(i),χ(j)), and its expression is:

[0024]

[0025] Where σ is the bandwidth parameter of the Gaussian kernel.

[0026] Furthermore, the following evaluation indicators are used to evaluate the optimization method, and the success of the optimization is determined based on the evaluation results; otherwise, the optimization is carried out again.

[0027]

[0028] Where x and y represent the images, μ is the image mean, and σ is the image variance. xy c1 and c2 are the covariances of the images x and y, respectively, and constants added to avoid division by zero.

[0029] Furthermore, the following evaluation indicators are used to evaluate the optimization method, and the success of the optimization is determined based on the evaluation results; otherwise, the optimization is carried out again.

[0030]

[0031] 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, respectively, and MAX is the maximum pixel value in the image.

[0032] Compared to classic subspace optimization methods, this invention retains the original mathematical structure of subspace optimization and, by introducing kernel methods, achieves a complementary advantage between regenerating kernel Hilbert space optimization and traditional subspace optimization algorithms. This algorithm model, while preserving the mathematical framework of kernel mapping, further enhances the generalization ability of subspace optimization, effectively compresses the size of the solution space, and inherits the advantages of kernel methods in high-dimensional feature extraction, thereby significantly improving the ability to express nonlinear feature data.

[0033] Meanwhile, kernel-based subspace optimization, through its efficient online incremental learning strategy, significantly alleviates the dependence on large-scale data volumes and computational resources. The introduction of kernel methods not only improves the efficiency of the learning process but also reduces the requirement for model size. Under the same scale conditions, the performance and accuracy of kernel-based subspace optimization algorithms far surpass those of traditional subspace optimization algorithms, providing a more efficient and accurate solution for processing and analyzing high-dimensional data. Attached Figure Description

[0034] Figure 1 This is a schematic diagram of the experimental results for inverse scattering imaging of irregular geometric shapes.

[0035] Figure 2 This is a schematic diagram of the experimental results of inverse scattering imaging of the MNIST handwritten digit dataset.

[0036] Figure 3 The improvement rate of the proposed method compared to the subspace optimization algorithm (SOM) and the traditional inversion algorithm (BP) under various experimental conditions is shown. Detailed Implementation

[0037] 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.

[0038] This invention introduces the implicit enhancement feature representation capability of kernel functions into the iteration of the subspace optimization algorithm, and performs joint optimization during the iteration process, utilizing the powerful approximation capabilities of kernel methods and optimization algorithms. The computation of each iteration is equivalent to the progressive state of the atomic space optimization system in the time dimension, and the transfer of iteration variables is equivalent to the state evolution process in the dynamic system, thus constituting a discrete approximation of the overall state evolution equation.

[0039] This approach not only preserves the mathematical structure of traditional subspace optimization algorithms but also incorporates the high-dimensional mapping capabilities of kernel space methods, enabling the system to adaptively capture nonlinear features in complex data scenarios. Furthermore, this layer-by-layer progressive optimization design significantly improves computational efficiency and optimization capabilities while maintaining the algorithm's physical interpretability.

[0040] The overall process of this invention and Figure 1 The pseudocode shown Figure 1 First, let T be the number of iterations for the subspace optimization algorithm, and let χ be the contrast of the generated matrix sequence. Next, choose an appropriate filter order n, and Divided into new sample sequences Using implicit mapping χ t Mapped to the regenerating kernel Hilbert space, it is represented as The weight vector Ω(i) in the regenerating kernel Hilbert space is initialized to 0, and the i-th iteration is performed. Then, the error is obtained by comparing the predicted and true values. After the i-th training step, the new weight vector Ω(i) can be represented as the historical samples. The system's predicted output y(i) is obtained by combining the historical error e(j) with the linear combination of the two factors.

[0041] In the experiment, the kernel subspace optimization algorithm updates the weight parameters iteratively by utilizing the similarity between the current sample and historical samples. The iterative process includes the following steps:

[0042] Step 1 (Data Preparation): Collect and prepare training data, and perform preprocessing.

[0043] Step 2 (Parameter Initialization): Initialize the parameters of the kernel subspace optimization algorithm, including contrast, Green's function value, singular value decomposition of Green's function, and step size.

[0044] Step 3 (Forward Propagation): Input the sample data into the kernel space optimization algorithm, and output the system's predicted value using historical data, current data, and weight parameters.

[0045] Step 4 (Error Calculation): Calculate the error between the predicted output and the true value, expressed as follows:

[0046] e(i) = d(i) - y(i) (7)

[0047] Step 5 (Update weight parameters using error): Update the weight coefficients in the system and record the error between the current sample's true value and the predicted value.

[0048] Step 6 (System Output): Iterate through the previous samples, and save the updated weight parameters each time. After a certain number of iterations, the system outputs the prediction result.

[0049] The experiment performed iso-inverse scattering imaging on datasets of irregular geometric shapes and handwritten characters, and compared the results with traditional Subspace Optimization (SOM) and Backpropagation (BP) algorithms. Evaluation metrics included metric 1 (Structural Similarity Simulation) and metric 2 (Peak Signal-to-Noise Ratio (PSNR), calculated as follows:

[0050]

[0051] 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.

[0052]

[0053] 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.

[0054] Specific experimental results are shown in Tables 1 to 4, and performance comparisons are shown in Tables 1 to 4. Figures 1 to 2 Performance improvements are shown in the appendix. Figure 3 The red solid and dashed lines represent the improvement rate of the proposed method compared to BP in evaluation metrics 1 and 2, respectively, while the black solid and dashed lines represent the improvement rate of the proposed method compared to SOM in evaluation metrics 1 and 2, respectively. The horizontal axis is from left to right... Figure 1 and Figure 2 The experimental objects in the diagram correspond to square, annular, circular, Austrian graphic, and handwritten fonts 1, 2, 3, and 4, respectively. The results show that, in different datasets, the method proposed in this invention achieves higher output accuracy and better stability compared to the classical BP method and the classical subspace optimization method.

[0055] The entire pseudocode process of this invention is shown in the table below, where the matrix after discretization of the Green's function in region D is G. D The discretized matrix of the Green's function from region D to the receiver is Gs, J + For deterministic induced current;

[0056]

[0057]

[0058] In view of the above situation and in order to overcome the shortcomings of the prior art, the innovation of the present invention lies in:

[0059] 1. Kernel adaptive filtering is applied to solve the full-wave inverse scattering problem, and the eigenmap of the kernel is used to better capture the nonlinear relationship in full-wave inverse scattering.

[0060] 2. Establish an adaptive learning mechanism for the subspace optimization algorithm to dynamically adapt to the complexity of full-wave inverse scattering.

[0061] 3. An integrated kernel method is used to enhance robustness while maintaining the complete physical model of the subspace optimization algorithm, providing better detection performance than traditional subspace optimization algorithms.

[0062] The above description of specific embodiments details the implementation process and application scenarios of the kernel subspace optimization algorithm, enabling those skilled in the art to implement the kernel-based subspace optimization algorithm according to the present invention. The above embodiments are merely illustrative of the technical solutions 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.

[0063] Table 1. Comparison of Algorithm Performance for Irregular Geometric Shapes (Evaluation Index 1)

[0064]

[0065] Table 2 Comparison of Algorithm Results for Irregular Geometric Shape Experiments (Evaluation Index 2)

[0066]

[0067] Table 3 Comparison of Algorithms in the MNIST Handwritten Font Experiment (Evaluation Metric 1)

[0068]

[0069] Table 4 Comparison of Algorithms in the MNIST Handwritten Font Experiment (Evaluation Metric 2)

[0070]

Claims

1. A kernel space optimization method for full-wave backscattering high-precision imaging, the method comprising: Step 1: define the iteration number of the kernel subspace optimization method as T, and the contrast of the generated matrix sequence χ is represented as Filter order n, will Divide into new sample sequences Use implicit mapping Map χ t To Hilbert space, denoted as Step 2: initializing the weight vector Ω(0) = 0 in the reproducing kernel Hilbert space, and calculating the i-th iteration by the stochastic gradient descent method as follows: where e(i) is the prediction error, Ω(i) is the weight vector in the reproducing kernel Hilbert space, i represents the time step, d(i) represents the expected value of the i-th iteration, and χ(i) represents the input contrast vector of the i-th iteration; Step 3: calculating the instantaneous gradient at the current time step i to update the weight vector, and the weight at the time step i is as follows: where η is the step size; During the iteration process, Ω(i) is further represented as: Step 4: After training through the i-th time step, the new weight vector Ω(i) is expressed as a linear combination of the history samples and the history errors e(j); for a new input sample The output y(i) of the system is: Step 5: y(i) is expressed in the form of inner product of input samples in the Hilbert space; y(i) is obtained by the kernel least mean square method as follows: where κ(·) is the kernel function; Step 6: alternately iterating steps 2 to 5 to realize the learning and evolutionary iteration of the subspace optimization algorithm in the Hilbert space.

2. The nuclear space optimization method for full-wave inverse scattering high-precision imaging of claim 1, wherein, The kernel function is κ(χ(i), χ(j)), and the expression is as follows: where σ is the bandwidth parameter of the Gaussian kernel.

3. The nuclear space optimization method for full-wave inverse scattering high-precision imaging of claim 1, wherein, The optimization method is evaluated by using the following evaluation index, and whether the optimization is successful is determined according to the evaluation result, otherwise the optimization is re-performed; where x and y represent the images, μ is the mean of the images, σ is the variance of the images, σ xy is the covariance of the images x and y, and c1, c2 are constants added to avoid division by zero.

4. The nuclear space optimization method for full-wave inverse scattering high-precision imaging of claim 1, wherein, The optimization method is evaluated by using the following evaluation index, and whether the optimization is successful is determined according to the evaluation result, otherwise the optimization is re-performed; where m and n are the dimensions of the image, I(i,j) and K(i,j) are the pixel values of the original image and the reconstructed image at the pixel position (i,j), and MAX is the maximum pixel value in the image.

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