Frequency binning cumulative frequency hopping inversion method fusing wavelength correlation weighting strategy

Abstract

The invention relates to a frequency binning cumulative frequency hopping inversion method fusing a wavelength correlation weighting strategy, and the method introduces the wavelength correlation weighting strategy which gives a higher weight to a low-frequency component on the basis of a multi-frequency cross correlation contrast source inversion frame, and improves the robustness of an iterative inversion algorithm when a high-contrast target is processed. Furthermore, a frequency binning mechanism and a cumulative frequency hopping mechanism are fused in a contrast updating process, so that the computational efficiency advantage of multi-frequency synchronous inversion and the inherent robustness of progressive frequency hopping inversion can be effectively integrated. By adopting the method, efficient, reliable and universal electromagnetic quantitative inversion of a high-contrast target body can be realized.

Description

Technical Field

[0001] The present application relates to the technical field of electromagnetic inversion imaging, and in particular to a frequency binning accumulation frequency hopping inversion method integrating a wavelength-dependent weighting strategy. Background Art

[0002] In the field of electromagnetic inverse scattering research, electromagnetic quantitative inversion aims to reconstruct the dielectric parameters of a target (equivalent to the contrast function) from limited electromagnetic field measurement data. The core challenge lies in effectively addressing the inherent ill-posedness and nonlinearity of electromagnetic inverse scattering. From the perspective of iterative optimization theory, ill-posedness and nonlinearity can be explained by the dense distribution of local extreme values ​​in the objective function, which severely hinders the identification of the global optimal solution.

[0003] Existing electromagnetic quantitative inversion methods can be divided into three categories based on their basic principles: linear quantitative inversion methods, inversion methods based on iterative optimization, and neural network-driven methods. Among them, linear quantitative inversion methods are only valid under the weak scattering assumption and are only applicable to low-contrast targets. They cannot achieve reliable quantitative inversion of high-contrast targets. Inversion methods based on iterative optimization address the nonlinearity and ill-posedness of the inverse scattering problem by optimizing the iterative framework. Although they can achieve high-precision reconstruction in general inverse scattering scenarios, they have the inherent defect of high computational complexity. Although neural network-driven methods can reduce runtime costs by pre-calculating the computational burden of the training phase, their system portability is still severely restricted due to dataset dependence and domain adaptation limitations.

[0004] Despite significant progress in existing electromagnetic quantitative inversion techniques, two key challenges—ill-posedness and nonlinearity—remain fundamentally unresolved. This is particularly true for the quantitative inversion of high-contrast targets. When electromagnetic waves penetrate high-contrast targets, the spatial distribution of the electric field becomes highly complex, leading to even more complex spatial characteristics in the contrast source distribution, exacerbating the nonlinearity and ill-posedness of the electromagnetic inverse scattering problem. Furthermore, the computational complexity of electromagnetic quantitative inversion algorithms continues to be a key bottleneck. Summary of the Invention

[0005] Based on this, it is necessary to provide a frequency binning cumulative frequency hopping inversion method that integrates a wavelength-related weighting strategy to address the above technical problems, which can provide an efficient, reliable and universal electromagnetic quantitative inversion solution for high-contrast targets.

[0006] A frequency binning cumulative frequency hopping inversion method integrating a wavelength-dependent weighting strategy, the method comprising:

[0007] The spatial perturbation of dielectric parameters caused by the target in the two-dimensional electromagnetic inverse scattering problem detection scene is defined as contrast, and the secondary excitation source generated by the target scattering is defined as the contrast source; the dielectric parameters include relative permittivity and conductivity;

[0008] The two-dimensional electromagnetic inverse scattering problem is iteratively solved using a multi-frequency cross-correlation contrast source inversion framework, and the contrast source cost function and contrast cost function within the framework are obtained.

[0009] Based on the wavelength-dependent weighting strategy, the low-frequency components in the two types of cost functions are given higher weights, and the weighted contrast source cost function and the weighted contrast cost function are obtained.

[0010] The weighted contrast source cost function and the weighted contrast cost function are minimized iteratively in turn through the gradient descent algorithm to achieve contrast source and contrast updates, and the relative dielectric constant and conductivity of the target body are reconstructed according to the optimal contrast output by the iterative convergence to complete the electromagnetic quantitative inversion of the target body; wherein, in the contrast update process, the frequency components from the lowest frequency bin to the highest frequency bin are gradually incorporated through the frequency binning and cumulative frequency hopping mechanism, and all the incorporated frequency components are retained in each iteration process to jointly update the weighted contrast cost function.

[0011] In one embodiment, the dielectric parameter spatial disturbance caused by a target in a two-dimensional electromagnetic inverse scattering problem detection scene is defined as contrast, and the secondary excitation source generated by the target scattering is defined as a contrast source, including:

[0012] In the two-dimensional electromagnetic inverse scattering problem detection scenario based on transverse magnetic polarization waves, the transmitting and receiving antennas are arranged in a circular array around the target and form a detection domain. Located in the detection domain Inversion domain is the area of ​​the target to be imaged; the transmitting and receiving antennas adopt a multi-input and output transceiver configuration, the receiving antenna position is marked by p∈{1,2,3,…,P}, the transmitting antenna position is marked by q∈{1,2,3,…,Q}, and the number of receiving antennas is much greater than the number of transmitting antennas;

[0013] Assume that the background medium in the two-dimensional electromagnetic inverse scattering problem detection scene has no dispersion effect and the background dielectric parameters are completely known. The background medium and the target body are defined to have the same vacuum magnetic permeability μ0. The spatial disturbance of the dielectric parameters caused by the target body is defined as the contrast Among them, ∈ i =ε-iσ / ω i and Represent the complex dielectric constant distribution of the target and background medium respectively, i 2 =1 represents a complex unit; ε and ε bg Denote the dielectric constants of the target and background medium, σ and σ respectively bg Represent the conductivity of the target and background medium respectively; ω i is the angular frequency, and its time harmonic factor is agreed to be exp(iω it), where t is time; for a transverse magnetic polarization wave, its electric field degenerates into a scalar that satisfies the following wave equation:

[0014]

[0015] in, is the Laplace operator represents the wave number, It is defined as a contrast source equivalent to the secondary excitation source generated by the target body scattering. and denote the scattered electric field component and the total electric field component respectively, the subscript i denotes the frequency index and i∈{1,2,…,N f}, N f is the total number of frequencies.

[0016] In one embodiment, a multi-frequency cross-correlation contrast source inversion framework is used to iteratively solve a two-dimensional electromagnetic inverse scattering problem, and a contrast source cost function and a contrast cost function within the framework are obtained, including:

[0017] The two-dimensional electromagnetic inverse scattering problem is defined as The inverse problem of reconstructing the contrast distribution, in which the electromagnetic forward operator is used to calculate the electric field generated by the contrast source using the frequency domain finite difference algorithm. The relational expression is:

[0018]

[0019] in, j p,i , χ i and are the vectorized contrast source, contrast and total electric field, ⊙ represents the point-by-point multiplication operator, A i is the stiffness matrix in the frequency domain finite difference algorithm;

[0020] Using multi-frequency cross-correlation contrast source inversion framework to compare electromagnetic forward operators The inverse problem of reconstructing the contrast distribution is solved iteratively, and the contrast source cost function and contrast cost function within the framework are obtained, which are expressed as:

[0021]

[0022] Among them, the contrast source cost function The data mismatch residual ρ p,i , residual γ of the state equation p,i and the cross-correlation residual ξ p,i Composition; contrast cost function The residual γ of the state equation p,i and the cross-correlation residual ξp,i Composition; Symbol: = indicates the definition; subscript p indicates the receiving antenna position index, N src Indicates the number of receiving antennas; and Represent the inversion domain and detection domain on norm; and is a normalization factor used to weight the residual components of different frequencies.

[0023] In one embodiment, the normalization factor and Respectively expressed as:

[0024]

[0025] Among them, y p,i represents the measured data, χ i and They are the contrast and incident electric field represented by vectors respectively; the subscript i represents the frequency index and i∈{1,2,…,N f}, N f is the total number of frequencies; ⊙ represents the point-by-point multiplication operator.

[0026] In one embodiment, a wavelength-dependent weighting strategy is used to assign higher weights to low-frequency components in the two types of cost functions, thereby obtaining a weighted contrast source cost function and a weighted contrast cost function, including:

[0027] The wavelength-dependent weighting strategy is defined as: for N in the antenna operating frequency band f discrete frequencies whose wavelengths are arranged in the order that Then the frequency λ i The corresponding weight coefficient w i Expressed as:

[0028]

[0029] Among them, n≥0 is an exponential parameter, and its specific value is determined by numerical experiments; for low-frequency components with longer wavelengths, its corresponding weight coefficient w i The value of is higher;

[0030] Based on the weight coefficient w i Compare the source cost functions separately With contrast cost function Improvements are made to obtain the weighted contrast source cost function and weighted contrast cost function Respectively expressed as:

[0031]

[0032] In one embodiment, the above method further includes: based on the weight coefficient w i Comparison of source cost functions With contrast cost function The normalization factor in and Improve and obtain the weighted normalization factor and Respectively expressed as:

[0033]

[0034] Among them, y p,i represents the measured data, χ i and They are the contrast and incident electric field represented by vectors respectively; the subscript i represents the frequency index and i∈{1,2,…,N f}, N f is the total number of frequencies; the subscript p represents the receiving antenna position index, N src represents the number of receiving antennas; ⊙ represents the point-by-point multiplication operator; and Represent the inversion domain and detection domain on norm.

[0035] In one embodiment, the weighted contrast source cost function and the weighted contrast cost function are sequentially minimized iteratively by a gradient descent algorithm to achieve contrast source and contrast updates, including:

[0036] First, the weighted contrast source cost function is minimized iteratively using the gradient descent algorithm to update the contrast source, and the total electric field, state equation residual, and cross-correlation residual are updated based on the updated contrast source.

[0037] Then, based on the updated contrast source, total electric field, state equation residual and cross-correlation residual, the gradient descent algorithm is used to iteratively minimize the weighted contrast function to achieve contrast update.

[0038] In one embodiment, during the contrast update process, frequency components from the lowest frequency bin to the highest frequency bin are gradually incorporated through frequency binning and cumulative frequency hopping mechanisms, and all incorporated frequency components are retained in each iteration to jointly update the weighted contrast cost function, including:

[0039] In the contrast iterative update process, the continuous spectrum is divided into K adjacent sub-bands from low frequency to high frequency based on the frequency binning mechanism, and the current lowest frequency component is sequentially selected in each sub-band to form a frequency bin containing K components. Assuming that each sub-band contains L frequency components, L independent frequency bins are finally generated, which can be expressed as:

[0040] B l ={f l ,f l+L ,f l+2L ,…,f l+(K-1)L};

[0041] Among them, B l represents the lth frequency bin, f l represents the lth frequency component, l=1,2,…,L;

[0042] Based on the cumulative frequency hopping mechanism, when the contrast is initially updated, only the frequency components in the lowest frequency bin B1 are used to participate in the weighted contrast cost function update; in the subsequent iterative update process of the contrast, the frequency components from the lowest frequency bin B1 to the highest frequency bin B1 are gradually included. L The frequency components in are included, and all the included frequency components are retained in each iteration to jointly update the weighted contrast cost function.

[0043] In one embodiment, during the accumulation of frequency bins, the weighted contrastive source cost function remains constant and continues to utilize all available frequency components.

[0044] In one embodiment, the cumulative frequency hopping mechanism satisfies the following frequency hopping constraint: during the contrast update process, the number of iterations for accumulating L frequency bins does not exceed N max / 3; where N max The maximum number of iterations required to obtain a satisfactory inversion result for the target body, N max The value of depends on the contrast, geometry and noise level of the target.

[0045] The frequency-binned cumulative frequency-hopping inversion method, which incorporates a wavelength-dependent weighting strategy, builds on the multi-frequency cross-correlation contrast source inversion framework by introducing a wavelength-dependent weighting strategy that assigns higher weights to low-frequency components. This higher weighting effectively suppresses unstable oscillations caused by high-frequency noise, enhancing the robustness of the iterative inversion algorithm when dealing with high-contrast targets. Furthermore, the frequency binning and cumulative frequency-hopping mechanism are integrated into the contrast update process, effectively combining the computational efficiency advantages of multi-frequency synchronous inversion with the inherent robustness of progressive frequency-hopping inversion. Frequency binning allows for simultaneous consideration of the reconstruction characteristics of different sub-bands from the initial contrast update stage, avoiding significant deviations between the contrast distribution derived from low-frequency components and that corresponding to high-frequency characteristics, thereby improving the reliability of the inversion. The cumulative frequency-hopping mechanism continuously retains low-frequency components in the contrast update as it progresses from low to high frequency bands, effectively maintaining the stability of the iterative inversion process and avoiding optimization divergence. This significantly alleviates the local minimum trapping problem in the iterative inversion process, effectively addressing the inherent ill-posedness and nonlinearity of the electromagnetic inverse scattering problem. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 1 is a flow chart of a frequency binning cumulative frequency hopping inversion method integrating a wavelength-dependent weighting strategy in one embodiment;

[0047] Figure 2 A schematic diagram of a two-dimensional electromagnetic inverse scattering problem detection scenario in one embodiment;

[0048] Figure 3 Schematic diagram of a framework of a multi-frequency cross-correlation contrast source inversion algorithm based on a wavelength-dependent weighting strategy in one embodiment;

[0049] Figure 4 Schematic diagram of a framework of a frequency binning cumulative frequency hopping inversion algorithm integrating a wavelength-dependent weighting strategy in one embodiment;

[0050] Figure 5 A schematic diagram of a simulation experiment detection configuration in one embodiment;

[0051] Figure 6 : is a schematic diagram of peak signal-to-noise ratio comparison of different values ​​of the WWn-CC-CSI algorithm in the inversion of the target "Austria" with different contrasts in one embodiment; wherein, Figure 6 (a) Comparison of peak signal-to-noise ratio curves of the WWn-CC-CSI algorithm with different values ​​in the target inversion of the Austria-Diel-e9 dataset; Figure 6 (b) Comparison of peak signal-to-noise ratio curves of the WWn-CC-CSI algorithm with different values ​​in the target inversion of the Austria-Diel-e15 dataset; Figure 6(c) Comparison of peak signal-to-noise ratio curves of the WWn-CC-CSI algorithm with different values ​​in the target inversion of the Austria-Diel-e15-e21-e9 dataset;

[0052] Figure 7 : is a schematic diagram of comparing peak signal-to-noise ratio curves of different inversion algorithms in inverting the "Austria" target with different contrasts in one embodiment; wherein, Figure 7 (a) Comparison of peak signal-to-noise ratio curves of different inversion algorithms in target inversion of the Austria-Diel-e9 dataset; Figure 7 (b) Comparison of peak signal-to-noise ratio curves of different inversion algorithms in target inversion of the Austria-Diel-e15 dataset; Figure 7 (c) Comparison of peak signal-to-noise ratio curves of different inversion algorithms in target inversion of the Austria-Diel-e15-e21-e9 dataset. DETAILED DESCRIPTION

[0053] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0054] In one embodiment, Figure 1 As shown, a frequency binning cumulative frequency hopping inversion method integrating a wavelength-dependent weighting strategy is provided, comprising the following steps:

[0055] Step 1: Define the spatial disturbance of dielectric parameters caused by the target body in the two-dimensional electromagnetic inverse scattering problem detection scene as contrast, and define the secondary excitation source generated by the target body scattering as the contrast source; wherein the dielectric parameters include relative permittivity and conductivity.

[0056] Step 2: Use the multi-frequency cross-correlation contrast source inversion framework to iteratively solve the two-dimensional electromagnetic inverse scattering problem and obtain the contrast source cost function and contrast cost function within the framework.

[0057] In step 3, based on the wavelength-dependent weighting strategy, a higher weight is assigned to the low-frequency components in the two types of cost functions to obtain the weighted contrast source cost function and the weighted contrast cost function.

[0058] In step 4, the weighted contrast source cost function and the weighted contrast cost function are minimized iteratively in turn through the gradient descent algorithm to achieve contrast source and contrast updates, and the relative dielectric constant and conductivity of the target body are reconstructed according to the optimal contrast output by the iterative convergence to complete the electromagnetic quantitative inversion of the target body; wherein, in the contrast update process, the frequency components from the lowest frequency box to the highest frequency box are gradually incorporated through the frequency binning and cumulative frequency hopping mechanism, and all the incorporated frequency components are retained in each iteration process to jointly update the weighted contrast cost function.

[0059] The frequency binning and cumulative frequency hopping inversion method, which incorporates a wavelength-dependent weighting strategy, introduces a wavelength-dependent weighting strategy within the multi-frequency cross-correlation contrast source inversion framework. This recalibrates the cost function using wavelength-adaptive weighting coefficients to prioritize low-frequency components, effectively suppressing unstable oscillations caused by high-frequency noise and enhancing the robustness of the iterative inversion algorithm when dealing with high-contrast targets. Furthermore, by integrating frequency binning and cumulative frequency hopping mechanisms into the contrast update process, the method effectively combines the computational efficiency advantages of multi-frequency synchronous inversion with the inherent robustness of progressive frequency hopping inversion. This method preserves low-frequency information during the high-frequency transition phase, significantly alleviating the local minimum trap problem in the iterative optimization process, thereby effectively addressing the inherent ill-posedness and nonlinear characteristics of the electromagnetic inverse scattering problem.

[0060] In one embodiment, step 1 includes:

[0061] In such Figure 2 In the two-dimensional electromagnetic inverse scattering problem detection scenario based on transverse magnetic polarized waves (TM waves), the transmitting and receiving antennas are arranged in a circular array around the target and form a detection domain. Located in the detection domain Inversion domain The target area to be imaged is represented by a multi-input / output (MIMO) configuration. The receiving antenna positions are labeled p∈{1,2,3,…,P}, and the transmitting antenna positions are labeled q∈{1,2,3,…,Q}. The number of receiving antennas is significantly greater than the number of transmitting antennas. The coordinate system follows the right-hand rule, with the z-axis perpendicular to the observation plane and pointing outward.

[0062] For the multi-frequency detection case, the background medium in the two-dimensional electromagnetic inverse scattering problem detection scene is assumed to have no dispersion effect and the background dielectric parameters are completely known. The background medium and the target are defined to have the same vacuum magnetic permeability μ0, and the spatial disturbance of the dielectric parameters caused by the target is defined as the contrast Among them, ∈ i =ε-iσ / ω i and Represent the complex dielectric constant distribution of the target and background medium respectively, i 2=1 represents a complex unit; ε and ε bg Denote the dielectric constants of the target and background medium, σ and σ respectively bg Represent the conductivity of the target and background medium respectively; ω i is the angular frequency, and its time harmonic factor is agreed to be exp(iω i t), where t is time; for a transverse magnetic polarization wave, its electric field degenerates into a scalar that satisfies the following wave equation:

[0063]

[0064] in, is the Laplace operator, represents the wave number, It is defined as a contrast source equivalent to the secondary excitation source generated by the target body scattering. and denote the scattered electric field component and the total electric field component respectively, the subscript i denotes the frequency index and i∈{1,2,…,N f}, N f is the total number of frequencies.

[0065] In one embodiment, step 2 includes:

[0066] The two-dimensional electromagnetic inverse scattering problem is defined as The inverse problem of reconstructing the contrast distribution, in which the electromagnetic forward operator is used to calculate the electric field generated by the contrast source using the frequency domain finite difference algorithm (FDFD). The relational expression is:

[0067]

[0068] in, j p,i , χ i and are the vectorized contrast source, contrast and total electric field, ⊙ represents the point-by-point multiplication operator, A i is the stiffness matrix in the frequency domain finite difference algorithm.

[0069] The multi-frequency cross-correlation contrast source inversion framework (hereinafter referred to as CC-CSI) is used to analyze the electromagnetic forward operator The inverse problem of reconstructing the contrast distribution is solved iteratively. CC-CSI is an existing nonlinear iterative inversion method derived from the paper "Inversion of Multifrequency Data With the Cross-Correlated Contrast Source Inversion Method." CC-CSI first updates the contrast source by minimizing the corresponding cost function using a gradient descent algorithm. It then updates the contrast parameters in a similar manner until convergence to the optimal contrast solution.

[0070] The contrast source cost function and contrast cost function within the multi-frequency cross-correlation contrast source inversion framework are further obtained and expressed as:

[0071]

[0072] Among them, the contrast source cost function The data mismatch residual ρ p,i , residual γ of the state equation p,i and the cross-correlation residual ξ p,i Composition; contrast cost function The residual γ of the state equation p,i and the cross-correlation residual ξ p,i Composition; Symbol: = indicates the definition; subscript p indicates the receiving antenna position index, N src Indicates the number of receiving antennas; and Represent the inversion domain and detection domain on norm; and is a normalization factor used to weight the residual components of different frequencies. The formal expression of the above three types of residuals is:

[0073] ρ p,i =y p,i -Φ p,i j p,i ;

[0074]

[0075] Among them, y p,i Represents the measurement data, Φ p,i is the measurement matrix, is the vectorized representation of the incident electric field.

[0076] In one embodiment, the normalization factor and Respectively expressed as:

[0077]

[0078] In one embodiment, step 3 includes:

[0079] The wavelength-dependent weighting strategy is defined as: for N in the antenna operating frequency band f discrete frequencies whose wavelengths are arranged in the order that Then the frequency λ i The corresponding weight coefficient w i Expressed as:

[0080]

[0081] Wherein, n≥0 is an exponential parameter, and its specific value is determined by numerical experiments. In this embodiment, n=4 is finally taken; according to the weight coefficient w i From the calculation expression, we can see that for the low-frequency component with a longer wavelength, the corresponding weight coefficient w i The value of is higher, therefore, the wavelength-dependent weighting strategy can achieve the optimization of low-frequency components, effectively suppress the unstable oscillation caused by high-frequency noise, and enhance the robustness of the iterative inversion algorithm when dealing with high-contrast targets.

[0082] Based on the weight coefficient w i Compare the source cost functions separately With contrast cost function Improvements are made to obtain the weighted contrast source cost function and weighted contrast cost function Respectively expressed as:

[0083]

[0084] Furthermore, based on the weight coefficient w i Comparison of source cost functions With contrast cost function The normalization factor in and Improve and obtain the weighted normalization factor and Respectively expressed as:

[0085]

[0086] Specifically, the inversion algorithm obtained by introducing the wavelength-dependent weighting strategy (WWn) based on CC-CSI is named: Multi-frequency cross-correlation contrast source inversion algorithm based on wavelength-dependent weighting strategy (hereinafter referred to as WWn-CC-CSI). The framework of WWn-CC-CSI is as follows: Figure 3 As shown in FIG, when n=0, it degenerates into the traditional CC-CSI algorithm.

[0087] In one embodiment, in step 4, the weighted contrast source cost function and the weighted contrast cost function are sequentially minimized and iterated by a gradient descent algorithm to achieve contrast source and contrast updates, including:

[0088] First, the gradient descent algorithm is used to iteratively minimize the weighted contrast source cost function to achieve contrast source update, and the total electric field, state equation residual and cross-correlation residual are updated according to the updated contrast source; then, based on the updated contrast source, total electric field, state equation residual and cross-correlation residual, the gradient descent algorithm is used to iteratively minimize the weighted contrast function to achieve contrast update.

[0089] In one embodiment, in step 4, during the contrast update process, frequency components from the lowest frequency bin to the highest frequency bin are gradually incorporated through frequency binning and cumulative frequency hopping mechanisms, and all incorporated frequency components are retained in each iteration to jointly update the weighted contrast cost function, including:

[0090] In the contrast iterative update process, the continuous spectrum is first divided into K adjacent sub-bands from low frequency to high frequency based on the frequency binning mechanism, and the current lowest frequency component is sequentially selected in each sub-band to form a frequency bin containing K components. Assuming that each sub-band contains L frequency components, L independent frequency bins are finally generated, which can be expressed as:

[0091] B l ={f l ,f l+L ,f l+2L ,…,f l+(K-1)L};

[0092] Among them, B l represents the lth frequency bin, f l Represents the lth frequency component, l = 1, 2,…, L.

[0093] Then, based on the cumulative frequency hopping mechanism (an incremental update mechanism), during the initial contrast update, only the frequency components in the lowest frequency bin B1 are used to participate in the weighted contrast cost function update; during the subsequent iterative updates of the contrast, the frequency components from the lowest frequency bin B1 to the highest frequency bin B L The frequency components in , and all included frequency components are retained in each iteration to jointly update the weighted contrast cost function. Among them, the weighted contrast source cost function remains unchanged during the accumulation of frequency bins and continues to utilize all available frequency components.

[0094] Furthermore, the above cumulative frequency hopping mechanism satisfies the following frequency hopping constraint: during the contrast update process, the number of iterations for accumulating L frequency bins does not exceed N max / 3. In other words, the frequency binning and cumulative frequency hopping mechanism can enable the algorithm to quickly locate the approximate area of ​​the global optimal solution, and more iterations should be allocated to refine the solution using all frequency bins. max The maximum number of iterations required to obtain a satisfactory inversion result for the target body, N max The value of depends on the contrast, geometry and noise level of the target (low signal-to-noise ratio data can easily lead to overfitting of the algorithm). In practical applications, a feasible solution is to empirically estimate N through pre-experimental inversion of similar targets. max value.

[0095] By further integrating frequency binning with the cumulative frequency hopping mechanism, compared to the traditional method of discarding low-frequency components after frequency hopping, cumulative frequency hopping can continuously retain low-frequency components for contrast updates as the frequency band advances from low to high. This effectively maintains the stability of the iterative inversion process and avoids optimization divergence. This is particularly important for processing broadband signals spanning multiple octaves and significantly enhances the robustness of the algorithm. Furthermore, frequency binning allows for simultaneous consideration of the reconstruction characteristics of different sub-bands from the initial stage, avoiding significant deviations between the contrast distribution derived from low-frequency components and that corresponding to high-frequency characteristics, thereby improving the reliability of the inversion.

[0096] Specifically, the inversion algorithm obtained by further integrating frequency binning and cumulative frequency hopping mechanism (FBCH) on the basis of WWn-CC-CSI is named frequency binning cumulative frequency hopping inversion algorithm integrated with wavelength-dependent weighting strategy (hereinafter referred to as FBCH-WWn-CC-CSI). The framework of FBCH-WWn-CC-CSI is as follows: Figure 4 As shown, Figure 4 The FBCH-CC-CSI in this paper is defined as an inversion algorithm based on CC-CSI, which only introduces frequency binning and cumulative frequency hopping. It is worth noting that the computational efficiency of FBCH-WWn-CC-CSI is slightly better than that of traditional iterative inversion methods that do not adopt this framework. This is because most of the iterative calculations only use a subset of frequency components, reducing computational complexity. The basic process of FBCH-WWn-CC-CSI is as follows:

[0097] Input: Φ p,i 、j p,i and

[0098] Output: χ i ;

[0099] 1. Initialize the comparison source: j p,i ;

[0100] 2. Calculate the initial total electric field: in,

[0101] 3. Initialize the WWn normalization factor:

[0102] 4. Initialize the contrast χ;

[0103] 5. Set the maximum number of iterations N max And initialize the iteration counter

[0104] 6. Divide the frequency bins B1, B2, ..., B L And initialize the frequency bin index to l=1;

[0105] 7. while do

[0106] 8. Update WWn normalization factor:

[0107] 9. Calculate data misfit residuals, state equation residuals, and cross-correlation residuals;

[0108] 10. Update the comparison source j through the gradient descent algorithm p,i (using all frequency components);

[0109] 11. Update the total electric field

[0110] 12. Recalculate the state equation residuals and cross-correlation residuals;

[0111] 13. If the frequency hopping constraint is met and l≤L then

[0112] 14. Perform cumulative frequency hopping: l = l + 1;

[0113] 15. end;

[0114] 16. Update the contrast χ by gradient descent algorithm i (Gradually use B1 to B L frequency bins);

[0115] 17. Iteration counter increments:

[0116] 18. end;

[0117] 19. Output the reconstructed relative dielectric constant and conductivity

[0118] The basic process of FBCH-WWn-CC-CSI naturally inherits the basic process architecture of CC-CSI. The initialization strategy and update steps of the contrast source and contrast parameters are consistent with those in CC-CSI.

[0119] In order to further verify the inversion performance of FBCH-WWn-CC-CSI provided by this application, a simulation experiment was further conducted. The simulation experiment detection configuration is as follows: Figure 5 As shown in the figure, 12 transmitters and 120 receivers are evenly distributed in concentric circles on a 3-meter radius. For each transmitter, the receiver data within the 30° angular domain centered on the transmitter is excluded, and only the measurement values ​​within the remaining 300° angular domain are retained. Therefore, the measurement data dimension of TM polarization inversion is 12×101×N f The simulation environment uses perfectly matched layers (PMLs) along the x and y boundaries to simulate a microwave anechoic chamber, while periodic boundary conditions (PBCs) are used in the z direction to maintain a two-dimensional configuration. The TM polarized incident wave is generated by a line source arranged along the z axis.

[0120] Scattering data is generated using a 5mm×5mm uniform grid, meeting criterion (where λ0 is the free-space wavelength). The scattered electric field is obtained by deducting the incident electric field from the total electric field. The inversion domain is set to [-0.5, 0.5]m × [-0.5, 0.5]m and discretized into a 32×32 coarse grid. A fast CC-CSI algorithm is used to ensure that the electric field calculation in the inversion process is derived from the analytical convolution of the Green's function, thereby maintaining computational accuracy on a coarse grid. The operating frequency band covers 200MHz to 1GHz, and nine frequency points are selected at equal wavelength intervals, corresponding to wavelengths of λ1 = 1.5m, λ2 = 1.35m, λ3 = 1.2m, ..., λ9 = 0.3m. This frequency selection strategy effectively utilizes the wavelength diversity within the target's resonant frequency band. The spectrum is divided into three sub-bands, each containing three discrete frequency points, forming three frequency bins: {f1, f4, f7}, {f2, f5, f8}, and {f3, f6, f9}. Unless otherwise stated, the system performs cumulative frequency modulation every 20,000 iterations during the calculation process.

[0121] To quantitatively evaluate the inversion performance of different algorithms, this application uses the peak signal-to-noise ratio (PSNR) as an evaluation indicator of numerical reconstruction accuracy. A larger value indicates a higher quantitative reconstruction accuracy. The MATLAB implementation of PSNR calculation is as follows:

[0122] psnr(Epsr,epsr_true,peakval_epsr);

[0123] Where epsr_true represents the true relative permittivity distribution, Epsr is the inversion algorithm's estimated value, and peakval_epsr corresponds to the true maximum permittivity. The PSNR calculation is confined to the target region [-0.4375, 0.4375]m × [-0.4375, 0.4375]m, discretized on a 0.01m × 0.01m grid. The estimated dielectric field Epsr is interpolated before calculation to ensure spatial registration and consistent discretization. In the experimental setup, complex Gaussian white noise with a signal-to-noise ratio (SNR) of 20dB was added to the observed data. This high SNR level was set to maintain a controlled noise contamination environment while enabling a comparative evaluation of the robustness of different inversion algorithms.

[0124] The experiment considered the widely used "Austria" standard test problem. The target to be reconstructed consisted of two disks and a ring. In the defined coordinate system, the z-axis coincided with the target axis: the two disks, each with a radius of 0.1 m, were located at (0.3, -0.15) m and (0.3, 0.15) m, respectively; the ring was centered at (-0.1, 0) m, with an outer diameter of 0.3 m and an inner diameter of 0.15 m. It is important to note that the inversion of the "Austria" profile is generally considered a challenging test case in existing literature.

[0125] This application uses this benchmark target to evaluate TM polarization simulation data generated under different contrast combinations. The specific simulation data sets selected are Austria-Diel-e9 (relative permittivity value of 9), Austria-Diel-e15 (relative permittivity value of 15), and Austria-Diel-e15-e21-e9 (relative permittivity of 15, 21, and 9, respectively). To facilitate quantitative comparison of inversion algorithm performance, the PSNR value is calculated only considering the relative permittivity. Therefore, all cylindrical components in the test configuration are assumed to be lossless media (zero conductivity).

[0126] First, the WWn-CC-CSI algorithm is applied to three simulation data sets. The performance of the algorithm under different n values ​​is compared and analyzed by comparing the evolution of PSNR with the number of iterations. Figure 6 (a) to Figure 6 Figure (c) shows the comparison of the peak signal-to-noise ratio curves of the WWn-CC-CSI algorithm with different values ​​in the target inversion of these three simulation data sets, where the value of n ranges from 0 to 6 (n = 0 corresponds to the traditional CC-CSI algorithm). Through curve comparison analysis, it can be seen that the empirical optimal value is determined to be n = 4.

[0127] It is worth noting that for low contrast targets (such as ε r=3), n = 0 exhibited optimal performance at very low contrast. However, even under these low-contrast conditions, selecting n = 4 still yielded satisfactory reconstruction results. Given that practical targets often have high dielectric constants, this application considers n = 4 to be the empirically optimal choice for universal applications.

[0128] After completing the evaluation of the WWn-CC-CSI algorithm, n = 4 was selected as the optimal parameter value. The FBCH-WW4-CC-CSI and FBCH-CC-CSI algorithms were then applied to the inversion of the three simulation data sets. Figure 7 (a) to Figure 7 (c) shows the comparison of PSNR curves of four algorithms (FBCH-WW4-CC-CSI, WW4-CC-CSI, FBCH-CC-CSI and CC-CSI). Figure 7 It shows that the FBCH-WW4-CC-CSI and WW4-CC-CSI provided by this application are significantly better than FBCH-CC-CSI and CC-CSI in all simulation data sets. In particular, Figure 7 (a) It is verified that the introduction of WW4 and FBCH mechanisms can significantly improve the CC-CSI inversion performance.

[0129] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0130] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.

Claims

1. A frequency binning cumulative frequency hopping inversion method integrating wavelength-dependent weighting strategy, characterized in that: The method comprises: The spatial perturbation of dielectric parameters caused by the target in the two-dimensional electromagnetic inverse scattering problem detection scene is defined as contrast, and the secondary excitation source generated by the target scattering is defined as the contrast source; wherein the dielectric parameters include relative permittivity and conductivity; The two-dimensional electromagnetic inverse scattering problem is iteratively solved using a multi-frequency cross-correlation contrast source inversion framework, and the contrast source cost function and contrast cost function within the framework are obtained. Based on the wavelength-dependent weighting strategy, the low-frequency components in the two types of cost functions are given higher weights, and the weighted contrast source cost function and the weighted contrast cost function are obtained. The weighted contrast source cost function and the weighted contrast cost function are sequentially minimized and iterated by the gradient descent algorithm to achieve contrast source and contrast updates, and the relative dielectric constant and conductivity of the target body are reconstructed according to the optimal contrast output by the iterative convergence, thereby completing the electromagnetic quantitative inversion of the target body; wherein, in the contrast update process, the frequency components from the lowest frequency bin to the highest frequency bin are gradually incorporated through the frequency binning and cumulative frequency hopping mechanism, and all the incorporated frequency components are retained in each iteration process to jointly update the weighted contrast cost function.

2. The method according to claim 1, characterized in that The dielectric parameter space disturbance caused by the target in the two-dimensional electromagnetic inverse scattering problem detection scene is defined as contrast, and the secondary excitation source generated by the target scattering is defined as the contrast source, including: In the two-dimensional electromagnetic inverse scattering problem detection scenario based on transverse magnetic polarization waves, the transmitting and receiving antennas are arranged in a circular array around the target and form a detection domain. Located in the detection domain Inversion domain is the area of ​​the target to be imaged; the transmitting and receiving antennas adopt a multi-input and output transceiver configuration, the receiving antenna position is marked by p∈{1,2,3,…,P}, the transmitting antenna position is marked by q∈{1,2,3,…,Q}, and the number of receiving antennas is much greater than the number of transmitting antennas; Assume that the background medium in the two-dimensional electromagnetic inverse scattering problem detection scene has no dispersion effect and the background dielectric parameters are completely known. The background medium and the target body are defined to have the same vacuum magnetic permeability μ0. The spatial disturbance of the dielectric parameters caused by the target body is defined as the contrast Among them, ∈ i =ε-iσ / ω i and Represent the complex dielectric constant distribution of the target and background medium respectively, i 2 =1 represents a complex unit; ε and ε bg Denote the dielectric constants of the target and background medium, σ and σ respectively bg Represent the conductivity of the target and background medium respectively; ω i is the angular frequency, and its time harmonic factor is agreed to be exp(iω i t), where t is time. For a transverse magnetic polarization wave, its electric field degenerates into a scalar that satisfies the following wave equation: in, is the Laplace operator, represents the wave number, It is defined as a contrast source equivalent to the secondary excitation source generated by the target body scattering. and denote the scattered electric field component and the total electric field component respectively, the subscript i denotes the frequency index and i∈{1,2,…,N f }, N f is the total number of frequencies.

3. The method according to claim 2, characterized in that The multi-frequency cross-correlation contrast source inversion framework is used to iteratively solve the two-dimensional electromagnetic inverse scattering problem, and the contrast source cost function and contrast cost function within the framework are obtained, including: The two-dimensional electromagnetic inverse scattering problem is defined as The inverse problem of reconstructing the contrast distribution, wherein when the frequency domain finite difference algorithm is used to calculate the electric field generated by the contrast source, the electromagnetic forward operator The relational expression is: in, j p,i , χ i and are the vectorized contrast source, contrast and total electric field, ⊙ represents the point-by-point multiplication operator, A i is the stiffness matrix in the frequency domain finite difference algorithm; Using multi-frequency cross-correlation contrast source inversion framework to compare electromagnetic forward operators The inverse problem of reconstructing the contrast distribution is solved iteratively, and the contrast source cost function and contrast cost function within the framework are obtained, which are expressed as: Among them, the contrast source cost function The data mismatch residual ρ p,i , residual γ of the state equation p,i and the cross-correlation residual ξ p,i Composition; contrast cost function The residual γ of the state equation p,i and the cross-correlation residual ξ p,i Composition; Symbol: = indicates the definition; subscript p indicates the receiving antenna position index, N src Indicates the number of receiving antennas; and Represent the inversion domain and detection domain on norm; and is a normalization factor used to weight the residual components of different frequencies.

4. The method according to claim 3, characterized in that The normalization factor and Respectively expressed as: Among them, y p,i represents the measured data, χ i and They are the contrast and incident electric field represented by vectors respectively; the subscript i represents the frequency index and i∈{1,2,…,N f }, N f is the total number of frequencies; ⊙ represents the point-by-point multiplication operator.

5. The method according to claim 3, characterized in that Based on the wavelength-dependent weighting strategy, the low-frequency components in the two types of cost functions are given higher weights, and the weighted contrast source cost function and the weighted contrast cost function are obtained, including: The wavelength-dependent weighting strategy is defined as: for N in the antenna operating frequency band f discrete frequencies whose wavelengths are arranged in the order that Then the frequency λ i The corresponding weight coefficient w i Expressed as: Among them, n≥0 is an exponential parameter, and its specific value is determined by numerical experiments; for low-frequency components with longer wavelengths, its corresponding weight coefficient w i The value of is higher; Based on the weight coefficient w i Compare the source cost functions separately With contrast cost function Improvements are made to obtain the weighted contrast source cost function and weighted contrast cost function Respectively expressed as:

6. The method according to claim 5, characterized in that The method further includes: based on the weight coefficient w i The cost function for the contrast source With contrast cost function The normalization factor in and Improve and obtain the weighted normalization factor and Respectively expressed as: Among them, y p,i represents the measured data, χ i and They are the contrast and incident electric field represented by vectors respectively; the subscript i represents the frequency index and i∈{1,2,…,N f }, N f is the total number of frequencies; the subscript p represents the receiving antenna position index, N src represents the number of receiving antennas; ⊙ represents the point-by-point multiplication operator; and Represent the inversion domain and detection domain on norm.

7. The method according to claim 5, characterized in that The weighted contrast source cost function and the weighted contrast cost function are sequentially minimized and iterated by a gradient descent algorithm to achieve contrast source and contrast update, including: First, the weighted contrast source cost function is minimized iteratively using a gradient descent algorithm to update the contrast source, and the total electric field, state equation residual, and cross-correlation residual are updated according to the updated contrast source; Then, based on the updated contrast source, total electric field, state equation residual and cross-correlation residual, a gradient descent algorithm is used to iteratively minimize the weighted contrast function to achieve contrast update.

8. The method according to claim 7, characterized in that In the contrast update process, frequency components from the lowest frequency bin to the highest frequency bin are gradually incorporated through frequency binning and cumulative frequency hopping mechanisms, and all incorporated frequency components are retained in each iteration to jointly update the weighted contrast cost function, including: In the contrast iterative update process, the continuous spectrum is divided into K adjacent sub-bands from low frequency to high frequency based on the frequency binning mechanism, and the current lowest frequency component is sequentially selected in each sub-band to form a frequency bin containing K components. Assuming that each sub-band contains L frequency components, L independent frequency bins are finally generated, which can be expressed as: B l ={f l ,f l+L ,f l+2L ,…,f l+(K-1)L }; Among them, B l represents the lth frequency bin, f l represents the lth frequency component, l=1,2,…,L; Based on the cumulative frequency hopping mechanism, when the contrast is initially updated, only the frequency components in the lowest frequency bin B1 are used to participate in the weighted contrast cost function update; in the subsequent iterative update process of the contrast, the frequency components from the lowest frequency bin B1 to the highest frequency bin B1 are gradually included. L , and retain all the included frequency components in each iteration to jointly update the weighted contrast cost function.

9. The method according to claim 8, characterized in that During the accumulation of frequency bins, the weighted contrastive source cost function remains constant and continues to utilize all available frequency components.

10. The method according to claim 8, characterized in that The cumulative frequency hopping mechanism satisfies the following frequency hopping constraints: During the contrast update process, the number of iterations for accumulating L frequency bins does not exceed N max / 3; where N max The maximum number of iterations required to obtain a satisfactory inversion result for the target body, N max The value of depends on the contrast, geometry and noise level of the target.

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