Untrained Metamaterial-Based Coded Aperture Imaging Optimization Model Based on Modified U-Net

Abstract

Metamaterial-based coded aperture imaging (MCAI) is a forward-looking radar imaging technique based on wavefront modulation. The scattering coefficients of the target can resolve as an ill-posed inverse problem. Data-based deep-learning methods provide an efficient, but expensive, way for target reconstruction. To address the difficulty in collecting paired training data, an untrained deep radar-echo-prior-based MCAI (DMCAI) optimization model is proposed. DMCAI combines the MCAI model with a modified U-Net for predicting radar echo. A joint loss function based on deep-radar echo prior and total variation is utilized to optimize network weights through back-propagation. A target reconstruction strategy by alternatively using the imaginary and real part of the radar echo signal (STAIR) is proposed to solve the DMCAI. It makes the target reconstruction task turn into an estimation from an input image by the U-Net. Then, the optimized weights serve as a parametrization that bridges the input image and the target. The simulation and experimental results demonstrate the effectiveness of the proposed approach under different SNRs or compression measurements.

1. Introduction

The radar system is capable of all-day and all-weather active detection and imaging. However, the azimuth resolution of the radar system is mainly limited by the physical antenna aperture, as defined by the Rayleigh diffraction limit. Generally, there are two approaches to enhance the azimuth resolution of radar systems. The synthetic aperture radar (SAR) system [1] utilizes the relative motion between the radar system and the target to accumulate the aperture, but it cannot achieve forward-looking imaging. The phased array radar (PAR) system [2] forms a narrow beam by increasing the number of antenna array elements, but it also increases the system costs and complexity.

Electromagnetic metamaterials [3,4] are artificially structured materials composed of sub-wavelength units that possess significant electromagnetic properties not found in conventional materials. Programmable metamaterials, in particular, allow for real-time controllable modulation of electromagnetic waves. Benefiting from the recent advances in digital metamaterials, the coded aperture imaging technique based on wavefront modulation provides a novel strategy for achieving high-resolution forward-looking radar imaging. The metamaterial-based coded aperture imaging (MCAI) radar system utilizes a digital metamaterial array antenna to modulate the radar signal wavefront. This system produces a spatiotemporal-independent radiation field over the target area. The receiver captures the radar echo reflected by the target. For the target reconstruction, the scattering coefficients of the target can be resolved as an inverse problem by using the pseudo randomness of the radar echo signals.

However, the radiation field cannot achieve complete spatiotemporal independence when each unit of the array antenna is independently modulated. As a result, extensive measurements are required to ensure high-resolution imaging, which, in turn, leads to increased time consumption. To address this issue, reconstruction algorithms based on the compressive sensing (CS) theory have been proposed to improve the imaging efficiency. Both the orthogonal matching pursuit (OMP) algorithm [5,6] and the sparse Bayesian learning (SBL) algorithm [7,8] make use of the sparsity of the target image as a prior for image reconstruction. However, the imaging results in the high-frequency band exhibit dense scattering centers, which cannot satisfy the sparsity prior. The Tikhonov regularization algorithm can be applied to address ill-posed problems in imaging reconstruction by constructing an optimization objective function with penalty terms (i.e., the L2 term) [9], but it is susceptible to noise interference.

Recently, neural-network-based methods have been utilized for target reconstruction [10,11,12], leveraging their inherent robustness to noise. Researchers have applied them to the field of radar imaging to enhance reconstruction results. A generative adversarial network (GAN) can effectively improve the reconstruction quality of degraded signals for MCAI after training by massive paired echo-image training data [13]. However, the acquisition of a sufficient number of target images for network training is often challenging, hindering progress in this field. Although some methods use generated training data to reduce the reliance on real training data, their performance is greatly affected by the generated scene [14,15,16].

To address this issue, training-free neural network-based methods are needed for target reconstruction. Ulyanov et al. proposed the untrained deep image prior (DIP) framework [17], which effectively captures a substantial amount of low-level statistical information from images to tackle standard inverse problems [18] such as denoising [19] and super resolution [20]. The most significant advantage of the DIP framework is that it eliminates the requirement for a large dataset for training the neural network. DIP has been applied in radar imaging for SAR image denoising and inpainting [21,22]. However, these applications are still limited to image-to-image tasks.

In this paper, an untrained deep radar-echo-prior-based MCAI (DMCAI) optimization model is proposed to reconstruct the scattering coefficients of the target with different sparsity. It combines the MCAI model with a modified U-Net and utilizes the deep neural network as a handcrafted prior, thereby eliminating the need for a large dataset while achieving high-resolution reconstruction. The joint loss function is composed of deep-radar echo prior and total variation. On top of that, we propose a target reconstruction strategy by alternatively using the imaginary and real part of the radar echo signal (STAIR) to solve the DMCAI. We update the U-Net weights with the Adam optimizer [23] by calculating the gradient of the loss function for each weight in the U-Nets. The optimal weights serve as a mapping from a randomly generated image to the target image. STAIR reduces the utilization of echo data by half in each iteration, enhancing the efficiency of the optimization process. The simulation and experimental results demonstrate that the proposed method outperforms traditional approaches under different SNRs and compression measurement scenarios.

The structure of this article is outlined as follows. Section 2 proposes a modified U-Net based untrained optimization model named as DMCAI. Then, a STAIR algorithm is proposed to solve DMCAI. Section 3 and Section 4 present the simulation and experimental results, respectively, to demonstrate the superiority of the proposed method. Section 5 draws the conclusion.

2. Theoretical Analysis

2.1. MCAI System Structure and Imaging Model

The structure of the MCAI system is shown in Figure 1. According to the research in [24], a 1-bit programmable meta-surface is utilized to randomly modulate the phase of the transmitted signal by either $0°$ or $180°$. This modulation enables the formation of a spatially and temporally independent radiation field distribution in the target area. The target region is segmented into grid cells, with the scattering centers of the target being represented by the grid center points.

The imaging plane is divided into $M$ grid cells. In this system, the stepped-frequency waveform (SFW) is transmitted from the Tx, and the transmitted pulse signal is a group of $N$ pulse trains, where the carrier frequency of each pulse train is as follows:

$$f_n=f_0+n\cdot \Delta f, n=0,1,\dots N-1.$$

(1)

Each pulse width is $t_p$, and the pulse repetition period is $T_P$. Therefore, the bandwidth of the pulse trains that composed of $N$ pulses is:

$$B=(N-1)\Delta f.$$

(2)

In a pulse train cycle, the radar signal can be expressed as:

$$S_t\left(t,f_n\right)=A\exp \left(\mathrm{j}2\mathsf{\pi }{f}_{n}t\right), n{T}_p\le t\le n{T}_p+t_p,$$

(3)

where $S_t\left(t\right)$ is the transmitted signal of the radar system, $f_n$ is the center frequency of the $n$th pulse, and $A$ is the amplitude of the signal. The transmitted signal propagates freely to the surface of metamaterials through space. The metamaterial array has an $I$ coding unit. The electromagnetic wave signal reaching the $i$th coding unit can be expressed as:

$$S_t\left(t,f_n,{\mathrm{code}}_i\right)=A\exp \left[\mathrm{j}2\mathsf{\pi }{f}_n\left(t-\frac{d_{Tx,{\mathrm{code}}_i}}{c}\right)\right],$$

(4)

where ${\mathrm{code}}_i$ represents the $i$th $\left(i=1,2,\dots I\right)$ coding unit of the meta-surface. $d_{T_x,{\mathrm{code}}_i}$ represents the signal propagation distance from transmitter to the ${\mathrm{code}}_i$. After modulation by metamaterial antenna array, the radiation field signal at the $m$th $\left(m=1,2,\dots M\right)$ target grid cell is:

$$\begin{array}{ll}{S}_t\left(t,f_n,m\right)& ={\displaystyle \sum _{i=1}^I{S}_t\left(t-\frac{d_{{\mathrm{code}}_i,m}}{c},f_n,{\mathrm{code}}_i\right)}\exp \left(\mathrm{j}{\phi }_{n,{\mathrm{code}}_i}\right)\\ & ={\displaystyle \sum _{i=1}^I\exp \left[\mathrm{j}2\mathsf{\pi }{f}_n\left(t-\frac{d_{Tx,{\mathrm{code}}_i}+d_{{\mathrm{code}}_i,m}}{c}\right)\right]}\exp \left(\mathrm{j}{\phi }_{n,{\mathrm{code}}_i}\right),\end{array}$$

(5)

where ${\phi }_{n,{\mathrm{code}}_i}$ is the phase modulation factor of the carrier frequency $f_n$, corresponding to the $i$th coding unit of the meta-surface, and its value is randomly taken as 0 or $\mathsf{\pi }$. $d_{{\mathrm{code}}_i,m}$ represents the signal propagation distance from the $i$th coding unit to the $m$th target grid cell.

After scattering by the target, the echo signal can be expressed as:

$$S_r\left(t,f_n\right)={\displaystyle \sum _{m=1}^M{S}_t\left(t-\frac{d_{m,Rc}}{c},f_n,m\right)}\cdot {\beta }_m,$$

(6)

where ${\beta }_m$ represents the scattering coefficient corresponding to the $m$th grid. $d_{m,Rc}$ represents the signal propagation distance from the $m$th target grid cell to the receiver. Then, the echo signal can be expressed in detail as:

$$S_r\left(t,f_n\right)={\displaystyle \sum _{m=1}^M\left\{{\displaystyle \sum _{i=1}^I\exp \left[\mathrm{j}2\mathsf{\pi }{f}_n\left(t-\frac{d_{Tx,{\mathrm{code}}_i}+d_{{\mathrm{code}}_i,m}+d_{m,Rc}}{c}\right)\right]}\exp \left(\mathrm{j}{\phi }_{n,{\mathrm{code}}_i}\right)\cdot {\beta }_m\right\}}.$$

(7)

If each pulse is sampled at the $t_s$ time after it has transmitted, then the sampled signal is:

$$S_r\left(t_s,f_n\right)={\displaystyle \sum _{m=1}^M\left\{{\displaystyle \sum _{i=1}^I\exp \left[\mathrm{j}2\mathsf{\pi }{f}_n\left(t_s-\frac{d_{Tx,{\mathrm{code}}_i}+d_{{\mathrm{code}}_i,m}+d_{m,Rc}}{c}\right)\right]}\exp \left(\mathrm{j}{\phi }_{n,{\mathrm{code}}_i}\right)\cdot {\beta }_m\right\}}.$$

(8)

As a result, at the sampling time, the electromagnetic wave signal at ${\beta }_m$ is:

$$S\left(t_s,f_n,m\right)={\displaystyle \sum _{i=1}^I\exp \left[\mathrm{j}2\mathsf{\pi }{f}_n\left(t_s-\frac{d_{Tx,{\mathrm{code}}_i}+d_{{\mathrm{code}}_i,m}+d_{m,Rc}}{c}\right)\right]}\exp \left(\mathrm{j}{\phi }_{n,{\mathrm{code}}_i}\right).$$

(9)

After $N$ times of sampling and receiving, the mathematical model of MCAI can be expressed as:

$$\left[\begin{array}{c}{S}_r(t_s,f_0)\\ S_r(t_s,f_1)\\ ⋮\\ S_r(t_s,f_{N-1})\end{array}\right]=\left[\begin{array}{cccc}S(t_s,f_0,1)& S(t_s,f_0,2)& \dots & S(t_s,f_0,M)\\ S(t_s,f_1,1)& S(t_s,f_1,2)& \cdots & S(t_s,f_1,M)\\ ⋮& ⋮& ⋮& ⋮\\ S(t_s,f_{N-1},1)& S(t_s,f_{N-1},2)& \dots & S(t_s,f_{N-1},M)\end{array}\right]•\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ ⋮\\ {\beta }_M\end{array}\right]+\left[\begin{array}{c}{\omega }_1\\ {\omega }_2\\ ⋮\\ {\omega }_N\end{array}\right].$$

(10)

The above formula can be abbreviated as:

$${\mathit{S}}_r=\mathit{S}\cdot \mathit{\beta }+\mathit{\omega },$$

(11)

where ${\mathit{S}}_r={\left[S_r\left(t_s,f_0\right),S_r\left(t_s,f_1\right),\cdots ,S_r\left(t_s,f_n\right),\cdots ,S_r\left(t_s,f_N\right)\right]}^T$, $\mathit{\beta }={\left[{\beta }_1,{\beta }_2,\cdots ,{\beta }_m,\cdots {\beta }_M\right]}^T$, and $\mathit{\omega }={\left[{\omega }_1,{\omega }_2,\cdots ,{\omega }_n,\cdots ,{\omega }_N\right]}^T$ denote the echo signal vector, the target scattering coefficient vector, and the additive noise, respectively. Therefore, the reference matrix $\mathit{S}$ in the imaging mathematical model is:

$$\mathit{S}=\left[\begin{array}{cccc}S(t_s,f_0,1)& S(t_s,f_0,2)& \dots & S(t_s,f_0,M)\\ S(t_s,f_1,1)& S(t_s,f_1,2)& \cdots & S(t_s,f_1,M)\\ ⋮& ⋮& ⋮& ⋮\\ S(t_s,f_{N-1},1)& S(t_s,f_{N-1},2)& \dots & S(t_s,f_{N-1},M)\end{array}\right].$$

(12)

2.2. DMCAI Optimization Model

A classical approach to solving the target reconstruction problem involves converting the inverse problem into a constrained optimization problem [25], which can be expressed as:

$$\begin{array}{l}\tilde{\mathit{\beta }}=\arg \min {\Vert \mathit{\beta }\Vert }_1 \\ \mathrm{subject} \mathrm{to}{\Vert {\mathit{S}}_r-\mathit{S}\mathit{\beta }\Vert }_2^2<{\epsilon }^2,\end{array}$$

(13)

where ${\epsilon }^2$ is the variance of noise. In practical situations, statistical methods can be used to estimate the level of noise. The L1 norm constraint on $\mathit{\beta }$ means that Equation (13) has the property of a convex function, which can ensure the global optimal solution is obtained. Then, the problem becomes a basis pursuit denoise (BPDN) problem. When solving BPDN problems with constraints, the constrained formulation can be relaxed to an unconstrained one by using a loss function:

$$\tilde{\mathit{\beta }}=\arg \underset{\mathit{\beta }}{\min }{\Vert {\mathit{S}}_r-\mathit{S}\mathit{\beta }\Vert }_2^2+\lambda {\Vert \mathit{\beta }\Vert }_1,$$

(14)

where $\lambda$ is a parameter that balances between the constraints and target prior.

In addition, with the popularity of deep learning, end-to-end neural network-based (EENN) algorithms are proposed to solve the problem by establishing the relationship between the echo signal and the target image [26]. Thus, the target $\mathit{\beta }$ can be estimated directly from the echo signal ${\mathit{S}}_r$ by a well-trained neural-network ${\mathcal{N}}_{\theta }$:

$$\tilde{\mathit{\beta }}={\mathcal{N}}_{\theta }\left({\mathit{S}}_r\right),$$

(15)

where ${\mathcal{N}}_{\theta }$ is a deep network with parameters $\theta$. Both the classic method and the EENN algorithm have limitations. The classic method’s performance is impacted by the generic prior of the target, and the EENN algorithm requires a large amount of echo-target paired data to train the neural network.

Compared with the training-based EENN algorithm, the DIP framework utilizes a randomly initialized neural network to adapt a single degraded image. By bridging the gap between training-based methods and training-free methods that utilize handcrafted priors, DIP exhibits potential applications in radar imaging. It utilizes the deep network architecture to implicitly capture the image prior [17] and seems to have excellent results in standard inverse problems, which can be abstracted as a type of energy minimization problem:

$$x*=\underset{x}{\min }E\left(x;x_0\right)+R\left(x\right),$$

(16)

where $E\left(x;x_0\right)$ is a task-dependent data term, $x_0$ is a degraded image, and $R\left(x\right)$ is a regularizing term. The choice of $E\left(x;x_0\right)$ is determined by the application scenario. Additionally, choosing a regularizing term is often challenging because it needs to capture generic priors on natural images. Therefore, the DIP replaces the regularizing term with the implicit prior captured by a neural network. Reference [20] has proved that the DIP framework has advantages in ghost imaging (GI). Therefore, we argue that the proposed DMCAI optimization model has the potential to enhance the performance of target reconstruction. The objective function of the DMCAI changed from Equation (14) to the following:

$${\mathcal{N}}_{{\theta }^{*}}=\underset{\theta }{\arg \min }\mathbb{E}\left({\Vert \mathit{S}{\mathcal{N}}_{\theta }\left({\mathit{\beta }}_0\right)-{\mathit{S}}_r\Vert }^2\right)+R\left({\mathcal{N}}_{\theta }\left({\mathit{\beta }}_0\right)\right),$$

(17)

where ${\mathcal{N}}_{\theta }$ represents a modified U-Net defined by a set of weights $\theta$, and ${\mathit{\beta }}_0$ is a randomly generated image. U-Net is shown to be a classical network of image processing and has been widely used [27]; its visualization structure is shown in Figure 2. In image processing, a total variation (TV) regularization [28] term can be used as a penalty term in the loss function to smooth the generated image. Thus, we choose the TV term as a regularizing term $R\left(\right)$. As a result, the proposed DMCAI optimization model can be expressed as follows:

$$\begin{array}{l}\underset{\theta }{\min }\mathbb{E}\left({\Vert {\tilde{\mathit{S}}}_r-{\mathit{S}}_r\Vert }^2\right)+{\lambda }_{\mathrm{TV}}\cdot \mathrm{TV}\left({\mathcal{N}}_{\theta }\left({\mathit{\beta }}_0\right)\right)\\ \mathrm{subject} \mathrm{to} {\tilde{\mathit{S}}}_r=\mathit{S}\tilde{\mathit{\beta }},\tilde{\mathit{\beta }}={\mathcal{N}}_{\theta }\left({\mathit{\beta }}_0\right),\end{array}$$

(18)

where $\mathrm{TV}\left(\right)$ represents two-dimensional total variation and ${\lambda }_{\mathrm{TV}}$ stands for its strength. The goal of DMCAI is to recover a reconstruction image $\tilde{\mathit{\beta }}$ from a randomly generated image ${\mathit{\beta }}_0$, which means obtaining a well-estimated image $\tilde{\mathit{\beta }}$ from the U-Net after substituting the minimizer of Equation (18). Therefore, the predicted echo signal ${\tilde{\mathit{S}}}_r$ of the MCAI system will be the best approximation of the actual echo signal ${\mathit{S}}_r$ based on the DMCAI optimization criterion.

2.3. Modified U-Net Architecture

The U-Net was developed for biomedical image processing and cell detection, but has since become a widely used image-processing network structure. In order to match the target reconstruction task of MCAI, we have made some modifications to the classical U-Net architecture. To demonstrate the ability of DMCAI to recover targets from any image, we typically choose randomly generated images as inputs. At this point, the input image contains no information about the target, and an excessive introduction of image priors will affect the imaging quality. Therefore, we did not use the skip connection in the advanced U-Net structure.

The modified U-Net consists of a down-sampling path and an up-sampling path, and the operations for every feature map have been marked in Figure 2. Significantly, in processing feature maps with $3\times 3$ convolutional kernels and rectified linear units (ReLU), padding is applied to ensure the output feature maps maintain their original size. The connections between the feature maps that have the same size can integrate the information of features in different depths. As a parametrization of reconstructing a target from the input image, the U-Net architecture is robust enough to retain all target information in radar echoes.

2.4. Solution of the Proposed Model

The complete process of target reconstruction is shown in Figure 3. Initially, a random image ${\mathit{\beta }}_0$ is generated as an input of the U-Net to obtain a processed image. Subsequently, the mathematical model abstracted from the MCAI model is combined with the modified U-Net. With the inputs of the reference matrix and the processed image out of the U-Net, the MCAI module outputs a predicted echo signal. The loss function based on DMCAI is set as follows:

$$\tilde{\mathit{\beta }}={\mathcal{N}}_{\theta }\left({\mathit{\beta }}_0\right),{\tilde{\mathit{S}}}_r=\mathit{S}\tilde{\mathit{\beta }},{\mathcal{L}}_{{\theta }^{*}}={\mathrm{argmin}}_{\theta }{\Vert {\tilde{\mathit{S}}}_r-{\mathit{S}}_r\Vert }^2+{\lambda }_{\mathrm{TV}}\cdot \mathrm{TV}\left({\mathcal{N}}_{\theta }\left({\mathit{\beta }}_0\right)\right),$$

(19)

With the update of U-Net weights, the reconstructed image that makes the predicted echo signal the best approximation of the actual echo signal is searched in the observation space. This means the optimal weights of the modified U-Net are found based on the DMCAI optimization criterion.

Next, we consider solving the DMCAI model. According to the Equation (19), $\mathit{\beta }$ updated in step $k$ can be expressed as:

$${\mathit{\beta }}_k={\mathcal{N}}_{{\theta }_{k-1}}\left({\mathit{\beta }}_0\right),$$

(20)

where ${\mathcal{N}}_{{\theta }_{k-1}}$ represents the modified U-Net after the $k-1$th update, and the weight of the network is ${\theta }_{k-1}$. Specifically, ${\mathit{\beta }}_k,k=1,2,\cdots ,K$ is obtained by the modified U-Net. So, the predicted echo signal ${\tilde{\mathit{S}}}_r$ can be acquired by:

$${\tilde{\mathit{S}}}_r=\mathit{S}{\mathit{\beta }}_k.$$

(21)

Based on the constructed loss function, we update the network weights by the Adam [23] algorithm. Finally, when the weights converge to a satisfactory value, the target can be reconstructed by the following equation:

$${\mathit{\beta }}_{\mathrm{DMCAI}}={\mathcal{N}}_{{\theta }_K}\left({\mathit{\beta }}_0\right).$$

(22)

In this process, the network weights serve as a parametrization of the target reconstruction task. We found that the measurements $N$ greatly affect the speed of network convergence to the optimal weight. The complex radar echo signal can be expressed as:

$${\mathit{S}}_r=\mathit{A}\cdot e^{\mathrm{j}\mathit{\phi }}={\mathit{S}}_{r-\mathrm{real}}+\mathrm{j}\cdot {\mathit{S}}_{r-\mathrm{imag}}.$$

(23)

The target information is contained in both the real and imaginary parts of the complex signal. However, the two parts of the signal have very low correlation because of the phase difference. This conclusion has been proved by using the Monte Carlo method, as shown in Figure 4. If the two parts of the signal are stacked to solve the inverse problem, the amount of the radar signal data will be twice the number of the measurements $N^{\prime }=2N$. To reduce the calculation time, the STAIR algorithm is proposed to solve the DMCAI optimization model, which is summarized in Algorithm 1.

Algorithm 1 STAIR algorithm

Parameters: Iterations: $K$, Learning rate: $\alpha$, Adam (${\nabla }_{{\theta }_k}$, $\alpha$$, {\beta }_1$$, {\beta }_2$), $\eta$: 100 Input: Reference Matrix $\mathit{S}$, echo signal ${\mathit{S}}_r$

1: Initialize: Random initialization U-Net weights $\theta$,$\mathrm{randomly} \mathrm{generated} \mathrm{image} {\mathit{\beta }}_0$

2: for step $k=1,2,\dots ,K$ do

3:       if  $k \mathrm{mod} 2$= =1 then

4:         ${\mathit{S}}_r={\mathit{S}}_{r-\mathrm{real}}$, $\mathit{S}={\mathit{S}}_{\mathrm{real}}$

5:       else

6:         ${\mathit{S}}_r={\mathit{S}}_{r-\mathrm{imag}}$, $\mathit{S}={\mathit{S}}_{\mathrm{imag}}$

7:       end if

8:       ${\mathit{\beta }}_k={\mathcal{N}}_{{\theta }_{k-1}}\left({\mathit{\beta }}_0\right)$, ${\tilde{\mathit{S}}}_r=\mathit{S}{\mathit{\beta }}_k$, ${\mathcal{L}}_{{\theta }_k}=\mathrm{MSE}\left({\mathit{S}}_r-{\tilde{\mathit{S}}}_r\right)+{\lambda }_{\mathrm{TV}}\cdot \mathrm{TV}\left({\mathcal{N}}_{{\theta }_k}\left({\mathit{\beta }}_0\right)\right)$

9:       ${\nabla }_{{\theta }_k}=\nabla \left({\mathcal{L}}_{{\theta }_k}\right)$ // calculate the gradient of the loss function

10:    ${\theta }_k=\mathrm{Adam}\left({\nabla }_{{\theta }_k},\alpha ,{\beta }_1,{\beta }_2\right)$

10:    $\alpha =\alpha {\gamma }^{k/\eta }$

11: end for

Output: High-resolution imaging results ${\mathit{\beta }}_{\mathrm{DMCAI}}={\mathcal{N}}_{{\theta }_K}\left({\mathit{\beta }}_0\right)$

3. Simulation Results

The simulation results are carried out in this section to verify the superiority of the proposed method with different imaging conditions. The setup is shown in Figure 1 and the system parameters are given in

Table 1. Parameters used in the MCAI simulation experiment.

Parameters	Value
Center frequency $f_c$	340 GHz
Bandwidth	20 GHz
Size of metamaterial antenna array	$0.128 \mathrm{m}\times 0.128 \mathrm{m}$
Size of imaging plane	$0.448 \mathrm{m}\times 0.448 \mathrm{m}$
Imaging distance	$2 \mathrm{m}$
Number of metamaterial coding unit	$64\times 64$
Number of grid cells in the imaging plane	$64\times 64$
Sample numbers	2001

. In this simulation, four 2-D projections of planes and cars are selected as the extended target dataset and each image is set as $64\times 64$. Center binarization and edge weakening are used for the target due to its dense scattering center in the high-frequency band. Two typical reconstruction algorithms, the TVAL3 algorithm [28] and the SBL algorithm [8], were chosen to compete with the proposed method. The quality of reconstructed images can be assessed by calculating the mean squared error (MSE) and structural similarity (SSIM) index.

As is known, a smaller MSE indicates better image quality. The SSIM is a human perception-based measure that compares the image content by separating the contributions of luminance, contrast, and structure [29]. The two indexes can quantify the error and the similarity between reconstructed and ground-truth images, respectively.

Each index in the simulation quantitative analyses represents the average results of 50 Monte Carlo trials for a single target due to the random initialization of modified U-Net weights.

In the simulation, we first consider image reconstruction tasks for four different aircraft and vehicle targets. The imaging results are shown in Figure 5. The results show that the proposed method can effectively recover the target scattering coefficients for different targets and targets with different scattering distributions.

Benefiting from the data visualization advantages of deep learning, the output of U-Net during the iteration process is observable at all times. In the task of the simulation experiments shown in Figure 6, $K=300$ is a reasonable setting. As it should be, for reconstruction tasks of different sizes of targets, the number of iterations can be adjusted dynamically. At the same time, we set an iterative interruption condition for the deep radar echo prior to the loss function, which means interrupting the iteration when ${\Vert {\tilde{\mathit{S}}}_r-{\mathit{S}}_r\Vert }^2<0.01$.

To evaluate the effectiveness of the STAIR algorithm, we conducted a comparative experiment with the proposed STAIR algorithm. Additional target reconstruction by using the real-part signal, imaginary-part signal, and the overall signal respectively is performed.

Table 2. Quantitative comparison with different data uses.

Data Use	${\mathit{S}}_{\mathit{r}-\mathbf{real}}$	${\mathit{S}}_{\mathit{r}-\mathbf{imag}}$	${\mathit{S}}_{\mathit{r}}$	${\mathit{S}}_{\mathit{r}}$ & STAIR
MSE	0.00299	0.00279	0.00207	0.00198
SSIM	0.695	0.763	0.881	0.877
Time (s)	73.167	71.667	123.817	70.764

shows the MSE, SSIM, and time results of the compared methods on the plane target with 50 Monte Carlo trials. The results indicate that the proposed STAIR algorithm improves computational efficiency by reducing data usage during the iterative process while guaranteeing the reconstruction accuracy of the targets.

To investigate the effect of noise on the performance of the proposed method, the SNRs from −5 to 20 dB with 5 dB intervals were taken into consideration in the simulations with the measurements N = 0.5 M. The reconstruction results of the SBL algorithm, TVAL3 algorithm, and DMCAI are shown in Figure 7. The results show that the proposed method outperforms SBL and TVAL3 in terms of reconstruction performance, with lower MSE and higher structural similarity in different SNRs. The main reason for this is that parametrization presents a high impedance to noise. As a result, the proposed algorithm has better reconstruction performance with a handcrafted prior across all SNRs.

We also examine the reconstruction performance of DMCAI under compression measurements. The reconstruction results of the SBL algorithm, TVAL3 algorithm, and DMCAI are shown in Figure 8a with the SNR set as 20 dB. The target can be successfully recovered with the measurements of 0.1 M. The performance of the reconstruction results is improved with the increase of measurements. According to the results in Figure 8b,c, the proposed method has better reconstruction performance under compression measurement. In particular, it excels in recovering details of the aircraft target, including the tail and wings.

4. Experimental Results

To further verify the superiority of the DMCAI, an experiment was performed with an actual MCAI radar system. The experimental setup is shown in Figure 9. The metamaterial antenna works in the Ka frequency band and contains $32\times 32$ metamaterial units. The setting for the experiment was prepared to match the simulation scene. The parameters of the metamaterial antenna and imaging plane are listed in

Table 3. Parameters used in the experiment.

Parameters	Value
Center frequency $f_c$	28 GHz
Bandwidth	2 GHz
Size of metamaterial antenna array	$0.176 \mathrm{m}\times 0.176 \mathrm{m}$
Size of imaging plane	$0.6 \mathrm{m}\times 0.6 \mathrm{m}$
Imaging distance	$2 \mathrm{m}$
Number of metamaterial coding unit	$32\times 32$
Number of grid cells in the imaging plane	$6\times 6$
Sample numbers	401

.

The imaging distance is set to 2 m, and the target plane is divided into grid cells with a size of $0.1 \mathrm{m}\times 0.1 \mathrm{m}$. The target is shown on the right side of Figure 9. It consists of sparse iron sheets with a size of $0.5 \mathrm{m}\times 0.5 \mathrm{m}$. Since the target has only $6\times 6$ pixels, the modified U-Net’s dimension can be simplified to enhance the processing speed and reduce complexity.

For the MCAI radar system, the reference matrix is always precise in the simulation. However, it is not always the case in experiments. System errors can impact the accuracy of the reference matrix. As a result, a reference matrix ${\mathit{S}}^{\prime }$ with errors and system noises will affect the DMCAI optimization model, causing the reconstructed image domain to deviate from the target domain, as depicted in Figure 10. Distortion will occur between the actual image and the reconstructed image. Therefore, the accuracy of the reference matrix is required in solving the DMCAI optimization model.

Despite the gap between the approximated optimization model and the real one, the network structure is adequate enough to capture a significant amount of low-level statistical information from the radar echo. The experimental results of the proposed method utilizing an optimized reference matrix based on equivalent measurement of radiation fields [30] are depicted in Figure 11.

Sparseness measures are used to determine how much energy of a vector is concentrated in just a few of its components. The sparseness criterion [31] of the target is defined as the given vector:

$$\mathrm{Sparseness}\left(\mathit{\beta }\right)=\frac{\sqrt{M}-\left({\displaystyle \sum \left|{\mathit{\beta }}_m\right|}\right)/\sqrt{{{\displaystyle \sum \left|{\mathit{\beta }}_m\right|}}^2}}{\sqrt{M}-1}$$

(24)

The sparseness of the targets in the experiment is 0.8536, 0.3056, 0.2835, 0.0511. Compared with the traditional method, DMCAI has better reconstruction performance with lower MSE for targets with different sparseness than SBL and TVAL3. According to the experimental results, the DMCAI optimization model can achieve a high-quality reconstruction of the targets with different sparsity, and the connection between the random image (input image) and each target is established.

5. Discussion

When reconstructing the target, the image information contained in the deep radar echo prior is solidified in the parameterized structure of the network. Although neural networks have natural noise resistance, the noise will gradually appear on the output image as the predicted echo approaches the actual echo during the iteration, as shown in Figure 12. As a result, the mean square error between the output image and the actual image shows a trend of decreasing and then increasing during the iteration process. This will make the selection of iteration times a challenge.

The DMCAI optimization model, combined with imaging models, eliminates the requirement for training data. However, further research is needed to address the issues of imaging performance degradation and the selection of iteration times caused by a decrease in SNR.

The loss function in Equation (19) based on DMCAI has a balance parameter ${\lambda }_{\mathrm{TV}}$, which should be tuned for each task. It can be adjusted according to the imaging results.

Table 4. Results of our method for different values of ${\lambda }_{\mathrm{TV}}$ with N = 0.5 M.

	${\mathit{\lambda }}_{\mathbf{TV}}$	$0$	$10^{-6}$	$10^{-5}$	$2\times 10^{-5}$	$5\times 10^{-5}$	$10^{-4}$
SNR = 20 dB	MSE	0.0037	0.0034	0.0029	0.0030	0.0032	0.0033
	SSIM	0.7832	0.7966	0.8361	0.8555	0.8684	0.8585
SNR = 5 dB	MSE	0.0326	0.0326	0.0246	0.0195	0.0123	0.0084
	SSIM	0.2071	0.2089	0.2364	0.2577	0.3657	0.6586

shows the results of reconstructing the plane target for different values of ${\lambda }_{\mathrm{TV}}$ with N = 0.5 M. Each index in the simulation quantitative analyses represents an average result of 50 Monte Carlo trials for the plane target. The results did not show that MSE and SSIM have a significant correlation with ${\lambda }_{\mathrm{TV}}$. Even when reconstructing the same target under different SNR conditions, the optimal parameter ${\lambda }_{\mathrm{TV}}$ differs. As a consequence, to achieve satisfactory performance in practice, the parameter ${\lambda }_{\mathrm{TV}}$ still needs to be tuned instead of selecting an arbitrary value.

6. Conclusions

In this paper, an imaging mathematical model is established according to the MCAI system structure. Based on the deep radar echo prior, an untrained DMCAI optimization model is proposed to achieve a high-quality reconstruction of targets with different sparsity. Then, a STAIR algorithm is designed to solve the proposed model. The method is systematically analyzed by numerical simulation, and the effectiveness of the method is verified by experiments. The results show that the method could recover the target image from the random image under the DMCAI optimization criterion, and that it has advantages in solving the ill-posed inverse problem under compression measurement.