Fast non-uniform background linear electromagnetic imaging algorithm based on singular value decomposition

Abstract

The application discloses a fast non-uniform background linear electromagnetic imaging algorithm based on singular value decomposition and relates to the technical field of medical imaging, and comprises the following steps: S1, a data acquisition system is designed, and the data acquisition system comprises a receiving antenna and a transmitting antenna; S2, a non-uniform background improved algorithm based on a main current coefficient is used to solve electromagnetic characteristics of a target to be measured in a target region by using scattered electric field data collected by the receiving and transmitting antennas in the data acquisition system. The fast non-uniform background linear electromagnetic imaging algorithm based on singular value decomposition firstly performs numerical calculation on the influence of a background in a non-uniform background problem, eliminates the background influence in a Green function and an incident field by using a linear calculation mode, then performs singular value decomposition on the new Green function obtained by calculation, extracts the main current coefficient by using a truncation mode, and the algorithm speed is improved, the robustness is enhanced, and imaging can be performed under a higher contrast.

Description

Technical Field

[0001] This invention relates to the field of medical imaging technology, specifically to a fast linear electromagnetic imaging algorithm for non-uniform backgrounds based on singular value decomposition. Background Technology

[0002] Electromagnetic backscattering imaging plays a crucial role in non-destructive testing, biomedical imaging, and geophysical exploration. It utilizes the penetrating power of microwaves to acquire scattered fields from areas invisible to the naked eye via electromagnetic transceivers. Imaging algorithms are then used to numerically analyze the scattered fields of unknown scatterers, thereby obtaining the spatial distribution of the scatterer's electromagnetic properties. However, due to multiple scattering between the background and the target object, backscattering imaging in non-uniform backgrounds—such as through-wall imaging and non-destructive testing; quality assessment of fiber optic cable cores; and evaluation of dielectric waveguides (where the core layer can be considered an unknown scatterer and the outer cladding layer as the background)—is more challenging than imaging in uniform backgrounds. Existing nonlinear iterative algorithms, such as the Subspace-Based Optimization Method (SOM) and the Distorted Borniterative Method (DBIM), involve multiple iterations, resulting in high computational demands, slow computation speed, and lack of real-time performance. Existing linear non-iterative algorithms, such as the Berne approximation and Rytov approximation, have limited contrast ranges, large reconstructed model sizes, and low resolution and accuracy. The Back-Propagation Scheme (BPS) is only applicable to cases with low dielectric constant contrast between the imaging target and the background and a small noise range. In summary, existing algorithms suffer from drawbacks such as poor real-time performance, poor noise resistance, and narrow application scope.

[0003] To address the aforementioned shortcomings, we propose a linear approximation method improved by singular value decomposition. This method considers that the radiation current corresponding to larger singular values ​​plays a major role in the calculation, and reasonably truncates the eigenvalues, thereby improving the algorithm's speed, enhancing its robustness, and enabling imaging at higher contrast. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a fast linear electromagnetic imaging algorithm for non-uniform backgrounds based on singular value decomposition, thereby solving the problems mentioned in the background technology.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a fast linear electromagnetic imaging algorithm for non-uniform backgrounds based on singular value decomposition, comprising the following steps:

[0006] S1. Design a data acquisition system, which includes a receiving antenna and a transmitting antenna;

[0007] S2. An improved algorithm based on the non-uniform background of the main current coefficient is used to solve the electromagnetic properties of the target in the target area by using the scattered electric field data collected by the transceiver antenna in the data acquisition system.

[0008] In step S2, the calculation process of the non-uniform background improvement algorithm for the main current coefficient is as follows:

[0009] The electric field relationship within the imaging region is as follows:

[0010]

[0011]

[0012] Where r is the transmitting antenna T x Distance to the imaging area;

[0013] The scattering intensity contrast is defined as follows:

[0014]

[0015] in For vectors, It is Arrange them in a diagonal formation;

[0016] Green's function is as follows:

[0017]

[0018] in 'a' is the radius of the circle whose area is the same as the area of ​​the divided rectangular grid, and 'S' is the area of ​​the rectangular grid. Let r be the position of the q-th receiving antenna. n This represents the position of the nth grid cell.

[0019] When n≠n'

[0020]

[0021] When n = n'

[0022]

[0023] Internal scattering field The solution is as follows:

[0024]

[0025] The induced current can be expressed as the relationship between contrast and total field:

[0026]

[0027] in For vectors, It is Arranged into a diagonal array, in which For vectors, It is Arrange them in a diagonal formation;

[0028] Substituting equations 1.1 and 1.7 into equation 1.8, we get:

[0029]

[0030] The induced current in the imaging region can be directly calculated from the internal scattering field:

[0031]

[0032] According to the solution Calculate the scattered field at the receiving antenna. As shown below:

[0033]

[0034] right Singular value decomposition was performed to verify that left singular values ​​play a dominant role in the induced current.

[0035]

[0036] The induced current can be expressed in the following form:

[0037]

[0038] in It is a column vector of induced current efficiency;

[0039]

[0040] σ j express The j-th eigenvalue after singular value decomposition, Let H represent the eigenvector corresponding to the j-th eigenvalue, and let H represent the conjugate transpose.

[0041] The first L current inductance coefficients are dominant, therefore:

[0042]

[0043] Where “+” indicates the first L terms;

[0044] Considering the impact of non-uniform backgrounds, such as walls, on the internal imaging area, the contrast ratio is redefined, with the wall contrast ratio being... Target contrast is

[0045] Considering the intensity of the scattered field at the receiving antenna due to the wall itself.

[0046]

[0047] The wall was found to be against The impact is expressed as:

[0048]

[0049] After removing the linear effect of the wall Represented as:

[0050]

[0051] The incident field after linear removal of the wall's influence is represented as:

[0052]

[0053] Total field on each grid Represented as:

[0054]

[0055] Combining equations 1.15 and 1.8, we obtain the expression for the main current coefficient:

[0056]

[0057] In equation 1.21 Similar to the comparison in Equation 1.14, combined with the regularization term, i.e., the L2 regularization form, we obtain the loss function with contrast as the variable:

[0058]

[0059] in From the Singular value decomposition yields:

[0060]

[0061] For multiple transmit antennas, the loss function is written as follows:

[0062]

[0063] Where N inc This refers to the number of transmitting antennas;

[0064] The contrast of the target object is obtained by minimizing the loss function. L2 regularization results in a convex function, meaning the derivative is zero at the extreme points, which can be directly solved for.

[0065]

[0066] The parameter β is obtained by solving the L_curve method. First, SVD is performed on the coefficient terms in the loss function:

[0067]

[0068] Then, an iterative process is used to find a relatively suitable hyperparameter β.

[0069] To further optimize this technical solution, in step S1, the antenna bandwidth range of the receiving antenna and the transmitting antenna is within the range of 100MHz-10GHz, and N (N≥2) antennas are uniformly arranged in a ring. The distance between the antenna and the center is adjusted according to the size of the target object being measured. According to the imaging requirements, the object is placed in the far field region.

[0070] To further optimize this technical solution, in step S2, firstly, the influence of the background in the non-uniform background problem is numerically calculated, and the background influence is eliminated in the Green's function and the incident field through linear calculation; then, the new Green's function is decomposed into singular values, and the larger value of the induced current corresponding to the non-zero eigenvalue is considered to have a decisive effect on the overall calculation, and the main current coefficient is extracted by truncation.

[0071] Compared with existing technologies, this invention provides a fast linear electromagnetic imaging algorithm for non-uniform backgrounds based on singular value decomposition, which has the following advantages:

[0072] This fast linear electromagnetic imaging algorithm based on singular value decomposition for non-uniform backgrounds first numerically calculates the influence of the background on the non-uniform background problem, and eliminates the background influence in the Green's function and incident field through linear calculation. Then, singular value decomposition is performed on the calculated new Green's function, considering that the larger values ​​of the induced current corresponding to the non-zero eigenvalues ​​play a decisive role in the overall calculation. The main current coefficient is extracted by truncation, which improves the algorithm speed, enhances robustness, and enables imaging at high contrast. Attached Figure Description

[0073] Figure 1 This is a schematic diagram of the data acquisition system in a fast non-uniform background linear electromagnetic imaging algorithm based on singular value decomposition proposed in this invention. Detailed Implementation

[0074] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0075] Example 1:

[0076] A fast linear electromagnetic imaging algorithm for non-uniform backgrounds based on singular value decomposition includes the following steps:

[0077] S1. Design a data acquisition system, which includes a receiving antenna and a transmitting antenna.

[0078] like Figure 1 As shown, the antenna bandwidth of the receiving antenna and the transmitting antenna is in the range of 100MHz-10GHz, and N (N≥2) antennas are set in a uniform ring. The distance between the antenna and the center is adjusted according to the size of the target object being measured. According to the imaging requirements, the object is placed in the far field region.

[0079] S2. An improved algorithm based on the non-uniform background of the main current coefficient is used to solve the electromagnetic characteristics of the target in the target area by using the scattered electric field data collected by the transceiver antenna in the data acquisition system. First, the influence of the background in the non-uniform background problem is numerically calculated, and the background influence is eliminated in the Green's function and the incident field through linear calculation. Then, the new Green's function is decomposed into singular values. Considering that the larger values ​​of the induced currents corresponding to the non-zero eigenvalues ​​have a decisive effect on the overall calculation, the main current coefficient is extracted by truncation.

[0080] In step S2, the calculation process of the non-uniform background improvement algorithm for the main current coefficient is as follows:

[0081] The electric field relationship within the imaging region is as follows:

[0082]

[0083]

[0084] Where r is the transmitting antenna T x Distance to the imaging area;

[0085] The scattering intensity contrast is defined as follows:

[0086]

[0087] in For vectors, It is Arrange them in a diagonal formation;

[0088] Green's function is as follows:

[0089]

[0090] in 'a' is the radius of the circle whose area is the same as the area of ​​the divided rectangular grid, and 'S' is the area of ​​the rectangular grid. Let r be the position of the q-th receiving antenna. n This represents the position of the nth grid cell.

[0091] When n≠n'

[0092]

[0093] When n = n'

[0094]

[0095] Internal scattering field The solution is as follows:

[0096]

[0097] The induced current can be expressed as the relationship between contrast and total field:

[0098]

[0099] in For vectors, It is Arranged into a diagonal array, in which For vectors, It is Arrange them in a diagonal formation;

[0100] Substituting equations 1.1 and 1.7 into equation 1.8, we get:

[0101]

[0102] The induced current in the imaging region can be directly calculated from the internal scattering field:

[0103]

[0104] According to the solution Calculate the scattered field at the receiving antenna. As shown below:

[0105]

[0106] right Singular value decomposition was performed to verify that left singular values ​​play a dominant role in the induced current.

[0107]

[0108] The induced current can be expressed in the following form:

[0109]

[0110] in It is a column vector of induced current efficiency;

[0111]

[0112] σ j express The j-th eigenvalue after singular value decomposition, Let H represent the eigenvector corresponding to the j-th eigenvalue, and let H represent the conjugate transpose.

[0113] The first L current inductance coefficients are dominant, therefore:

[0114]

[0115] Where “+” indicates the first L terms;

[0116] Considering the impact of non-uniform backgrounds, such as walls, on the internal imaging area, the contrast ratio is redefined, with the wall contrast ratio being... Target contrast is

[0117] Considering the intensity of the scattered field at the receiving antenna due to the wall itself.

[0118]

[0119] The wall was found to be against The impact is expressed as:

[0120]

[0121] After removing the linear effect of the wall Represented as:

[0122]

[0123] The incident field after linear removal of the wall's influence is represented as:

[0124]

[0125] Total field on each grid Represented as:

[0126]

[0127] Combining equations 1.15 and 1.8, we obtain the expression for the main current coefficient:

[0128]

[0129] In equation 1.21 Similar to the comparison in Equation 1.14, combined with the regularization term, i.e., the L2 regularization form, we obtain the loss function with contrast as the variable:

[0130]

[0131] in From the Singular value decomposition yields:

[0132]

[0133] For multiple transmit antennas, the loss function is written as follows:

[0134]

[0135] Where N inc This refers to the number of transmitting antennas;

[0136] The contrast of the target object is obtained by minimizing the loss function. L2 regularization results in a convex function, meaning the derivative is zero at the extreme points, which can be directly solved for.

[0137]

[0138] The parameter β is obtained by solving the L_curve method. First, SVD is performed on the coefficient terms in the loss function:

[0139]

[0140] Then, an iterative process is used to find a relatively suitable hyperparameter β.

[0141] Example 2:

[0142] To verify the feasibility and practicality of the algorithm proposed in Example 1, simulation and experimental samples were tested. During the test, 12 plane wave incident sources were uniformly distributed on a circle outside the imaging region. The imaging region had a size of 2λ×2λ, divided into a 40×40 grid. The incident wave frequency was 2.4 GHz, and the obstacle was a square wall with an outer side length of 21 cm and a thickness of 1 cm. The scattered field was received by 24 receiving antennas uniformly distributed on a circle with a radius of 113 cm outside the imaging region. The measured scattered field matrix... The dimensions are 24×12. 10% white noise was added to the forward data of the simulation.

[0143] Verification Example 1:

[0144] The shape of the scatterer to be determined is "Austria", as shown in the first column of Table 1. Since the training set of the neural network uses the MNIST handwritten digit set, which does not contain cylindrical scatterers, this embodiment aims to test the generalization ability of the algorithm.

[0145] The two circles have centers at (0.2λ, 0.4λ) and (-0.2λ, 0.4λ) respectively, and both have a radius of 0.15λ. The annulus has a center at (0, -0.2λ), an inner radius of 0.2λ, and an outer radius of 0.4λ. The relative permittivity of the scatterer Austria is 2.0 + 0.5i, and the relative permittivity of the wall is also 2.0 + 0.5i.

[0146] Due to the effect of the wall and the complex structure of Austria, the impact of multiple scattering is significant, making reconstruction through the wall much more difficult than in any previous experiment.

[0147] Table 1 shows that the dielectric constant of the wall is 2 + 0.5i, and the dielectric constant of the target object is 2 + 0.5i.

[0148]

[0149] Verification Example 2:

[0150] To further verify the practicality of the proposed algorithm, experimental data was used for testing. In the configuration of the through-wall imaging system, the operating frequency was set to 2.4 GHz, and 24 antennas were uniformly distributed on a circle with a diameter of 113 cm, of which 12 served as transmitting antennas. The relative permittivity of the wall was 2 (PTFE), the relative permittivity of the cylindrical object was approximately 3, and the relative permittivity of the U-shaped scatterer (plexiglass) was approximately 3. To avoid interference from near-field coupling effects, the two receiving antennas adjacent to the transmitting antennas were closed. Therefore, the dimension of the final scattered field matrix was 21 × 12. Table 2 lists the relative permittivity reconstructed using the measured scattered field using the backpropagation algorithm and the proposed non-uniform background improvement algorithm based on the main current coefficient.

[0151] Table 2 Experimental Results

[0152]

[0153] The beneficial effects of this invention are:

[0154] This fast linear electromagnetic imaging algorithm based on singular value decomposition for non-uniform backgrounds first numerically calculates the influence of the background on the non-uniform background problem, and eliminates the background influence in the Green's function and incident field through linear calculation. Then, singular value decomposition is performed on the calculated new Green's function, considering that the larger values ​​of the induced current corresponding to the non-zero eigenvalues ​​play a decisive role in the overall calculation. The main current coefficient is extracted by truncation, which improves the algorithm speed, enhances robustness, and enables imaging at high contrast.

[0155] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0156] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A fast linear electromagnetic imaging algorithm for non-uniform backgrounds based on singular value decomposition, characterized in that, Includes the following steps: S1. Design a data acquisition system, which includes a receiving antenna and a transmitting antenna; S2. An improved algorithm based on the non-uniform background of the main current coefficient is used to solve the electromagnetic properties of the target in the target area by using the scattered electric field data collected by the transceiver antenna in the data acquisition system. In step S2, the calculation process of the non-uniform background improvement algorithm for the main current coefficient is as follows: The electric field relationship within the imaging region is as follows: Where r is the transmitting antenna Distance to the imaging area; The scattering intensity contrast is defined as follows: in For vectors, It is Arrange them in a diagonal formation; Green's function is as follows: in , where 'a' is the radius of the circle whose area is the same as the area of ​​the divided rectangular grid, and 'S' is the area of ​​the rectangular grid. Let q be the position of the receiving antenna. This represents the position of the nth grid cell. when hour, when hour, Internal scattering field The solution is as follows: The induced current can be expressed as the relationship between contrast and total field: in For vectors, It is Arranged into a diagonal array, in which For vectors, It is Arrange them in a diagonal formation; Substituting equations 1.1 and 1.7 into equation 1.8, we get: The induced current in the imaging region can be directly calculated from the internal scattering field: According to the solution Calculate the scattered field at the receiving antenna. As shown below: right Singular value decomposition was performed to verify that left singular values ​​play a dominant role in the induced current. The induced current can be expressed in the following form: in It is a column vector of induced current efficiency; express The j-th eigenvalue after singular value decomposition, Let H represent the eigenvector corresponding to the j-th eigenvalue, and let H represent the conjugate transpose. The first L current inductance coefficients are dominant, therefore: Where "+" indicates the first L terms; Considering the impact of a non-uniform background wall on the internal imaging area, the contrast ratio is redefined, with the wall contrast ratio being... The contrast of the target object is ; Considering the intensity of the scattered field at the receiving antenna due to the wall itself. The wall was found to be against The impact is expressed as: After removing the linear effect of the wall Represented as: The incident field after linear removal of the wall's influence is represented as: Total field on each grid Represented as: Combining equations 1.15 and 1.8, we obtain the expression for the main current coefficient: In equation 1.21 Similar to the comparison in Equation 1.14, combined with the regularization term, i.e., the L2 regularization form, we obtain the loss function with contrast as the variable: in From the Singular value decomposition yields: For multiple transmit antennas, the loss function is written as follows: in This refers to the number of transmitting antennas; The contrast of the target object is obtained by minimizing the loss function. L2 regularization results in a convex function, meaning the derivative is zero at the extreme points, which can be directly solved for. Among the parameters The loss function is obtained by L_curve method. First, SVD is performed on the coefficient terms of the loss function: Then, an iterative process is used to find a relatively optimal hyperparameter. .

2. The fast non-uniform background linear electromagnetic imaging algorithm based on singular value decomposition according to claim 1, characterized in that, In step S1, the antenna bandwidth of the receiving antenna and the transmitting antenna is within the range of 100MHz-10GHz, and N antennas are set in a uniform ring, where N≥2. The distance between the antenna and the center is adjusted according to the size of the target object being measured, and the object is placed in the far field region according to the imaging requirements.

3. The fast non-uniform background linear electromagnetic imaging algorithm based on singular value decomposition according to claim 1, characterized in that, In step S2, the influence of the background in the non-uniform background problem is first numerically calculated, and the background influence is eliminated in the Green's function and the incident field through linear calculation. Then, the new Green's function is decomposed into singular values. The larger values ​​of the induced current corresponding to the non-zero eigenvalues ​​are considered to have a decisive effect on the overall calculation, and the main current coefficient is extracted by truncation.

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