Electromagnetic Imaging of Uniaxial Objects by Two-Step Neural Network

Abstract

The integration of electromagnetic imaging technology with the Internet of Things plays an important role in fields as diverse as healthcare, geophysics, and industrial diagnostics. This paper presents a novel two-step neural network architecture to solve the electromagnetic imaging for uniaxial objects which can be used in the Internet of Things. We incident TM and TE waves to unknown objects and receive the scattered fields. In order to reduce the training difficulty, we first input the gathered scattered field information into a deep convolutional neural network (DCNN) to obtain the preliminary guess. In the second step, we feed the guessed image into the convolutional neural network (CNN) to reconstruct high-resolution images. Our numerical results demonstrate the real-time imaging capability of our proposed two-step method in reconstructing high-contrast scatterers.

1. Introduction

Electromagnetic imaging is an advanced technique that entails evenly distributing transmitter and receiver antennas around a scatterer widely employed in biomedical imaging [1], remote sensing [2], non-destructive testing [3], etc. By illuminating an unknown scatterer with electromagnetic waves and capturing the reflected scattered field, valuable information regarding its size, shape, and material composition can be reconstructed. Due to the highly nonlinear nature of inverse scattering problems (ISPs), iterative algorithms with regularization are frequently utilized to address them. Electromagnetic imaging algorithms generally fall into two categories: (1) Approximation methods [4,5]: Traditional techniques demonstrate significant inversion capabilities, particularly with small scatterers or low-contrast objects. Accuracy will diminish significantly when the size or contrast becomes larger due to the multiple scattering effect. (2) Iterative methods [6,7,8,9]: These methods can achieve higher accuracy by minimizing the mismatch error between inversion and measurement results. Considering the inherent ill-posedness of the ISPs, incorporating a regularization term into the cost function is crucial to stabilize the optimization process. However, challenges such as high computational costs and low iterative convergence will hinder instant inversion tasks.

Convolutional neural networks (CNNs) have been widely utilized in image and speech recognition recently, yielding promising results. There are three main categories of learning methods: (1) Direct learning methods [10,11,12,13,14,15,16]: These methods employ neural networks directly to learn wave equations and solve them. (2) Learning-assisted objective function methods [17,18,19,20,21]: Neural networks are used iteratively in the training process for image reconstruction. (3) Physics-assisted learning methods [22,23,24,25,26,27,28,29,30,31,32]: This method applies Maxwell’s equations to pre-calculate certain physical quantities. Neural networks are subsequently employed to compute more precise physical quantities to achieve better reconstruction of objects. In 2020, Shao introduced a two-stage model to solve ISPs. His approach involved developing an autoencoder to convert high-resolution vectors in the first stage, followed by creating a second neural network to correlate microwave signals with the compressed features. The numerical results indicated that the deep learning network developed by Shao could accurately identify objects of similar complexity levels across the entire domain of interest [14]. In 2021, Zhuo proposed a modified contrast scheme utilizing convolutional neural networks to handle the high nonlinearity present in electromagnetic imaging. The numerical findings suggested that this modified contrast scheme, when combined with convolutional neural networks, could get rid of nonlinearity effectively [28]. In 2022, Ye combined the visual geometric group and the total variation loss function to reconstruct biaxial objects with a generative adversarial network (GAN). The experimental results show that this method requires a long training time, but it has good image reconstruction capabilities [29]. In 2023, Xu combined a weighted loss function composed of adversarial loss, mean absolute percentage error (MAPE), and structural similarity (SSIM) into a generative adversarial network with a self-attention mechanism. The numerical results show that the proposed architecture exhibits good generalization ability and stability [31].

Recently, the convergence of electromagnetic imaging (EMI) technology with the Internet of Things (IoT) has sparked a wave of innovation across multiple sectors. This amalgamation offers unprecedented opportunities for real-time monitoring, predictive maintenance, and enhanced decision-making processes. As the world becomes increasingly interconnected, the integration of EMI with the IoT presents a transformative paradigm that transcends traditional boundaries. Electromagnetic imaging techniques, ranging from magnetic resonance imaging (MRI) to ground-penetrating radar (GPR), have long been instrumental in diverse fields such as healthcare, geophysics, and industrial diagnostics. These methods harness electromagnetic waves to probe, visualize, and analyze the internal structures and properties of materials and environments with remarkable precision. However, the advent of the IoT has infused this realm with a newfound vitality, enabling seamless connectivity, data aggregation, and intelligent decision-making capabilities. Figure 1 shows the diagram illustrates the schematic of electromagnetic imaging and the IoT.

Our contributions are summarized below:

• As far as we know, no two-stage network for reconstructing electromagnetic images of uniaxial objects has been published to date. We are the pioneers who rolled out this idea.
• The tensor formed by the TE wave in the cross-section may exhibit greater complexity compared to the TM case.
• The dielectric constant components distributed on various lateral directions for uniaxial objects may lead to the greater severity of the nonlinear problem. Consequently, regeneration by scattered fields will be relatively challenging for a TE scenario.

Section 1 analyzes techniques applied in the past to solve ISPs. Our theory and formula are elucidated in Section 2. Section 3 introduces the two-step neural network architecture. Section 4 presents the numerical results. Section 5 presents the conclusions.

2. Formula

This section presents the direct problem formulation for TM and TE waves via electromagnetic imaging. Figure 2 shows the dielectric scatterer located in the free space. We configure the transmitter and receiver around the domain of interest area for measurement. The position, shape, and size of the scatterers can be known from the measured scattered field. We define a Cartesian coordinate system to describe the dielectric coefficient distribution of the object by a diagonal matrix as follows:

$${\stackrel{=}{\epsilon }}_r={\left[\begin{array}{ccc}{\epsilon }_x\left(x,y\right)& 0& 0\\ 0& {\epsilon }_y\left(x,y\right)& 0\\ 0& 0& {\epsilon }_z\left(x,y\right)\end{array}\right]}_{xyz}$$

(1)

In the TM case, we assume the incident wave to be $e^{j\omega t}$ and the scattered field along the z-direction. The material of the scatterer is related to the x- and y-directions, so the irradiation of the TM-polarized waves will produce only TM-scattered waves. We define the incident wave as follows:

$${\overline{E}}^i\left(\stackrel{\rightharpoonup }{r}\right)=E_z^i\left(x,y\right)\widehat{z}=e^{-j{k}_0(xcos\varnothing +ysin\varnothing )}\widehat{z}$$

(2)

where $k_0$ represents the wave number in free space, and $\varnothing$ denotes the angle of incidence.

Figure 2. Schematic of dielectric scatterer.

Figure 2. Schematic of dielectric scatterer.

The TM wave contains the z-component scattered field only. Equations (3) and (4) represent the total field ${\overline{E}}_z$ = $E_z \widehat{\mathrm{z}}$ and the external scattered field ${\overline{E}}^s$ = $E_z^s \widehat{\mathrm{z}}$, respectively.

$$E_z\left(\overline{r}\right)={{\displaystyle \int }}_s^{ }G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_z\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_z^{ }\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }+E_z^i\left(\stackrel{\rightharpoonup }{r}\right), \stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \in S$$

(3)

$$E_z^s\left(\overline{r}\right)={{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_z\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_z^{ }\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }, \stackrel{\rightharpoonup }{r}\notin S, \stackrel{\rightharpoonup }{r}\prime \in S$$

(4)

In 2-D free space, the Green’s function is $G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)=-\frac{j{k}_0^2}{4}{H}_0^{\left(2\right)}\left(k_0\left|\stackrel{\rightharpoonup }{r}-\stackrel{\rightharpoonup }{r}\prime \right|\right)$. $H_0^{\left(2\right)}$ represents the zero-order Hankel function of the second kind.

In the case of TE, the material properties of the scatterer are related to the x- and y-directions. We define the incident wave as:

$$E_x^i\left(\stackrel{\rightharpoonup }{r}\right)=-sin\varnothing e^{-j{k}_0\left(xcos\varnothing +ysin\varnothing \right)}$$

(5)

$$E_y^i\left(\stackrel{\rightharpoonup }{r}\right)=cos\varnothing e^{-j{k}_0\left(xcos\varnothing +ysin\varnothing \right)}$$

(6)

The total field $\overline{E}\left(\stackrel{\rightharpoonup }{r}\right)=E_x\left(\stackrel{\rightharpoonup }{r}\right)\widehat{x}+E_y\left(\stackrel{\rightharpoonup }{r}\right)\widehat{y}$ and the external scattered field ${\overline{E}}^s\left(\overline{r}\right)=E_x^s\left(\overline{r}\right)\widehat{x}+E_y^s\left(\overline{r}\right)\widehat{y}$ can be defined as follows:

$$E_x\left(\stackrel{\rightharpoonup }{r}\right)=\left(\frac{{𝜕}^2}{𝜕x^2}+{k_0}^2\right)\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_x\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_x\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]+\frac{{𝜕}^2}{𝜕x𝜕y}\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_y\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_y\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]+{\stackrel{\rightharpoonup }{E}}_x^i\left(\stackrel{\rightharpoonup }{r}\right)$$

(7)

$$E_y\left(\stackrel{\rightharpoonup }{r}\right)=\frac{{𝜕}^2}{𝜕x𝜕y}\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_x\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_x\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]+\left(\frac{{𝜕}^2}{𝜕y^2}+{k_0}^2\right)\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_y\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_y\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]+{\stackrel{\rightharpoonup }{E}}_y^i\left(\stackrel{\rightharpoonup }{r}\right)$$

(8)

$$E_x^s\left(\stackrel{\rightharpoonup }{r}\right)=\left(\frac{{𝜕}^2}{𝜕x^2}+{k_0}^2\right)\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_x\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_x\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]+\frac{{𝜕}^2}{𝜕x𝜕y}\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_y\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E_y\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]$$

(9)

$$E_y^s\left(\stackrel{\rightharpoonup }{r}\right)=\frac{{𝜕}^2}{𝜕x𝜕y}\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_x\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E\widehat{x}\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]+\left(\frac{{𝜕}^2}{𝜕y^2}+{k_0}^2\right)\left[{{\displaystyle \int }}_{s}G\left(\stackrel{\rightharpoonup }{r},\stackrel{\rightharpoonup }{r}\prime \right)\left({\epsilon }_y\left(\stackrel{\rightharpoonup }{r}\prime \right)-1\right)E\widehat{y}\left(\stackrel{\rightharpoonup }{r}\prime \right)d{s}^{\prime }\right]$$

(10)

By first solving the total electric field E using Equations (3), (7), and (8) with the tensor distribution information, we can subsequently calculate the scattered field $E^s$ from Equations (4), (9), and (10). Further information on the Green’s functions can be found in [30].

In the numerical computations for the direct problem, we begin by dividing both the scatterer and the closed surface S into N small segments. This ensures that the dielectric coefficient and electric field remain constant within each segment. Let ${\epsilon }_{xn}$, ${\epsilon }_{yn}$, and ${\epsilon }_{zn}$ denote the dielectric coefficients corresponding to the n-th small segment in the x-, y-, and z-directions, respectively. To solve Equations (3)–(10) using the method of moments, we utilized pulse basis functions for expansion and Dirac delta functions for testing. These equations are subsequently transformed into matrix equations.

$$-\left(E_z^i\right)=\left(\left[G_1\right]\left[{\tau }_z\right]-\left[I\right]\right)\left(E_z\right)$$

(11)

$$\left(E_z^s\right)=\left[G_2\right]\left[{\tau }_z\right]\left(E_z\right)$$

(12)

$$\left(\begin{array}{c}-E_x^i\\ -E_y^i\end{array}\right)=\left\{\left[\begin{array}{cc}\left[G_3\right]& \left[G_4\right]\\ \left[G_4\right]& \left[G_5\right]\end{array}\right]\left[\begin{array}{cc}\left[{\tau }_x\right]& 0\\ 0& \left[{\tau }_y\right]\end{array}\right]-\left[\begin{array}{cc}\left[I\right]& 0\\ 0& \left[I\right]\end{array}\right]\right\}\left(\begin{array}{c}{E}_x\\ E_y\end{array}\right)$$

(13)

$$\left(\begin{array}{c}{E}_x^s\\ E_y^s\end{array}\right)=\left\{\left[\begin{array}{cc}\left[G_6\right]& \left[G_7\right]\\ \left[G_7\right]& \left[G_8\right]\end{array}\right]\left[\begin{array}{cc}\left[{\tau }_x\right]& 0\\ 0& \left[{\tau }_y\right]\end{array}\right]\right\}\left(\begin{array}{c}{E}_x\\ E_y\end{array}\right)$$

(14)

In the next section, we will utilize the measured scattered fields as input for a two-stage neural network to reconstruct electromagnetic images. The first stage neural network is trained to simulate Maxwell’s equations and swiftly reconstruct the initial image. We employ the second stage neural network to refine and reconstruct accurate electromagnetic images. This approach stands out for its high efficiency and distinction between previous imaging methods.

3. Two-Step Neural Network

Artificial intelligence technology is currently experiencing a boom with widespread applications in self-driving cars. Meanwhile, electromagnetic imaging technology, image processing, and speech recognition are also widely utilized, including CNN, DCNN, and GAN, among others. In this section, we implement a two-stage neural network to reconstruct electromagnetic images. Initially, we utilize the measured scattered field input to train and simulate Maxwell’s equations in the first-stage neural network. This may accelerate the reconstructed acquisition image. Subsequently, we feed the image estimated from the first stage into the neural network for the second stage of training, aiming to reconstruct a more accurate electromagnetic image. In our research, DCNN architecture, renowned for its effectiveness in processing and analyzing visual data, is employed in the first stage. Inspired by the human visual cortex, this architecture consists of convolutional layers interconnected by artificial neurons. Each neuron within a convolutional layer is dedicated to detecting a specific visual feature present in the input data. Figure 3 illustrates the architecture of the DCNN.

In the second stage, we enhance the reconstructed electromagnetic images by employing a CNN architecture, as illustrated in Figure 4. The proposed architecture consists of two halves: the left half comprises a sequence of a $3\times 3$ convolutional layer, a normalization layer, a linear correction layer, and a $2\times 2$ pooling layer. Conversely, the right half involves a $3\times 3$ convolutional layer, a normalization layer, a linear correction layer, and a $3\times 3$ up-convolutional layer, which are added iteratively. Finally, a $1\times 1$ convolutional layer acts as a fully connected layer, and the resulting average is input into the regression layer to compute the error value associated with the dielectric coefficient distribution.

4. Numerical Results

In our investigation, we operate under the assumption that anisotropic objects are situated within free space, surrounded by emitters and receivers distributed uniformly around the scattering object. Both TM and TE waves are emitted in diverse directions to illuminate the anisotropic objects, while we gather scattered field data from these unknown scatterers. Leveraging this received scattered field data, we employ a DCNN to estimate the initial permittivity. Subsequently, this estimate is fed into a CNN to reconstruct a precise electromagnetic image.

In our simulated environment, we partition the edge constant size of the scatterer into $\frac{0.2{\lambda }_0}{\sqrt{{\epsilon }_r}}$, where ${\epsilon }_r$ stands for the relative permittivity of the anisotropic object, and ${\lambda }_0$ represents the wavelength in free space. We vary the dielectric coefficient of the scatterer from 1 to 2.5, while maintaining the frequency of the incident wave at 3 GHz. We deploy 32 transmitter and receiver antennas evenly. To mimic real-world conditions, we introduce Gaussian noise of 5% and 20% into the simulated environment for analysis. We gauge the dielectric constant distribution using scatter field information through the DCNN in the first stage, i.e., to input the real and imaginary parts, respectively, into the DCNN. In the second stage, we use the preliminary image from the DCNN to reconstruct a more accurate dielectric constant distribution through a convolutional neural network architecture. In the artificial intelligence segment, we allocate 80% of the scattered field for the training set and the rest for the test set. We utilize ADAM uniformly to train the two-step module, setting the training parameters with a learning rate ranging from ${10}^{-3}$ to ${10}^{-4}$, and a maximum epoch of 40. For every optimization epoch, we ensure that the training data are shuffled.

We use the root mean square error and image similarity to verify the results of the numerical simulation as well as conduct comparison and analysis. The equations are as follows:

$$-\left(E_z^i\right)=\left(\left[G_1\right]\left[{\tau }_z\right]-\left[I\right]\right)\left(E_z\right)$$

(15)

$$\left(E_z^s\right)=\left[G_2\right]\left[{\tau }_z\right]\left(E_z\right)$$

(16)

${\stackrel{=}{\mathsf{\epsilon }}}_{\mathrm{r}}$ and ${\stackrel{=}{\mathsf{\epsilon }}}_{\mathrm{r}}^{\mathrm{r}}$ are the ground truth and reconstructed relative permittivity profiles, respectively, ${\mathrm{M}}_{\mathrm{t}}$ represents the number of tests conducted, and F denotes the Frobenius norm.

In this context, $\underset{\mathrm{y}}{˜}$ and y represent the reconstructed and true relative permittivity profiles, respectively. ${\mathsf{\mu }}_{\mathrm{y}}$ denotes the mean of y, ${\mathsf{\sigma }}_{\underset{\mathrm{y}}{˜}}^2$ signifies the variance of y, and ${\mathsf{\sigma }}_{\underset{\mathrm{y}}{˜}\mathrm{y}}$ represents the covariance of $\underset{\mathrm{y}}{˜}$ and y. To prevent division by zero, two small constraints ${\mathrm{C}}_1={\left({\mathrm{K}}_1\mathrm{D}\right)}^2$ and ${\mathrm{C}}_2={\left({\mathrm{K}}_2\mathrm{D}\right)}^2$ are added, where ${\mathrm{K}}_1=0.01$ and ${\mathrm{K}}_2=0.03$ serve as hyperparameters. Here, $\mathrm{D}$ signifies the dynamic pixel range for the image y [30].

4.1. Relative Permittivity between 1 and 1.5

In this scenario, we establish the permittivity distribution within the range of 1 to 1.5. We analyze 10 scatterers, each characterized by distinct permittivity distributions, and these scatterers have the capability to freely move among 50 different positions within the measurement area. We introduce 20% Gaussian noise into the simulated environment. Following the estimation of dielectric coefficient distributions from scattered field data using a DCNN, we proceed with reconstructing the electromagnetic image using a CNN. The results for TM and TE cases are visualized in Figure 5 and Figure 6, respectively. Figure 5a and Figure 6a display the ground truth distributions of ${\epsilon }_z$ and ${\epsilon }_x$ for the scatterers, while Figure 5b and Figure 6b depict the reconstructed images of ${\epsilon }_z$ and ${\epsilon }_x$, respectively.

Table 1. Performance of relative permittivity between 1 and 1.5 by TM and TE wave with 20% noise.

	${\mathit{\epsilon }}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{x}}$
RMSE	4.4%	3.5%
SSIM	96.1%	94.2%

summarizes the reconstruction performance metrics for ${\epsilon }_z$ and ${\epsilon }_x$.

4.2. Relative Permittivity between 1.5 and 2

In this scenario, we establish the permittivity distributions ranging from 1.5 to 2. We examine 10 scatterers with unique permittivity distributions, capable of freely moving across 50 positions within the measurement area. We proceed to estimate the dielectric coefficient distribution via the received scattered field information from the DCNN with 5% Gaussian noise added. This information is then inputted into the CNN for electromagnetic image reconstruction. Figure 7 and Figure 8 illustrate the respective TM and TE results. Figure 7a and Figure 8a display the ground truth of the ${\epsilon }_z$ and ${\epsilon }_x$ scatterers, respectively, while Figure 7b and Figure 8b display the image reconstruction of the ${\epsilon }_z$ and ${\epsilon }_x$ scatterers, respectively.

Table 2. Performance of relative permittivity range from 1.5 to 2 by TM and TE wave with 5% noise.

	${\mathit{\epsilon }}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{x}}$
RMSE	3.82%	3.85%
SSIM	95.7%	96.5%

gives the reconstruction performance of ${\epsilon }_z$ and ${\epsilon }_x$.

4.3. MNIST Dielectric Distribution between 7.5 and 8

Established in 1988 by the National Institute of Standards and Technology, the Modified National Institute of Standards and Technology Database (MNIST) is a sizable collection of handwritten data. Widely exploited in the realms of machine learning and deep learning, the MNIST dataset encompasses 70,000 grayscale images of handwritten numbers ranging from 0 to 9, each image measuring $28\times 28$ pixels. In this instance, we chose MNIST and define its dielectric distribution within the range of 7.5 to 8. In our simulation, we randomly select 1000 images to comprise the dataset and augment with 5% Gaussian noise. The DCNN is used prior to estimate the dielectric coefficient distribution from the scattered field information before inputting into a CNN for reconstructing the electromagnetic images. Figure 9a and Figure 10a display the ground truth of the ${\epsilon }_z$ and ${\epsilon }_x$ scatterers, respectively, while Figure 9b and Figure 10b display the image reconstruction of the ${\epsilon }_z$ and ${\epsilon }_x$ scatterers, respectively. The reconstruction performance of ${\epsilon }_z$ and ${\epsilon }_x$ is listed in

Table 3. Performance of dielectric distribution between 7.5 and 8 by TM and TE wave with 5% noise.

	${\mathit{\epsilon }}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{x}}$
RMSE	8.6%	8.9%
SSIM	82.8%	80.4%

.

4.4. Experimental Data Verification

We selected the training data to mirror case A and case B above, with the sole distinction being the relative permittivity of (1.3–3.3). Within these training data, 8 transmitter antennas were employed, yielding a total dataset of 15,120 (1890 × 8) scattered fields. We trained the two-step model with noiseless synthetic data and subsequently tested the network, implementing the measured experimental data sourced from the Fresnel Institute [32]. This dataset was used to validate the proposed two-step neural network for the TM and TE cases. The experimental setup included 8 transmitter antennas and 241 receiver antennas, positioned 1.67 m apart from the source and measurement object. A horn antenna was used to measure the scattered field, resulting in receivers not being evenly distributed around the transmitter. We specifically selected the FoamDielExt dataset for both the TM and TE cases. This dataset comprises large (SAITEC SBF 300) and small (Berlon) cylinders with diameters of 80 mm and 31 mm, respectively, and dielectric coefficients of ${\epsilon }_r=1.45\pm 0.15$ and ${\epsilon }_r=3\pm 0.3$ (ref. Figure 11). Here, “$\pm$” denotes the uncertainty range surrounding the experimental values, and $D_e$ represents the diameter. In our simulation, the scatterer was placed within a 320 × 320 mm domain. A 3 GHz wave frequency was incidented in both the TM and TE scenarios. The experimental data were normalized using the simulated scattered field received from the opposite side of the incident angle.

The reconstruction outcomes through the experimental data implementing the two-step neural networks for the TM and TE cases are illustrated in Figure 12.

Table 4. Performance of experimental data by TM and TE wave.

	${\mathit{\epsilon }}_{\mathit{z}}$	${\mathit{\epsilon }}_{\mathit{x}}$
RMSE	6.3%	4.5%
SSIM	60.5%	82.1%

displays the corresponding RMSE and SSIM values of satisfactory levels for 2 cases. Based on these results, we assert that our proposed method has been validated by the experimental data. Additionally, the two-step neural network proves to be more adept at handling highly nonlinear problems in either case.

5. Conclusions

In this paper, we deploy a two-stage neural network architecture to address the ISPs. We unify a DCNN and CNN to reconstruct the uniaxial objects. The findings demonstrate that the proposed two-stage neural network architecture can vividly handle more complex TE scenarios. Moving forward, we aim to further enhance the two-stage neural network architecture and apply it to even more challenging situations such as reconstructing buried objects in half-space. In addition to that, we also consider merging the enhanced two-stage neural network architecture with a Switch Transformer to improve the imaging.