Nuclear subspace optimization method for full-wave inverse scattering high-precision imaging

Abstract

The invention discloses a nuclear subspace optimization method for full-wave inverse scattering high-precision imaging, and relates to the field of machine learning and full-wave inverse scattering in order to solve the problem that an existing subspace-based optimization method is insufficient in precision in the field of full-wave inverse scattering imaging. According to the method, the robustness framework of a traditional subspace optimization method and the self-adaptive learning advantage of a kernel method are combined, iteration of an optimization method of an original subspace is transferred into a regeneration kernel Hilbert space, and evolution in the iteration process is learned with the minimum mean square error as the learning criterion. According to the kernel-based subspace optimization method, a mature and interpretable framework of a subspace optimization method is combined with the adaptive learning ability of a kernel method, a robust and universal algorithm is provided for full-wave inverse scattering in the fields of medical imaging, geophysical exploration, nondestructive testing and the like, and an accurate reconstruction result can be provided.

Description

Technical Field

[0001] The present invention relates to the fields of machine learning and full-wave inverse scattering. Background Art

[0002] The full-wave inverse scattering problem plays a vital role in fields such as medical imaging, geophysical exploration, and nondestructive testing. Accurately reconstructing the physical properties of the target from the scattered electromagnetic field has far-reaching significance for both scientific research and practical applications. Among the algorithms developed for this purpose, the subspace optimization method has become a widely used method due to its strong ability to handle data. However, in the full-wave inverse scattering problem with highly nonlinear data and significant noise, the traditional subspace optimization algorithm still faces some challenges. These limitations lead to reduced accuracy, slow convergence, and instability in the reconstruction process. In recent years, deep learning has become a powerful tool. Some technologies integrate deep neural network methods, which can achieve 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 usually raises concerns about interpretability. This uninterpretability makes the decision-making process of the algorithm difficult to trace, especially in the medical field, where diagnostic results need to be highly credible and transparent to ensure patient safety. At the same time, this lack of transparency not only undermines doctors' trust in the output of the method, but also may make it difficult to verify the reliability of the results in critical diagnostic scenarios. Therefore, how to improve the reconstruction accuracy and efficiency in highly nonlinear data and significant noise data while ensuring the interpretability of the method has become a method that is urgently needed.

[0003] As a classic adaptive data processing method, adaptive filtering method has the advantages of lightweight, real-time, adaptability, robustness and high precision, and has achieved great success in the fields of system identification, inverse modeling, prediction and interference elimination. Recently, a powerful adaptive filter called kernel adaptive filter is proposed to solve the highly nonlinear prediction problem by incorporating the kernel method into the adaptive filter framework.

[0004] The present invention combines the kernel adaptive filtering method with the subspace optimization method. It maps the data of traditional subspace optimization to the regenerated Hilbert space to obtain the high-dimensional information of the data, and uses the similarity between the current sample information and the historical information to update the weight coefficient. 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 above research background, the present invention proposes an interpretable kernel subspace optimization method with strong generalization ability. This method introduces a kernel into the traditional subspace optimization method. The kernel can map the data into the regenerated Hilbert kernel space and solve it implicitly. It has high-precision, strong robustness and strong adaptability in data processing under the condition of limited computing resources.

[0006] The present invention is applied to the field of inverse scattering imaging. The kernel subspace optimization algorithm can significantly improve imaging accuracy and computational efficiency by effectively extracting key information from complex scattering data. The algorithm can not only help accurately reconstruct the internal structure of the target object, but also has strong interpretability, helping personnel understand the contribution of different scattering features to the imaging results. Therefore, it is suitable for application fields such as medical imaging, geophysical exploration and non-destructive testing that require high-precision and high-reliability imaging.

[0007] The method first divides the input data of inverse scattering imaging into discrete grids of fixed size (e.g., 64x64 grids) for grid-by-grid processing. Based on the kernel subspace optimization algorithm, the detailed information and background features in the grid can be efficiently extracted, and the features of the scatterer can be dynamically updated using the kernel subspace optimization. The technical solution of the present invention is: a kernel subspace optimization method for full-wave inverse scattering high-precision imaging, the method 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 is expressed as The filter order n is Divide into new sample sequences Using implicit mapping χ t Mapped to Hilbert space, it is expressed as

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

[0010]

[0011] 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;

[0012] Step 3: Calculate the instantaneous gradient of the current time step i to update the weight vector. The weight of time step i is 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 the i-th time step, the new weight vector Ω(i) is represented as a historical sample and a linear combination of historical errors e(j); for a new input sample The output y(i) of the system is:

[0018]

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

[0020]

[0021] Among them, κ(·) is the kernel function;

[0022] Step 6: Alternately iterate the above steps 2 to 5 to realize the learning iterative evolution of the subspace optimization algorithm in the Hilbert space.

[0023] Furthermore, the kernel function is κ(χ(i),χ(j)), expressed as:

[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 whether the optimization is successful is determined based on the evaluation results, otherwise the optimization is performed again;

[0027]

[0028] Where x and y represent images, μ is the mean of the image, σ is the variance of the image, and σ xy is the covariance of images x and y, c 1 , c 2 is a constant added to avoid division by zero.

[0029] Furthermore, the following evaluation indicators are used to evaluate the optimization method, and whether the optimization is successful is determined based on the evaluation results, otherwise the optimization is performed again;

[0030]

[0031] 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 pixel position (i,j), and MAX is the maximum pixel value in the image.

[0032] Compared with the classical subspace optimization method, the present invention retains the original mathematical structure of subspace optimization, and by introducing the kernel method, realizes the complementary advantages of reproducing kernel Hilbert space optimization and traditional subspace optimization algorithms. On the basis of retaining the mathematical framework of kernel mapping, the algorithm model further strengthens the generalization ability of subspace optimization, effectively compresses the scale of the solution space, inherits the advantages of the kernel method in high-dimensional feature extraction, and thus significantly improves the expression ability of nonlinear feature data.

[0033] At the same time, kernel-based subspace optimization greatly alleviates the dependence on large-scale data and computing resources through an efficient online incremental learning strategy. The introduction of kernel methods not only improves the efficiency of the learning process, but also weakens the demand for model size. Under the same scale conditions, the performance and accuracy of the kernel-based subspace optimization algorithm model far exceed the traditional subspace optimization algorithm, providing a more efficient and accurate solution for the processing and analysis of high-dimensional data. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 Schematic diagram of the experimental results of inverse scattering imaging of irregular geometric shapes.

[0035] Figure 2 Schematic diagram of the experimental results of inverse scattering imaging of the MNIST handwritten digit dataset.

[0036] Figure 3 It is the improvement rate of the proposed method compared with the subspace optimization algorithm (SOM) and the traditional inversion algorithm (BP) under various experimental conditions. DETAILED DESCRIPTION

[0037] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0038] The present invention introduces the kernel function to enhance the expressive power of implicit features in the iteration of the subspace optimization algorithm, and jointly optimizes in the iterative process, utilizing the powerful approximation ability of the kernel method and the optimization algorithm. The calculation of each iteration is equivalent to the progressive state of the atomic space optimization system in the time dimension, and the transmission of the iterative variable is equivalent to the process of state evolution in the dynamic system, thus forming a discrete approximation of the overall state evolution equation.

[0039] Through this method, the kernel subspace optimization algorithm not only retains the mathematical structure of the traditional subspace optimization algorithm, but also combines the high-dimensional mapping capability of the kernel space method, so that the system can adaptively capture nonlinear features in complex data scenarios. At the same time, this design based on inter-layer progressive optimization enables the kernel subspace optimization algorithm to significantly improve the computational efficiency and optimization capabilities while maintaining the physical interpretability of the algorithm.

[0040] The overall process of the present invention is Figure 1 The pseudo code shown Figure 1 First, we define the number of iterations of the subspace optimization algorithm as T, and generate the contrast χ of the matrix sequence, which can be expressed as Secondly, select the appropriate filter order n, Divide into new sample sequences Using implicit mapping χ t Mapped to the reproducing kernel Hilbert space, expressed as Initialize the weight vector Ω(i) in the reproducing kernel Hilbert space to 0 and perform the i-th iteration. Then, the error is obtained by comparing the predicted value with the true value. After the i-th step of training, the new weight vector Ω(i) can be expressed as the historical sample and the linear combination of the historical error e(j). Finally, we get the predicted output y(i) of the system.

[0041] In the experiment, the kernel subspace optimization algorithm is used to update the weight parameters through iteration and using the similarity between the current sample and the historical sample. The iteration 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 the Green's function, and step size.

[0044] Step 3 (forward propagation): Input the sample data into the kernel subspace optimization algorithm, and output the predicted value of the system through historical data, current data and weight parameters.

[0045] Step 4 (error calculation): Calculate the error between the predicted output and the true value. The expression is as follows:

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

[0047] Step 5 (update weight parameters using errors): Update the weight coefficients in the system using the error between the actual value and the predicted value of the current sample and record them.

[0048] Step 6 (system output): Iterate through the previous samples, and the updated weight parameters are saved each time. After a certain number of iterations, the system outputs the prediction results.

[0049] The experiment conducted equal inverse scattering imaging tasks on irregular geometric figures and handwritten font data sets, and compared the traditional subspace optimization algorithm (SOM) and traditional inversion algorithm (BP) distribution. The evaluation indexes used were evaluation index 1 (structural similarity SSIM) and evaluation index 2 (peak signal-to-noise ratio PSNR). The calculation method of the evaluation index is as follows:

[0050]

[0051] Where μ is the mean of the image, σ is the variance of the image, and σ xy is the covariance of images x and y, c 1 , c 2 is a constant 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 ​​of the original and reconstructed images at pixel position (i,j), and MAX is the maximum pixel value in the image.

[0054] The specific experimental results are shown in Tables 1 to 4, and the performance comparison is shown in Figure 1 to Figure 2 , performance improvement see attached Figure 3 The red solid line and dotted line represent the improvement rate of the proposed method compared with BP in evaluation index 1 and evaluation index 2, respectively, while the black solid line and dotted line represent the improvement rate of the proposed method compared with SOM in evaluation index 1 and evaluation index 2. Figure 1 and Figure 2 The experimental objects in correspondence represent square, ring, circle, Austrian figure, handwriting font 1, 2, 3, 4. The results show that in different data sets, the method proposed by the present invention has higher output accuracy and better stability compared with the classical BP method and the classical subspace optimization method.

[0055] The entire pseudo code process of the present invention is shown in the following table, where the matrix after the discretization of the Green function in region D is G D , the discretized matrix of the Green function from region D to the receiver is Gs, J + is the deterministic induced current;

[0056]

[0057]

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

[0059] 1. Kernel adaptive filtering is applied to solve the full-wave inverse scattering problem, and the kernel's characteristic mapping 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. Integrating kernel methods to enhance robustness while maintaining the complete physical model of the subspace optimization algorithm provides better detection performance than traditional subspace optimization algorithms.

[0062] Through the description of the above specific implementation methods, the implementation process and application scenarios of the kernel-based subspace optimization algorithm are described in detail, so that technicians can implement the kernel-based subspace optimization algorithm according to the content of the present invention. The above embodiments are only used to illustrate the technical scheme of the present invention and do not limit the protection scope of the present invention. Technicians in the relevant field can make various deformations and modifications according to the content of the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

[0063] Table 1 Comparison of various algorithms in irregular geometric figure experiments (evaluation index 1)

[0064]

[0065] Table 2 Comparison of various algorithms in irregular geometric figure experiment (evaluation index 2)

[0066]

[0067] Table 3 Comparison of algorithms in MNIST handwritten font experiment (evaluation index 1)

[0068]

[0069] Table 4 Comparison of algorithms in MNIST handwritten font experiment (evaluation index 2)

[0070]

Claims

1. A nuclear subspace optimization method for full-wave inverse scattering high-precision imaging, the method comprising: 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 is expressed as The filter order n is Divide into new sample sequences Using implicit mapping χ t Mapped to Hilbert space, it is expressed as Step 2: Initialize the weight vector Ω(0) = 0 in the reproducing kernel Hilbert space. By the stochastic gradient descent method, the i-th iteration is calculated as: 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: Calculate the instantaneous gradient of the current time step i to update the weight vector. The weight of time step i is as follows: Where η is the step size; During the iteration process, Ω(i) is further expressed as: Step 4: After training at the i-th time step, the new weight vector Ω(i) is represented as a historical sample and the linear combination of the historical error e(j); for a new input sample The output y(i) of the system is: Step 5: y(i) is expressed in Hilbert space as the inner product of the input samples; by the kernel least mean square method y(i) is: Among them, κ(·) is the kernel function; Step 6: Alternately iterate the above steps 2 to 5 to realize the learning iterative evolution of the subspace optimization algorithm in the Hilbert space.

2. A method for optimizing the nucleus space for full-wave inverse scattering high-precision imaging according to claim 1, characterized in that: The kernel function is κ(χ(i),χ(j)), and the expression is: Where σ is the bandwidth parameter of the Gaussian kernel.

3. The kernel space optimization method for full-wave inverse scattering high-precision imaging according to claim 1, characterized in that: The following evaluation indicators are used to evaluate the optimization method, and whether the optimization is successful is determined based on the evaluation results, otherwise the optimization is restarted; Where x and y represent images, μ is the mean of the image, σ is the variance of the image, and σ xy is the covariance of images x and y, c1, c2 are constants added to avoid division by zero.

4. The kernel space optimization method for full-wave inverse scattering high-precision imaging according to claim 1, characterized in that: The following evaluation indicators are used to evaluate the optimization method, and whether the optimization is successful is determined based on the evaluation results, otherwise the optimization is restarted; 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 pixel position (i,j), and MAX is the maximum pixel value in the image.

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