1. Introduction

Electromagnetic (EM) inverse scattering imaging is a process to reconstruct the electrical parameters and spatial distribution of targets from the received scattering field. In recent years, this technique has been widely applied, including nondestructive testing [1], urban riot security checks, targets’ through-wall imaging [2], and ground-penetrating radar [3].

Traditional EM inverse scattering imaging methods, such as the back projection (BP) method, linear sampling method (LSM) [4], and the contrast source inversion (CSI) [5] method, struggle to break the Rayleigh limit and realize super-resolution imaging of subwavelength targets. At present, super-resolution imaging is an active research field, and its principal methods can be divided broadly into mathematical methods and physical methods. Most physical super-resolution methods attempt to use evanescent waves, such as near-field imaging [6] and super-oscillation [7]. In these approaches, lenses, slabs, or meta-surfaces are arranged near the objects to recover the evanescent wave component [8–12]. The evanescent wave is the high-order spectral component of the electric field, which contains fine information of targets [13]. Physical methods use a variety of approaches to enable the receivers to receive as much fine information carried by the evanescent wave component as possible; however, these techniques cannot be conveniently applied to all practical situations. For example, the recovery range of a perfect lens is limited to the position of symmetric points [11,13]. The imaging area is thus limited, making this approach challenging to apply to wide-range target imaging.

Mathematical methods include the regularization method, compressed sensing (CS) method, neural network method, etc. The CS method [14,15] uses sparse priori information combined with an iterative optimization algorithm to solve the inverse scattering problem (ISP), thus imaging targets with spatial sparsity [16] and improving the imaging resolution to a certain extent. However, the CS method is not suitable for targets without sparse characteristics. With the help of LHM, the restriction on the target sparsity can be relaxed to some extent. However, the CS method still cannot image the wide-range targets even under the LHM environment. Therefore, a more powerful mathematical method is needed.

With the development of deep learning, the neural network approach, as a powerful mathematical method, has been used to solve the electromagnetic ISP. Neural networks mainly use nonlinear characteristics to realize inverse scattering imaging. In 2019, Li et al. [17,18] proposed a DNN-based nonlinear electromagnetic inverse scattering method (DeepNIS), including a multilayer complex-valued residual convolutional neural network (CNN) module, which served to approximately characterize the multi-scattering physical mechanism. Wei et al. applied the U-Net framework [19] to the ISP and proposed three imaging schemes [20] to illustrate the relationship between deep learning (DL) and nonlinear electromagnetic inverse scattering problems (EMISPs). Their research team subsequently proposed a new approach based on the U-Net framework to solve the ISP with phaseless data by using contrast sources as inputs [21]. However, information loss usually occurs during the process of receiving the scattered field, and it is difficult to recover this part of the super-resolution information during follow-up processing or network training.

Therefore, by combining the physical method of left-handed material (LHM) slab assistance and the neural network method, this study proposes a new super-resolution imaging method. Different from conventional methods such as CS method, there are no restrictions imposed on the targets as well as wide-range targets can be imaged, improving the imaging resolution and efficiency at the same time. The main innovations of this paper are listed as follows:

2. Inverse scattering reconstruction model based on the LHM slab

2.1 Mathematical model of the inverse scattering problem

Consider the inverse scattering theory for a two-dimensional (2D) situation with a LHM slab. As shown in Fig. 1, the whole xoy-plane is divided into three regions according to the horizontal coordinate, namely Region 1, Region 2, and Region 3, the x-coordinate range of which is x < d₁, d₁≤ x ≤ d₂, x > d₂, respectively. The target scatterers are distributed in region D, and the characteristic parameters of the target remain unchanged along the z-axis. Region D is a rectangular part of region 3, which we also call the computational domain, or region of interest (ROI). The radiation sources and receivers are evenly distributed in region C, which is several wavelength distances away from the region D. Region C is a rectangular part of region 1, which is also called the observation domain. The LHM slab is placed in region 2.

 

Fig. 1. Diagram of electromagnetic inverse scattering model

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The relative permittivity and conductivity of the target can be expressed as ε_r and σ. Assume that there are N_t radiation sources located at ρ_p, where p = 1,2,…,N_t and ρ_p belongs to region C. For each radiation source, there are N_r receivers located at ρ_q to receive the scattered field, where q = 1,2,…,N_r. Based on the volume equivalence principle, the electromagnetic scattering problem can be expressed in the form of an integral equation as:

The target contrast distribution can be expressed as:

Equation (1) is called the state equation. The scattered field in region C can be written as:

2.2 Solution of the forward scattering process with a LHM slab

In this paper, a perfect lens made of LHM is chosen to realize the recovery of far-field fine image information. We first briefly introduce the negative refractive property of LHM. When both the permittivity ε and magnetic permeability µ are negative, the wave vector k is opposite to the direction of the power flow, then we have k = ω(εµ)^1/2< 0. According to the law of causation [22], we can finally get the refractive index n = (εµ)^1/2< 0. In this way we can use the LHM slab to construct the perfect lens and focus the electromagnetic wave. First, the forward scattering process with the existence of a LHM slab must be considered, i.e., the solution of the state equation in Eq. (4).

2.2.1 Numerical solution of the state equation

As shown in Fig. 1, in the 2D case, the radiation source is an infinitely long line source along the z-axis. The LHM slab is located from x = d₁ to d₂, and d = d₂−d₁ (d₂> d₁). We first calculate the state Eq. (4), i.e., the incident field distribution in region D.

We assume that the radiation source is a line source with amplitude I located at ρ₀= (x₀,y₀). The incident wave propagates along the direction ρ, thus the incident electric field of region 1 can be expressed in the form of a spectral domain integral:

Furthermore, the incident electric field of region 3 produced by the radiation source can be expressed in the form of a spectral domain integral as:

For the propagating wave component |k_y|<|k_i|, the wave number in the propagating direction can be expressed as ${k_{1x}} = {(k_1^2 - k_y^2)^{{1 / 2}}}$ in a normal medium and ${k_{2x}} ={-} {(k_2^2 - k_y^2)^{{1 / 2}}}$ in the LHM. For the evanescent wave component |k_y|>|k_i|, the wave number in the propagation direction is ${k_{1x}} ={-} j{(k_y^2 - k_1^2)^{{1 / 2}}}$ in a normal medium and ${k_{2x}} ={-} j{(k_y^2 - k_2^2)^{{1 / 2}}}$ in the LHM.

The electric field distribution obtained from Eq. (8) is the incident electric field of region D produced by the radiation source. To solve state Eq. (4), we must also solve the scattering electric field of region D. Based on the volume equivalence principle, the scattered field generated by the target can be considered as the radiation field of the equivalent current J_eq, as shown in Fig. 2. The target’s equivalent current J_eq(ρ´) can be written as:

 

Fig. 2. Diagram of equivalent current scattering

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The target is located in region 3, and the scattering field of region 3 generated by the equivalent current J_eq can be expressed as:

Secondly, the equivalent current is expressed as the pulse basis function, i.e., ${{\boldsymbol J}_{eq}}({\boldsymbol{\rho}^{\prime}}) = \sum\nolimits_{n = 1}^{{N_{eq}}} {\alpha ({{{\boldsymbol{\rho}^{\prime}}}_n}) \cdot \Pi ({{{\boldsymbol{\rho}^{\prime}}}_n})}$, where α(ρ´_n) is the undetermined coefficient of the equivalent current and N_eq is the number of discrete grids occupied by the target. By substituting the basis function into Eq. (12) and using the point matching function δ(ρ−ρ_m) as the weight function, we obtain:

The electric field generated by the unit-amplitude contrast source at ρ´_n in region D can be obtained from Eq. (13). Based on state Eq. (4), the matrix G_D is used to represent the discretized operator, and the n´-th column element ${g}_{n^{\prime}}^D$ can be expressed as:

2.2.2 Solution of the measurement equation

In Eq. (5), G_C is the measurement matrix, which represents the electric field distribution of region C as generated by the unit-amplitude contrast source. There are N_r receivers placed in region C; thus, G_C is a matrix of size N_r× N.

The scattered electric field received by the receivers located at (x_r,y_r) in region 1 is equivalent to the superposition of the electric field generated by linear current sources with a magnitude of α(ρ´_n) in region 1. Based on measurement Eq. (5) and the equivalent current amplitude coefficient α(ρ´_n) solved by Equation(13), the received scattered electric field at position (x_r,y_r) can be expressed as:

Accordingly, the n´-th column in the observation matrix G_C can be expressed as:

3. Imaging methods with the existence of a LHM slab

Based on the above forward scattering process, the scattered field in the presence of a LHM slab is received, and the evanescent wave component is supplemented using physical methods. The measured scattered field data is used to reconstruct the electrical parameters and spatial distribution of targets. choose different imaging methods to realize the reconstruction targets.

3.1 BP method and CSI method

Back-projection (BP) method is an imaging method based on matching filtering, with low imaging accuracy and large reconstruction error, but its simple calculation process makes it a common method to solve the initial value of contrast distribution. Due to the strong nonlinearity of ISP, the process of solving contrast distribution ${{\chi }_0}$ is generally divided into two steps. Taking the q-th radiation as example, we first obtain the initial value of contrast source with BP method.

Contrast source inversion (CSI) method is a kind of classical nonlinear inverse scattering method, which not only makes use of measurement equation to reconstruct the contrast source, but also adds the error constraint of the state equation as a regularization term into the cost function. Thus, the cost function of CSI method is the sum of measurement error and state error [5]:

Obviously, F is a quadratic function of w_q and has a higher degree of nonlinearity about ${\chi }$. Obtained by Eq. (17), w_q,0 is used as the initial value of CSI iteration, and the contrast distribution is iteratively solved.

3.2 Compressed sensing method

Faced with the ISP, only optimization algorithms such as the CSI method can be used to iteratively update the contrast source for the reason that the number of measurement equations is far less than the number of unknowns. Generally, the Region D will define as a large area to ensure that the target is within it; thus, the target distribution is sparse compared to the size of the Region D. On this basis, we can apply the CS method to solve EMISPs. However, CS theory is only suitable for linear models, so we use contrast source $w = \chi {E^{tot}}$ as an intermediate variable and apply the CS method to solve EMISPs. Additionally, based on the multi-measurement vector (MMV) model, the joint sparse (JS) method is used to reconstruct the JS contrast source W [16].

We use the JS reconstruction algorithm to precisely recover W under the MMV model, from which the exact position of the target can be extracted as strong prior information. We can then obtain the mask of the target location and add it to the subsequent iterative CSI solutions to achieve perfect reconstruction of the target’s contrast distribution.

Based on the MMV analysis, the JS contrast source is sparse. Accordingly, the objective function can describe the number of rows containing non-zero elements, and the JS ISP can be expressed as [16]:

3.3 Neural network method

In the above traditional methods, the LHM slab’s assistance in recovering the evanescent wave component has been analyzed. However, the iterative process is very time-consuming with a limited imaging range of targets. Even if the source and receiver arrays are widened laterally, there is still little improvement in the resolution for normal-width targets. Moreover, the CS method is a linear solution method and only sparse targets can be handled. In contrast, the neural network method has a strong advantage in solving nonlinear problems by learning the complex nonlinearity of scattering process and extract high-dimensional features. In the following, the physical method of LHM-assisted recovery of evanescent wave components and the mathematical neural network method are combined to realize super-resolution imaging of target scatterers.

During network training, we directly use the network to extract super-resolution information carried by the evanescent wave components. Based on the BP method, the initial value of the contrast distribution calculated from the scattered field with the existence of a LHM slab is taken as the network input, and the true contrast distribution of the spatial targets is taken as the training label. The training process can be expressed as the minimization problem of the objective function:

In Eq. (23), parameters such as ${\boldsymbol e}_q^{sca}$ and ${{\boldsymbol G}_C}$ have been calculated in the presence of a LHM slab in Section 2.2, which includes the recovered evanescent wave component. The output of the network is the target contrast distribution, and the cost function is the mean square error (MSE) function, which is calculated at the network’s output layer. The cost function calculation is represented by specific physical parameter symbols; for example, the network’s output is denoted as $\hat{{\boldsymbol \chi }}$, and the label is denoted as ${{\boldsymbol \chi }^{std}}$. The MSE cost function can thus be expressed as:

 

Fig. 3. Convolutional neural network structure diagram

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On the other hand, the expansive path aims to recover the spatial information lost during the contracting path, whose structure corresponds to the contracting path. Instead of using max pooling for down-sampling in the contracting path, the expansive path employs a 3 × 3 up-convolution, also known as transposed convolution or deconvolution. Moreover, the expansive path utilizes two concatenations. These concatenations involve combining the feature maps from the contracting path with the correspondingly cropped feature maps from the expansive path. This step helps to preserve important local information and improve spatial resolution. The specific structure of which is shown in Fig. 3.

4. Simulation results and analysis

4.1 Description of the experimental configuration

In this paper, the incident wave is set to a 2D TMz cylindrical wave with a frequency of f = 3 GHz and a wavelength of λ=0.1 m. The x-coordinate of the center region C is x = 0, 4λ away from that of region D, and the radiation sources and receivers are evenly distributed in array form in region C. The LHM slab is located between regions C and D, and the x-axis values occupied by it range from d₁=λ to d₂= 3λ. The relative dielectric constant and the relative permeability of the LHM slab are set to ε_r2= –1 and µ_r2= –1. In practice, loss will inevitably occur due to the presence of a LHM slab, so the conductivity σ is set as 10⁻²⁰S/m.

The target can be located anywhere in region D, whose size is λ×λ and center is at d₃= 4λ, and the target’s relative dielectric constant is between 1.5 and 3. Considering the grid's discrete accuracy is λ/20, we obtain region D with a discrete size of 20 × 20. The radiation sources and receivers are evenly distributed in region C, which forms a rectangle centered at the coordinate origin. Its longitudinal height is the same as that of region D and its width is related to the target distribution range. The distribution of the radiation source and receiving station are specified in different experiments. We mainly use the received scattered field at the receiving station to reconstruct the target.

4.2 Experiment results of traditional methods

4.2.1 Imaging results of narrow-range targets by traditional methods

First, we discuss the accuracy of super-resolution imaging with the assistance of a LHM slab. Region C of this experiment is a rectangular region centered at the coordinate origin, with a width of λ/5 and a length of λ, and transmitters and receivers are evenly distributed in it. Two grid targets are placed lengthwise at x = 4λ, where the target distance is d_t. The CSI method is used to image the targets, results of which are shown in Fig. 4.

 

Fig. 4. The accuracy of super-resolution imaging

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As shown in Fig. 4, Fig. 4(a)-(d) illustrate the true contrast distribution of the targets, d_t of which are λ/20, λ/10, λ/10, and λ, respectively, and Fig. 4(e)-(h) are the CSI results after 600 iterations. When the distance ${d_t}$ between the two target grids is less than d_t =λ/10, the CSI method with a LHM slab cannot distinguish the two grids even after 600 iterations. However, when without the LHM slab, it can hardly reconstruct the target until the two grids of the target are about λ apart. By comparing Fig. 4(f) and Fig. 4(g), the CSI method with a LHM slab performs much better than the free space configuration. Therefore, we conclude that the imaging effect can be greatly improved with the assistance of a LHM slab, and the imaging resolution can reach λ/10.

4.2.2 Imaging results of wide-range targets by traditional methods

In Section 4.2.1, we use traditional methods to image narrow-range targets. Based on the results, the imaging effect with a LHM slab assistance is very good, and super-resolution imaging for the scenes can be performed satisfying the complementary medium request. However, not all imaging targets meet the conditions of narrow range and spatial sparsity; accordingly, in this section, we further improve the inverse scattering imaging method. First, we use traditional methods to image different targets. Region C of this experiment is a square region centered at the coordinate origin, both the width and length of which are λ, and transmitters and receivers are evenly distributed in it. The imaging scenes are the same as those described in section 4.2.1 except that the horizontal distribution range of targets has been extended. The imaging results are displayed in Fig. 5.

 

Fig. 5. Reconstruction results of targets by the traditional methods

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Figure 5(a)-(c) illustrate the standard contrast distribution of targets, Fig. 5(d)-(f) show the reconstruction results of the CSI method after 600 iterations with a LHM slab, and Fig. 5(g)-(i) show the reconstruction results of the CS method with a LHM slab. Figure 5(a), (b), (c) are different kinds of targets respectively, a strip-shaped target with a spacing of λ/20, an array-shaped target with the same spacing of λ/20 and a digit-shaped target based on the MNIST dataset [24].

When we use CSI method to reconstruct the above targets, the final results after 600 iterations are shown in Fig. 5(d), (e), (f), which take 31.01 seconds, 89.42 seconds and 344.88 seconds respectively, closely related to the complexity of the target. It can be found that CSI method can hardly realize the reconstruction, no matter in the position, shape or parameters of targets. Figure 5(g), (h), (i) are the reconstruction results of CS method. With the help of spatial position mask, we can easily reconstruct targets with sparse space characteristics, like Fig. 5(g). However, for other targets without such sparsity, there still exit some errors in the reconstructed shape and parameters, such as Fig. 5(h), (i). It takes 31.01 seconds, 89.42 seconds and 344.88 seconds for the reconstruction with CS method, respectively. The reasons for the reconstruction failure are analyzed in section 4.2.3a.

Our findings indicate that even if the observation domain is widened to encompass more radiation sources and receivers, traditional methods still cannot reconstruct the target. The CSI method can only make a rough image of the target shape with blurred edges and missing or redundant mesh. Once the grid number occupied by the target exceeds the upper limit of the grid number satisfying the space-sparsity condition, the CS method cannot generate a mask or image the target. At this point, even if the evanescent wave component is recovered with the assistance of LHM, it is still challenging for traditional methods to achieve the requirements of super-resolution imaging. Therefore, it is necessary to estimate the sparsity of the target, i.e., determine what kind of target can be successfully solved by traditional methods, to select the most appropriate imaging scheme.

4.2.3 Results analysis of traditional methods

In this section, we discuss the successful reconstruction condition [25,26] in CS method, which can be measured by the correlation of the measurement matrix G_C, which can be expressed as the column vector ${{\boldsymbol G}_C} = [\begin{array}{{cccc}} {{\boldsymbol g}_1^C}&{{\boldsymbol g}_2^C}& \cdots &{{\boldsymbol g}_N^C} \end{array}]$. The degree of linear correlation between columns can be determined by ${Err _{\textrm{corr }}}$ [25] [26]:

 

Fig. 6. Linear dependent error corresponding to the rectangular substructure

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Without the assistance of LHM, the correlation error corresponding to the rectangular substructure in free space is found to be very small, indicating a strong correlation between grids. In this case, it is difficult to distinguish between adjacent grids. While with the assistance of the LHM slab, the magnitude of the correlation error greatly increases, thus indicating that the LHM helps to recover super-resolution information, consistent with the experimental results presented in Section 4.2.1. In the case where the correlation error corresponding to the evanescent wave component is very large, the evanescent wave component carries sufficient super-resolution information and can easily be used to distinguish adjacent grids. The method proposed in this paper primarily uses the evanescent wave component to achieve super-resolution imaging.

Due to the significant variability in the longitudinal values, we present ${Err _{\textrm{corr }}}$ in logarithmic form in Fig. 6. As shown in the locally enlarged detail diagram in Fig. 6, the linear correlation error between the columns in G_C of the evanescent wave component with the assistance of the LHM slab gradually decreases with increasing distance when x > 4λ. This change occurs because when the distance to the target is very close, the evanescent wave component carries extensive super-resolution information and can easily distinguish the adjacent grids; thus, the correlation between grids in this case is very weak. However, with increasing distance, the information carried by the evanescent wave decreases exponentially, and the correlation between grids gradually strengthens, hence the linear correlation error decreases with increasing distance.

Based on the analysis of Spark(G_C) above, when the discretization size is λ/20, Spark(G_C)≈5 in free space. Accordingly, the CS method can be successfully reconstructed when the occupied grid number target x < 4. With the assistance of LHM, when the number of objects x < 14, the CS method can be successfully reconstructed. Note that the analysis conducted here is only in the physical sense, and in reality, Spark (G_C) may not be exactly equal to the above inference. However, the analysis from a purely physical perspective successfully explains the influence of LHMs on the target number in successful reconstruction, which has a range of implications for subsequent algorithm improvement.

As discussed in sections 4.2.1 and 4.2.2, the transverse imaging resolution of a target can be improved to some extent by broadening the array of radiation sources and receivers. However, when the transverse distribution range of the observation domain is widened to a certain extent, the imaging of the target cannot be further improved. We accordingly analyze this phenomenon based on the properties of the complementary medium [27].

When the thickness of the LHM slab is d=|d₂–d₁|, according to Eq. (8), the property of the complementary medium without loss can be expressed as:

Equation (26) shows that the electric field received at (x₀+ 2d,y₀) is equal to the electric field at (x₀,y₀) generated by the irradiated source. It has previously been proven that the cancellation property is effective for a complementary medium with any parameter distribution [27,28]. Thus, for different combinations of normal media and LHMs, the receivers in the far field can be configured to perfectly receive the super-resolution information of the target’s near field, as shown in Fig. 7.

 

Fig. 7. Schematic diagram of spectral domain analysis in complementary media. (a) Lossless medium, (b) lossy medium, and (c) focusing of multiple radiation sources.

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In the case of lossless LHM, the theoretical focus point is shown in Fig. 7(a). As shown, due to the negative refraction characteristic of the LHM, various spectral components are focused at x = d₃. When the spectral domain component |k_y|>|k₁|, k_y is the evanescent wave component, such as k_y= ±k_1b, etc. This type of component carries target super-resolution information in the propagation process and decreases exponentially as the propagation distance increases. The LHM slab amplifies the amplitude of the wave component. When the spectral domain component |k_y|<|k₁|, k_y is the propagating wave component, such as k_y= ±k_1a, etc. In this instance, the LHM performs phase compensation for this wave component type. However, as shown in Fig. 7(b), it is impossible to achieve a truly lossless environment in practice, and the attenuation of higher-order spectral components is more pronounced during the propagation process. In this case, it is difficult for each order spectral component to focus perfectly at the theoretical position x = d₃.

Figure 7(c) shows two groups of radiation sources, receivers, and targets, which separately satisfy the complementarity theorem. Positions x = d₃ and x = d´₃ correspond to the locations of the radiation sources, receivers, and targets, thus satisfying the position correspondence of the complementarity theorem. The analysis performed for a lossless medium, based on Eq. (8), can be expressed as:

For the propagating wave component, ${k_{1x}} = {(k_1^2 - k_y^2)^{1/2}}$, and for the evanescent wave component, ${k_{1x}} ={-} j{(k_y^2 - k_1^2)^{1/2}}$. As shown, the second term of Eq. (27) is the evanescent wave component, which decreases exponentially with increasing x while x ≥ d₂. When the corresponding target is reached, the field component of x = d₃ (or d´₃) is equal to the field component of x = d₀ (or d´₀). Thus, when the two radiation sources (A and B) are placed horizontally side-by-side at x = d₀ and x = d´₀, the evanescent wave component arriving at x = d₃ will be generated simultaneously. The evanescent wave amplitude of source B will be greater than that of source A; however, it does not carry the corresponding super-resolution information of source A and will thus negatively affect the focus of the target located at x = d₃. The focus of a perfect lens holds only at that point or column. When the source array is further broadened laterally, this negative influence will become increasingly significant, and even the magnitude of the non-active component amplitude will cover the effective field component. Therefore, super-resolution imaging of wide-range targets cannot be improved by broadening the source and receiver array. The experimental results presented in Section 4.2.2 are consistent with these theoretical analysis results.

For targets that do not meet the successful reconstruction condition, with the assistance of LHM to recover the evanescent wave component, we extract the shape, position, and parameter information of the target from the received scattered field by using the powerful nonlinear fitting ability of the neural network method and finally realize super-resolution reconstruction of general targets. The results of this analysis are presented in the next section.

4.3 Experimental results of the neural network method

In this paper, a convolutional neural network is used for training, the input of which is the initial target contrast values calculated by the BP method. We use the MNIST dataset, taking handwritten digits as target scatterers, where the target contrast can take any value between 0.5 and 2. Region D is the square centered at (d₃,0) with a side length of λ, so the size of region D after discretization is 20 × 20. The number of radiation sources and receivers in region C is 10 × 10, forming a square centered at (0,0) with side length λ. We use the initial target contrast value calculated by the BP method as the network input, so the input size and output size are both 20 × 20. A total of 25 000 datasets were used for this training process, 80% of which were used as training sets and 20% were used as verification sets. A further 1000 datasets are used as test sets. The error curve of the training process is shown in Fig. 8.

 

Fig. 8. Network training error curve

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In this paper, the workstation used for network training has an Intel Xeon Gold 6230R CPU @ 2.10 GHz. As shown in Fig. 8, the network training process tends to converge after 20 epochs. To ensure imaging accuracy, we took the network results after the 30^th training epoch for imaging. The training time for 30 epochs is around seven hours. Note that the datasets used for network training all were ideal, i.e., without noise.

First, a discrete distributed target is used to determine the neural network’s reconstruction resolution based on the LHM slab. Figure 9(a) and (b) show the labels of the discretely distributed target and the super-resolution reconstruction result. Besides, the reconstruction time is about 0.016 seconds. Compared with the traditional methods of Fig. 5, the reconstruction speed is greatly improved. Overall, the results demonstrate that the neural network method can essentially achieve super-resolution imaging of the target with a resolution of λ/20.

 

Fig. 9. Imaging resolution of network method

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Second, the trained network is used to image the target scatterers in the test sets, with the imaging results displayed in Fig. 10. As shown in Fig. 10, the neural network method can perform super-resolution imaging on general targets without spatial sparsity. Figure 10(a)-(d) are labels of different targets, and Fig. 10(e)-(h), Fig. 10(i)-(l) and Fig. 10(m)-(p) are reconstruction results of the neural network method under the condition of no noise, 40 dB and 30 dB respectively. In this experiment, the time required to reconstruct 1000 test sets in different conditions is around 13 seconds, and the average time for each target is 0.013seconds.

 

Fig. 10. Neural network method reconstruction results

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Next, the statistical error of the imaging is quantitatively expressed. The MSE calculation for the relative dielectric constant of the target can be expressed as [29]:

Table 1. The imaging error calculated under different SNRs

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In summary, under the condition of no noise, the contour of the imaging result is essentially consistent with that of the target with only slight errors in the reconstruction parameters. When the SNR is 40 dB, the imaging results show some edge blur but the neural network method can still image the targets successfully. However, when the signal-to-noise ratio is 30 dB or less, the edge blur effect in the imaging results will be worsened. This indicates that the neural network method proposed in this paper has a certain anti-noise ability, whereas the CS method is extremely sensitive to noise—even a small noise content in the scattered field will cause large interference during mask generation. Based on the analysis in this paper, LHMs are used to assist the recovery of evanescent wave component, and the BP method is used to calculate the initial value of the target contrast distribution, reducing the influence of noise to a certain extent. Overall, the neural network super-resolution imaging method based on LHM proposed in this paper has some anti-noise performance.

In addition, we tested the generalization performance of the method proposed in this paper. To do so, we used the trained network to reconstruct letter-shaped targets, the results of which are shown in Fig. 11.

 

Fig. 11. Imaging results of the neural network method

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Figure 11 includes the reconstruction results of the neural network under two different scenarios, where Fig. 11(a)-(d) are the labels of targets’ contrast distribution, Fig. 11(e)-(h) are the reconstruction results with a LHM slab and Fig. 11(i)-(l) are the reconstruction results of free space. Compared with Fig. 11(i)-(l), the results of free space, Fig. 11(e)-(h) have a more accurate imaging effect, which indicates that the network trained with the assistance of a LHM slab has a better generalization performance. The quantitative errors are shown in Table 2 based on Eq. (28).

Table 2. The imaging error of letter-shaped targets

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In conclusion, the trained network can accurately image the letter-shaped targets, indicating that the neural network method proposed in this paper has good generalization performance while also achieving accurate imaging.

5. Conclusion and discussion

To overcome the limitations of traditional methods, such as the CSI method’s long calculation time and low imaging accuracy and the CS method’s requirement for targets with spatial sparsity, this paper proposes a super-resolution LHM slab-assisted imaging method based on a neural network. For simple targets with spatial sparsity, using a LHM slab to recover the evanescent wave components can effectively improve the imaging accuracy of traditional methods. For a more complex target, this paper takes advantage of the physical characteristics of LHMs by fully collecting the scattered field carrying super-resolution information of the target before network training and also utilizes the powerful nonlinear fitting ability of the neural network method to obtain the inverse scattering process from the received scattered field to calculate the target contrast distribution. This study’s experimental results show that the proposed method can achieve efficient and accurate super-resolution imaging of the target with good generalization ability and some anti-noise performance.

It should be noted that it is difficult for target imaging with the assistance of a LHM slab to achieve 360-degree of all-round reconstruction. Therefore, the inverse scattering imaging scene with a cylinder-shaped LHM will be considered, which is further applicable to the super-resolution imaging of targets in general scenes.