A High-Resolution and Robust Microwave Correlation Imaging Method Based on URRF Using MC-AAMPE Algorithm

Abstract

This manuscript presents a novel framework for high-resolution and robust microwave correlation imaging. In order to generate a more diverse random radiation field distribution, the unified random radiation field (URRF) model is proposed. The URRF model can accurately characterize the joint random modulation in the signals’ phase, amplitude, and frequency. Furthermore, we build a parametric imaging model based on URRF which clearly describes the relationship between the image to be reconstructed and the signals by the URRF model. By using this imaging model, the reconstruction of an image is converted into solving a multi-parameter optimization problem with multiple constraints. To solve this optimization problem with high efficiency and accuracy, the model-constrained adaptive alternating multiple parameter estimation (MC-AAMPE) algorithm is proposed. This algorithm decomposes the high-dimensional multi-parameter optimization problem into several sub-optimization problems. The renewing solutions to these sub-optimization problems make the multi-parameter optimization converge to the image of the target and the parameters of clutter and noise, which are all unknown before the solution. In comparison with the existing methods, the proposed scheme generates images with higher resolution and is more robust under noise conditions. Extensive simulation experiments confirmed the effectiveness and robustness of the proposed method.

1. Introduction

Microwave correlation imaging is a novel imaging technique based on wavefront modulation and spatiotemporal stochastic radiation fields [1,2,3]. By performing random modulation on the transmitted signal to generate stochastic radiation field distributions, microwave correlation imaging enhances the capability to obtain scene information. Through correlation processing, it extracts and decouples target information within the beam, resulting in high-resolution images. Compared to traditional imaging methods, microwave correlation imaging has significant advantages, including surpassing the limitations of antenna aperture, independence from relative motion for imaging, high-resolution, and strong anti-interference capabilities [4,5,6]. This makes it suitable for various fields such as staring imaging of geosynchronous orbit satellites and near-space vehicles, earth observation by distributed satellites, and anti-terrorism security screening [7,8,9,10].

Despite significant progress in microwave correlation imaging, there are still certain challenges. The microwave correlation imaging method transmits a set of temporal and spatial random signals via a radar array, constructing a radiation field with a random distribution in time and space. Through correlation processing, it extracts and decouples target information within the beam. Due to the spatiotemporal randomness of the radiation field, each received echo contains the information of the same target and the information of different radiation fields. By using information fusion processing technology for correlation processing, spatial resolution surpassing the antenna aperture can be achieved. The performance of the microwave correlation imaging method demands high spatiotemporal randomness of the radiation field. However, in practice, the construction method of the radiation field is often relatively simple, and factors such as the number of signals and the number of array elements may also restrict the generation of random radiation fields [11]. At the same time, microwave correlation imaging uses the electromagnetic field distribution with random fluctuation in time and space to obtain high-spatial-resolution decoupling of target information, and the fluctuation characteristics of the radiation field formed in different parts of the space are different; the echo signal accumulation is limited compared to SAR/ISAR imaging techniques [12,13,14]. Consequently, the signal-to-noise ratio (SNR) has a significant impact on the quality of microwave correlation imaging [15].

To enhance the performance of microwave correlation imaging, two main approaches have been proposed: (1) optimizing the construction methods of the random radiation field [16,17,18,19], and (2) improving imaging performance through correlation processing techniques, such as correlation technology [20,21,22,23], compressed sensing technology [24,25,26,27,28,29,30,31,32,33], or deep learning technology [34,35,36,37,38].

In terms of optimizing the construction methods of the random radiation field, Ferri et al. [16] conducted experiments using the single classical source of pseudothermal speckle light divided by a beam splitter to achieve high-resolution ghost imaging and ghost diffraction. Hunt et al. [19] proposed the use of apertures designed with metamaterials for compressive microwave imaging, utilizing a waveguide metamaterial aperture to perform compressive image reconstruction of two-dimensional sparse static images and video scenes in the K-band, employing frequency diversity to avoid mechanical scanning. Cheng et al. [39] proposed a scheme of random frequency modulation array radar coincidence imaging using frequency hopping waveforms. Quan et al. [4] proposed a new microwave forward-looking correlation imaging method based on the random radiation field, which used the random phase shift of the phased array radar to form a two-dimensional random radiation field that is spatially and temporally incoherent. Chen et al. [40] proposed an azimuth phase modulation method, which can effectively solve the forward-looking imaging problem of airborne single-channel forward-looking radar on high-speed platforms. Mao et al. [41] proposed stochastic radiation radar imaging based on the 2D amplitude–phase orthogonal distribution array, which uses the 2D amplitude–phase orthogonal array to form a spatial–time stochastic radiation field.

Enhancing correlation imaging performance with correlation techniques, compressive sensing, or deep learning methods has garnered widespread attention, and numerous studies have already been published. Li et al. [20] proposed an improved correlation algorithm, successfully enhancing imaging resolution and performance. Zhu et al. [22] theoretically derived the second-order correlation method for microwave correlation imaging, achieving a resolution approximately 55% of that of coherent array radar, significantly improving imaging performance. Zhao et al. [10] introduced a new method of distributed space-borne satellite staring imaging, combining distributed networking theory with correlation algorithms to achieve two-dimensional imaging of targets, offering higher imaging efficiency compared to classical methods. Mao et al. [42] proposed a generalized adaptive asymptotic minimum variance estimator that relies on a normalized projection array model, which can achieve fast angular super-resolution reconstruction of real aperture radar with high-dimensional data. Zhang et al. [15], Li et al. [43], Zhang et al. [44], and Xu et al. [45] combined microwave correlation imaging with the compressed sensing model, introduced target sparse prior information and scene prior information into the imaging model, and used compressed sensing theory to achieve high-performance microwave correlation imaging. Sihna et al. [38] successfully solved the problem of computational imaging by training deep neural networks (DNNs). He et al. [46], Wang et al. [47], and Li et al. [48] also applied deep learning technology to the image reconstruction of correlated imaging. They learned and extracted features from a large amount of data through deep neural networks, which significantly improved the quality and speed of image reconstruction.

Although these studies have made some progress in improving the performance of microwave correlation imaging, there are still some challenges. Most of the optimizing construction methods of the random radiation field require complex hardware system support, which is costly, and most of the current microwave correlation imaging is based on a single modulation, such as random frequency modulation or random phase modulation. This single modulation method may be effective in certain specific environments, but in more complex application scenarios, it may not be able to give full play to its advantages. Combining novel image reconstruction signal processing techniques, such as correlation processing, can enhance super-resolution performance, but the improvement is still limited. Compressive sensing techniques rely on the selection of multiple parameters, and improper parameter selection can lead to a decrease in imaging quality. Deep learning techniques require substantial training data to enhance model generalization due to the variability in data distribution across different application scenarios caused by random radiation fields. Additionally, they may struggle to perform well in new scenarios. At the same time, the microwave correlation imaging performance of the existing methods still needs to be improved at low SNR, and more efficient noise suppression methods are needed.

To address the aforementioned issues, this manuscript proposes a high-resolution and robust microwave correlation imaging method based on URRF using the MC-AAMPE algorithm.

(1)

Firstly, we analyze the array antenna far-field patterns and their spatiotemporal characteristics under phase modulation, amplitude modulation, and frequency modulation. Based on this, we propose a novel URRF that integrates phase, amplitude, and frequency modulation. This URRF model enhances the flexibility and adaptability of the random radiation field.

(2)

Secondly, we consider the impact of the URRF on imaging performance and establish a parametric model for microwave correlation imaging based on the URRF, including both a 2D imaging model and a three-dimensional (3D) imaging model.

(3)

Finally, we transform the high-resolution image reconstruction into a multi-parameter optimization problem of sparse representation. This optimization incorporates prior information by considering scene noise and image distribution characteristics. Based on this, we propose the MC-AAMPE algorithm. Under the parametric model for microwave correlation imaging, the high-dimensional multivariate optimization problem is decomposed into several sub-optimization problems. In each iteration of the alternating estimation process, we fix the non-estimated parameters and optimize the current estimated parameter. By gradually estimating noise variance, Laplace scale coefficient, and the reconstructed image, we approach the optimal solution of the overall optimization problem. The continuous estimating of scene noise and image statistical distribution parameters gradually enhances the sparsity of the reconstructed images, effectively distinguishing the effects of signal and noise when solving the optimization problem. The parameters do not need to be set manually, which enhances adaptability. The proposed method can reconstruct high-resolution images and achieve high-performance imaging results while reducing the impact of noise, showing good robustness under noise conditions.

This manuscript is organized as follows. Section 2 introduces the construction of the URRF model integrating phase modulation, amplitude modulation, and frequency modulation. Additionally, we establish the parametric imaging model and propose the MC-AAMPE algorithm. Section 3 provides the analysis and discussion of the experimental results, verifying the effectiveness of the proposed method. Section 4 concludes the manuscript.

2. Methods

2.1. Imaging Mechanism and URRF Model

Section 2.1.1 introduces the imaging mechanism based on the random radiation field. In Section 2.1.2, we establish the URRF integrating phase, amplitude, and frequency modulation, which is spatially and temporally incoherent.

2.1.1. Imaging Mechanism Based on the Random Radiation Field

Traditional real aperture imaging technology relies on large aperture antennas to synthesize narrow beams, thereby improving spatial angular resolution. However, this method has limitations in distinguishing multiple targets within a single beam. The main problem comes from the equivalent phase center of the antenna, resulting in little change in the space–time radiation field of the entire imaging area. Therefore, the scattering field information is still essentially uniform, and we lack sufficient details to effectively distinguish multiple targets.

Microwave correlation imaging uses a specific modulation waveform to induce random modulation of the wavefront, resulting in a random radiation field that is irrelevant in time and space. Figure 1 compares the random modulation wavefront with the conventional plane wavefront. The radiation pattern of the traditional plane wavefront is almost linearly correlated, which means that additional measurements will not significantly enrich the information. On the contrary, the randomly modulated wavefront will introduce spatial diversity, resulting in significant electromagnetic changes in the signals received from targets of different beams. This spatial diversity enriches the information available for target resolution in the same beam [15,39].

In microwave correlation imaging, multiple transmitting array elements emit electromagnetic waves that are modulated randomly. These waves combine incoherently in space to form a spatio-temporal random radiation field within the target region. Due to the spatial diversity introduced by the random modulations, targets at different positions produce echoes with distinct modulation characteristics. This variation across the field increases the potential for target resolution. The imaging process involves integrating and correlating multiple echo signals with the constructed spatio-temporal random radiation field. By utilizing the multiple modulation characteristics of the scattering echo, the system can effectively extract the high-resolution spatial distribution information of the target from the received signal [18]. This method can improve the target resolution and imaging performance in the detection plane. Figure 2 illustrates the principle of microwave correlation imaging.

Microwave correlation imaging overcomes the limitations of traditional antenna apertures by using the space–time random radiation field and advanced correlation algorithm. Using a random modulation wavefront overcomes the limitations of traditional real aperture imaging technology, enabling higher spatial angular resolution and more accurate target recognition ability.

2.1.2. URRF Integrating Multiple Modulations

In the current microwave correlation imaging technology, the random radiation field generally adopts a single type of random modulation, such as random frequency or random phase modulation. The microwave correlation imaging method based on single modulation has limitations.

In order to solve this problem, we analyze the far-field pattern and space–time characteristics of phased array radar under phase, amplitude, and frequency modulation. Based on this analysis, we propose the URRF model integrating phase, amplitude, and frequency modulation. The model can realize more diverse random radiation field distribution and improve the flexibility and adaptability of random radiation field.

Figure 3 illustrates the schematic diagram of the phased array antenna element arrangement. The coordinate system O-xyz is established, with the phased array antenna arranged in the Oxy plane, containing $M\times N$ antenna elements. There are M antenna elements in the x direction and N antenna elements in the y direction. Each element is labeled by $\left(m,n\right)$, where $m\left(0\le m\le M-1\right)$ and $n\left(0\le n\le N-1\right)$ represent the indices along the x-axis and y-axis, respectively.

The reference array element $\left(0,0\right)$ is at point O, the line between the far-field observation point P and point O is OP, the angle between the projection of OP on the Oxz plane and the z-axis is $\theta$, and the angle between the projection of OP on the Oxz plane and OP is $\beta$. The electromagnetic performance of the array antenna [49] can be expressed as

$$E\left(\theta ,\beta \right)={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{f}_{mn}\left(\theta ,\beta \right)I_{mn}\exp \left(j{\phi }_{mn}\right)\exp \left(jk{r}_{mn}·r_0\right)}}$$

(1)

where $f_{mn}\left(\theta ,\beta \right)$ is the pattern of the array element, which represents the far-field radiation distribution of the array element, and is related to the physical structure and excitation form of the element itself—for simplicity, we assume it is consistent across all array elements; $I_{mn}$ and ${\phi }_{mn}$ are the excitation amplitude and phase, respectively; $k=2\pi /\lambda$ is the free-space wavenumber; and $\lambda$ denotes the wavelength that is related to the frequency of the antenna. $r_{mn}=\left(m-1\right)d_{x}x+\left(n-1\right)d_{y}y$ is the vector from the coordinate origin to the phase center of the array element, where $d_x$ and $d_y$ are the element intervals in the x and y directions, respectively. $x$ and $y$ are the unit vectors in the x and y directions, respectively; and $r_0$ is the unit vector that points in the observation direction.

• Random Radiation Field Based on Phase Modulation

Assuming the phase of an array element is a function of time, the phase of an array element can be expressed as

$${\phi }_{mn}\left(t\right)=-\left(k\left(\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0+\left(n-1\right)d_y\sin {\beta }_0\right)+\Delta {\varphi }_{mn}\left(t\right)\right)$$

(2)

where $\Delta {\varphi }_{mn}\left(t\right)$ is a stochastic process, which represents the additional random phase added to the array element. For example, they can be set to be independent of each other and obey uniform distribution $\Delta {\varphi }_{mn}\left(t\right)\sim U\left(0,2\pi \right)$. $\left({\theta }_0,{\beta }_0\right)$ is the antenna beam direction.

Assuming the amplitude of an array element is uniform, such as $I_{mn}=1$, the far-field radiation pattern of the phased array radar can be expressed as a function of $\theta$, $\beta$ and time $t$:

$$\begin{array}{cc}\hfill F_p\left(\theta ,\beta ;t\right)& ={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}\exp \left(jk\left(\left(m-1\right)d_x\sin \theta \cos \beta +\left(n-1\right)d_y\sin \beta \right)+j{\phi }_{mn}\left(t\right)\right)}}\hfill \\ & ={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}\exp \left(\begin{array}{l}jk\left(\left(m-1\right)d_x\sin \theta \cos \beta -\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0\right)\\ +jk\left(\left(n-1\right)d_y\sin \beta -\left(n-1\right)d_y\sin {\beta }_0\right)\\ -j\Delta {\varphi }_{mn}\left(t\right)\end{array}\right)}}\hfill \end{array}$$

(3)

Observing Equation (3), it can be seen that due to the random distribution characteristics of $\Delta {\varphi }_{mn}\left(t\right)$, the fields of the array elements no longer coherently combine in space. With random phase modulation applied to each array element, the far-field radiation pattern $F_p\left(\theta ,\beta ;t\right)$ of the phased array radar is in the form of random fluctuations.

Based on random phase modulation, the orthogonality between the transmitted signals of multiple array elements can be easily realized. The transmitted signals of each array element are modulated by independent random signals. There are $M\times N$ groups of independent random phases, denoted as $\left\{\Delta {\varphi }_{mn}\left(t\right),0\le m\le M-1,0\le n\le N-1\right\}$, and the cross-correlation function of random phase modulation can be expressed as

$$\begin{array}{cc}\hfill R_{\Delta \varphi }\left(m_1{n}_1,m_2{n}_2;t_1,t_2\right)& =E\left[\Delta {\varphi }_{m_1{n}_1}\left(t+t_1\right)\Delta {\varphi }_{m_2{n}_2}^{\ast }\left(t+t_2\right)\right]\hfill \\ & =\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}{\displaystyle {\int }_{-T/2}^{T/2}\Delta {\varphi }_{m_1{n}_1}\left(t+t_1\right)\Delta {\varphi }_{m_2{n}_2}^{\ast }\left(t+t_2\right)}dt\hfill \\ & =\delta \left(m_1{n}_1-m_2{n}_2;t_1-t_2\right)\hfill \end{array}$$

(4)

where $E\left(\cdot \right)$ denotes the averaging operation, and ${\left[\cdot \right]}^{\ast }$ represents the conjugate transpose operation. Furthermore, the cross-correlation function of the radiation pattern with random phase modulation can be calculated as

$$\begin{array}{cc}\hfill R_{F_p}\left(t_1,t_2\right)& =E\left[F_p\left(\theta ,\beta ;t_1\right){F_p}^{\ast }\left(\theta ,\beta ;t_2\right)\right]\hfill \\ & =R_{\Delta \varphi }\left(t_1,t_2\right)\hfill \\ & =\delta \left(t_1-t_2\right)\hfill \end{array}$$

(5)

From the above derivation, it can be seen that random phase modulation can achieve the orthogonality of transmitted signals between multiple array elements, and the temporal incoherence of the far-field radiation pattern.

• Random Radiation Field Based on Amplitude Modulation

Assume that the excitation amplitude of the array element is a stochastic process $I_{mn}\left(t\right)$ varying with time. For example, they can be set to be independent of each other and obey uniform distribution $I_{mn}\left(t\right)\sim U\left(-1,1\right)$. The phase of array element can be represented as

$${\phi }_{mn}\left(t\right)=-\left(k\left(\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0+\left(n-1\right)d_y\sin {\beta }_0\right)\right)$$

(6)

The far-field radiation pattern of the phased array radar under random amplitude modulation can be expressed as:

$$\begin{array}{cc}\hfill F_a\left(\theta ,\beta ;t\right)& ={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{I}_{mn}\left(t\right)\exp \left(jk\left(\left(m-1\right)d_x\sin \theta \cos \beta +\left(n-1\right)d_y\sin \beta \right)+j{\phi }_{mn}\left(t\right)\right)}}\hfill \\ & ={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{I}_{mn}\left(t\right)\exp \left(\begin{array}{l}jk\left(\left(m-1\right)d_x\sin \theta \cos \beta -\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0\right)\\ +jk\left(\left(n-1\right)d_y\sin \beta -\left(n-1\right)d_y\sin {\beta }_0\right)\end{array}\right)}}\hfill \end{array}$$

(7)

It can be seen that due to the random distribution characteristics of $I_{mn}\left(t\right)$, the random amplitude modulation is performed on each array element, and the far-field radiation pattern $F_a\left(\theta ,\beta ;t\right)$ of the phased array radar is in the form of random fluctuations.

Based on random amplitude modulation, the orthogonality between the transmitted signals of multiple array elements can be easily realized. The transmitted signals of each array element are modulated by independent random signals. There are $M\times N$ groups of independent random amplitudes, expressed as $\left\{I_{mn}\left(t\right),0\le m\le M-1,0\le n\le N-1\right\}$, and the cross-correlation function of random amplitude modulation can be expressed as

$$\begin{array}{cc}\hfill R_I\left(m_1{n}_1,m_2{n}_2;t_1,t_2\right)& =E\left[I_{m_1{n}_1}\left(t+t_1\right)I_{m_2{n}_2}^{\ast }\left(t+t_2\right)\right]\hfill \\ & =\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}{\displaystyle {\int }_{-T/2}^{T/2}{I}_{m_1{n}_1}\left(t+t_1\right)I_{m_2{n}_2}^{\ast }\left(t+t_2\right)}dt\hfill \\ & =\delta \left(m_1{n}_1-m_2{n}_2;t_1-t_2\right)\hfill \end{array}$$

(8)

The cross-correlation function of the far-field pattern under random amplitude modulation can be written as

$$\begin{array}{cc}\hfill R_{F_a}\left(t_1,t_2\right)& =E\left[F_a\left(\theta ,\beta ;t_1\right){F_a}^{\ast }\left(\theta ,\beta ;t_2\right)\right]\hfill \\ & =R_I\left(t_1,t_2\right)\hfill \\ & =\delta \left(t_1-t_2\right)\hfill \end{array}$$

(9)

It can be seen that the random amplitude modulation can achieve the orthogonality of the transmitted signals between multiple array elements and the time irrelevance of the far-field radiation pattern.

• Random Radiation Field Based on Frequency Modulation

Assuming that the frequency modulation of the array element is a stochastic process $\Delta f_{mn}\left(t\right)$ that changes with time, Equation (1) can be rewritten as

$$E\left(\theta ,\beta \right)={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{f}_{mn}\left(\theta ,\beta \right)I_{mn}\exp \left(j{\phi }_{mn}\right)\exp \left(\begin{array}{l}j\frac{2\pi }{c}\left(f_c+\Delta f_{mn}\left(t\right)\right)\\ \times \left(\begin{array}{l}\left(m-1\right)d_x\sin \theta \cos \beta \\ +\left(n-1\right)d_y\sin \beta \end{array}\right)\end{array}\right)}}$$

(10)

where $f_c$ represents the carrier frequency, and $c$ is the speed of light. The phase of the array element can be represented as

$${\phi }_{mn}\left(t\right)=-\left(k\left(\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0+\left(n-1\right)d_y\sin {\beta }_0\right)\right)$$

(11)

Assuming the amplitude of the array element is same, such as $I_{mn}=1$, the far-field radiation pattern of the phased array radar can be rewritten as

$$\begin{array}{cc}\hfill F_f\left(\theta ,\beta ;t\right)& ={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}\exp \left(j\frac{2\pi }{c}\left(f_c+\Delta f_{mn}\left(t\right)\right)\times \left(\begin{array}{l}\left(m-1\right)d_x\sin \theta \cos \beta \\ +\left(n-1\right)d_y\sin \beta \end{array}\right)+j{\phi }_{mn}\left(t\right)\right)}}\hfill \\ & ={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}\exp \left(j\frac{2\pi }{c}\left(f_c+\Delta f_{mn}\left(t\right)\right)\times \left(\begin{array}{l}\left(\begin{array}{l}\left(m-1\right)d_x\sin \theta \cos \beta \\ -\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0\end{array}\right)\\ +\left(\begin{array}{l}\left(n-1\right)d_y\sin \beta \\ -\left(n-1\right)d_y\sin {\beta }_0\end{array}\right)\end{array}\right)\right)}}\hfill \end{array}$$

(12)

The random distribution characteristics of $\Delta f_{mn}\left(t\right)$ make the field of the array element no longer coherently combined in space. The array element performs random frequency modulation, so that the far-field radiation pattern $F_f\left(\theta ,\beta ;t\right)$ of the phased array radar exhibits a random fluctuation form.

Under random frequency modulation, the orthogonality between the transmitted signals of multiple array elements can be easily realized. The independent random signals are used to modulate the transmitted signals of the array element. There are $M\times N$ groups of independent random frequencies, which can be expressed as $\left\{\Delta f_{mn}\left(t\right),0\le m\le M-1,0\le n\le N-1\right\}$; the cross-correlation function of random frequency modulation can be written as

$$\begin{array}{cc}\hfill R_{\Delta f}\left(m_1{n}_1,m_2{n}_2;t_1,t_2\right)& =E\left[\Delta f_{m_1{n}_1}\left(t+t_1\right)\Delta f_{m_2{n}_2}^{\ast }\left(t+t_2\right)\right]\hfill \\ & =\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}{\displaystyle {\int }_{-T/2}^{T/2}\Delta f_{m_1{n}_1}\left(t+t_1\right)\Delta f_{m_2{n}_2}^{\ast }\left(t+t_2\right)}dt\hfill \\ & =\delta \left(m_1{n}_1-m_2{n}_2;t_1-t_2\right)\hfill \end{array}$$

(13)

The cross-correlation function of the far-field pattern under random frequency modulation can be expressed as

$$\begin{array}{cc}\hfill R_{F_f}\left(t_1,t_2\right)& =E\left[F_f\left(\theta ,\beta ;t_1\right){F_f}^{\ast }\left(\theta ,\beta ;t_2\right)\right]\hfill \\ & =R_{\Delta f}\left(t_1,t_2\right)\hfill \\ & =\delta \left(t_1-t_2\right)\hfill \end{array}$$

(14)

Observing the above derivation, it can be seen that the random frequency modulation can achieve the orthogonality of the transmitted signals between the array elements, and the time irrelevance of the far-field pattern.

• URRF Model

It can be seen that random phase modulation, random amplitude modulation, and random frequency modulation can obtain the orthogonality of the transmitted signal between the array elements and the time irrelevance of the far-field pattern, which has two-dimensional irrelevance in space and time. Based on this, we propose a URRF model that combines phase, amplitude, and frequency modulation. This URRF model can be specifically expressed as

$$F_U\left(\theta ,\beta ;t\right)={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{I}_{mn}\left(t\right)\times \exp \left(\begin{array}{l}j\frac{2\pi }{c}\left(f_c+\Delta f_{mn}\left(t\right)\right)\times \left(\begin{array}{l}\left(\begin{array}{l}\left(m-1\right)d_x\sin \theta \cos \beta \\ -\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0\end{array}\right)\\ +\left(\begin{array}{l}\left(n-1\right)d_y\sin \beta \\ -\left(n-1\right)d_y\sin {\beta }_0\end{array}\right)\end{array}\right)\\ -j\Delta {\varphi }_{mn}\left(t\right)\end{array}\right)}}$$

(15)

The URRF model proposed in this manuscript integrates phase modulation, amplitude modulation, and frequency modulation, enabling a more diverse distribution of random radiation field. This approach enhances the random radiation field’s ability to deal with a wider range of environmental conditions and application scenarios.

2.2. High-Resolution and Robust Microwave Correlation Imaging

High-resolution and robust microwave correlation imaging based on the URRF is proposed in this section, and some key procedures are discussed in detail. Based on the URRF model proposed in Section 2.1, the challenges of establishing 2D and 3D parametric imaging models and noise interference are crucial for achieving high-performance imaging. To address these issues, a novel processing framework has been proposed in this manuscript, as illustrated in Figure 4.

To reconstruct the 2D image based on the URRF, a 2D parametric imaging model based on the correlation between the isometric wavefront echo information and the spatio-temporal incoherent URRF is established. The model clearly reveals the relationship between the image to be reconstructed and the URRF, and provides a reliable model support for image reconstruction.

To obtain the 3D configuration of the target and acquire more comprehensive target information, this manuscript establishes a 3D parametric imaging model. The model reveals the correlation between the echo information from different distance wavefront planes and the URRF. It aims to solve the imaging of relatively stationary targets or radar forward-looking 3D imaging. It is worth noting that the model does not depend on Doppler frequency shift and does not require a large number of radar observation angles for data accumulation.

To address the issues of limited noise suppression ability and insufficient resolution in current microwave correlation imaging algorithms, this manuscript proposes the MC-AAMPE algorithm. The proposed algorithm decomposes the high-dimensional multivariate optimization problem into several sub-optimization problems. The proposed method achieves more accurate high-resolution image reconstruction and more robust noise suppression.

2.2.1. 2D Imaging Model Based on the URRF

The principle of 2D imaging based on microwave correlation is shown in Figure 5. The coordinate system O-xyz is established in the direction of the radar line of sight. The scene reference center is located at the origin O, and the phased array radar reference center is located on the z-axis. Assume that the transmission signal of the phased array radar is the linearly frequency-modulated signal [28]:

$$s_t\left(\widehat{t},t_l\right)=\mathrm{rect}\left({\displaystyle \frac{\widehat{t}}{T_p}}\right)\exp \left(j2\pi f_{c}t\right)\times \exp \left(j\pi \gamma {\widehat{t}}^2\right)$$

(16)

where $t=\widehat{t}+t_l$, $\widehat{t}$ is the fast time, $t_l=l\times T_r$ is the slow time, $l$ represents the number of pulse repetitions, $T_r$ represents the pulse repetition period, $\mathrm{rect}\left(·\right)$ is the rectangular window function, $T_p$ is pulsewidth, $f_c$ represents the carrier frequency, $\gamma =B/T_p$ is the chirp rate of the signal, and $B$ is the bandwidth.

If the target is composed of $P$ scattering points, and the range from the p-th $\left(1\le p\le P\right)$ scattering point to the phased array radar is $R_p\left(t_l\right)$, the baseband echo of the scene can be represented as

$$s_r\left(\widehat{t},t_l\right)={\displaystyle \sum _{p=1}^P\begin{array}{l}{a}_p{F}_U\left({\theta }_p,{\beta }_p;t_l\right)rect\left(\frac{\left(\widehat{t}-\frac{2R_p\left(t_l\right)}{c}\right)}{T_p}\right)\\ \times \exp \left(-j2\pi f_c\frac{2R_p\left(t_l\right)}{c}\right)\exp \left(j\pi \gamma {\left(\widehat{t}-\frac{2R_p\left(t_l\right)}{c}\right)}^2\right)\end{array}}$$

(17)

where $a_p$ is the reflectivity coefficient corresponding to the p-th scattering point; ${\theta }_p$ is the angle between the projection of the line connecting the p-th scattering point and the radar center in the Oyz plane and the z-axis. ${\theta }_p$ is denoted as the azimuth angle; ${\beta }_p$ is the angle between the projection of the line connecting the p-th scattering point and the radar center in the Oyz plane and the line connecting the p-th scattering point and the radar center, and ${\beta }_p$ is the pitch angle; $F_U\left({\theta }_p,{\beta }_p;t_l\right)$ is the radiation pattern of l-th pulse, which is the URRF proposed in Section 2.1. $F_U\left({\theta }_p,{\beta }_p;t_l\right)$ remains constant during each pulse transmission and reception process, while varying between pulses to provide multiple independent random samples.

The reference signal is

$$\begin{array}{c}{s}_{ref}\left(\widehat{t},t_l\right)=rect\left({\displaystyle \frac{\widehat{t}-\frac{2R_{ref}}{c}}{T_{ref}}}\right)\exp \left(j2\pi f_c\left(t-{\displaystyle \frac{2R_{ref}}{c}}\right)\right)\\ \times \exp \left(j\pi \gamma {\left(\widehat{t}-{\displaystyle \frac{2R_{ref}}{c}}\right)}^2\right)\end{array}$$

(18)

where $R_{ref}$ is the reference range. $T_{ref}$ is the reference signal pulsewidth, which is longer than $T_p$. After the dechirping process, the echo signal can be written

$$s_d\left(\widehat{t},t_l\right)={\displaystyle \sum _{p=1}^P\begin{array}{l}{a}_p{F}_U\left({\theta }_p,{\beta }_p;t_l\right)rect\left(\frac{\left(\widehat{t}-\frac{2R_p}{c}\right)}{T_p}\right)\times \exp \left(-j2\pi f_c\frac{2R_{\Delta p}}{c}\right)\\ \times \exp \left(j\frac{4\pi \gamma }{c^2}{\left(R_{\Delta p}\right)}^2\right)\times \exp \left(-j\frac{4\pi }{c}\gamma \left(\widehat{t}-\frac{2R_{ref}}{c}\right)R_{\Delta p}\right)\end{array}}$$

(19)

where $R_{\Delta p}=R_p-R_{ref}$. Taking the Fourier transform of Equation (19) along the fast time, we can obtain

$$S_d\left(f,t_m\right)={\displaystyle \sum _{p=1}^P\begin{array}{l}{a}_p{F}_U\left({\theta }_p,{\beta }_p;t_l\right)\sin c\left[T_p\left(f+\frac{2\gamma }{c}{R}_{\Delta p}\right)\right]\times \exp \left(-j2\pi f_c\frac{2R_{\Delta p}}{c}\right)\\ \times \exp \left(-j\frac{4\pi \gamma }{c^2}{\left(R_{\Delta p}\right)}^2\right)\times \exp \left(-j\frac{4\pi f}{c}{R}_{\Delta p}\right)\end{array}}$$

(20)

Observing Equation (20), the dechirp received signal becomes a sinc function with a very narrow width in the range frequency domain, and its peak is located at $f=-2\gamma R_{\Delta p}/c$. when the target at distance $R_{\Delta p}$ is compensated, only the phase at $f=-2\gamma R_{\Delta p}/c$ needs to be compensated. Then the last two phase terms in Equation (20) can be written as

$$\Delta \Phi =-{\displaystyle \frac{4\pi \gamma }{c^2}}{\left(R_{\Delta p}\right)}^2-{\displaystyle \frac{4\pi f}{c}}{R}_{\Delta p}={\displaystyle \frac{\pi f^2}{\gamma }}$$

(21)

The matching function is constructed, which can be expressed as

$$S_c\left(f\right)=\exp \left(-j{\displaystyle \frac{\pi f^2}{\gamma }}\right)$$

(22)

After removing the RVP term and the range profile deviation term, we can obtain

$$S_{dc}\left(f,t_l\right)={\displaystyle \sum _{p=1}^P{a}_p{F}_U\left({\theta }_p,{\beta }_p;t_l\right)\sin c\left[T_p\left(f+\frac{2\gamma }{c}{R}_{\Delta p}\right)\right]\times \exp \left(-j2\pi f_c\frac{2R_{\Delta p}}{c}\right)}$$

(23)

where $\sin c\left[T_p\left(f+{\displaystyle \frac{2\gamma }{c}}{R}_{\Delta p}\right)\right]$ is the range envelope of the p-th scattering point. Each discretely sampled range cell represents all scattering point echoes of the equidistant wavefront plane, and changes with the modulation of each URRF.

The observed scene and the unified random radiation pattern are discretized. $K=N_{\theta }\times N_{\beta }$ is the total number of imaging grid units obtained after the observed scene is discretized at equal intervals according to the angle; $N_{\theta }$ is the number of azimuth grids; $N_{\beta }$ is the total number of pitch grids; ${\theta }_k$ and ${\beta }_k$ are the azimuth and pitch angles of the k-th $k\left(1\le k\le K\right)$ scattering cell, respectively; the size of each scattering cell is $\Delta \theta \times \Delta \beta$. The 2D parametric imaging model based on microwave correlation can be expressed as

$$\begin{array}{l}{S}_{dc}^q=F_U{E}^{q}a+n\\ S_{dc}^q={\left(vec{\left(S_{dc}^q\left(t_1\right),S_{dc}^q\left(t_2\right),\cdots S_{dc}^q\left(t_L\right)\right)}^T\right)}_{L\times 1}\\ F_U={\left[\begin{array}{c}{F}_U\left(\theta ,\beta ;t_1\right)\\ F_U\left(\theta ,\beta ;t_2\right)\\ ⋮\\ F_U\left(\theta ,\beta ;t_L\right)\end{array}\right]}_{L\times K}\\ F_U\left(\theta ,\beta ;t_l\right)={\left[F_U\left({\theta }_1,{\beta }_1;t_l\right),F_U\left({\theta }_2,{\beta }_2;t_l\right),\cdots ,F_U\left({\theta }_K,{\beta }_K;t_l\right)\right]}_{1\times K} \left(\begin{array}{l}1\le l\le L,\\ t_1\le t_l\le t_L\end{array}\right)\\ F_U\left({\theta }_k,{\beta }_k;t_l\right)={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{I}_{mn}\left(t_l\right)\times \exp \left(\begin{array}{l}j\frac{2\pi }{c}\left(f_c+\Delta f_{mn}\left(t_l\right)\right)\\ \times \left(\begin{array}{l}\left(\begin{array}{l}\left(m-1\right)d_x\sin \theta \cos \beta \\ -\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0\end{array}\right)\\ +\left(\begin{array}{l}\left(n-1\right)d_y\sin \beta \\ -\left(n-1\right)d_y\sin {\beta }_0\end{array}\right)\end{array}\right)\\ -j\Delta {\varphi }_{mn}\left(t_l\right)\end{array}\right)}}\left(\begin{array}{l}1\le k\le K,\\ {\theta }_1\le {\theta }_k\le {\theta }_K,\\ {\beta }_1\le {\beta }_k\le {\beta }_K\end{array}\right) \\ E^q=\mathrm{diag}{\left(\exp \left(-j2\pi f_c{\displaystyle \frac{2R_{\Delta p}^q}{c}}\right)\right)}_{K\times K}\end{array}$$

(24)

where $S_{dc}^q$ is the echo vector of q-th range slice corresponding to the imaging plane after dechirping under the URRF; $L$ is the number of pulses; $a$ is the two-dimensional image to be reconstructed; $F_U$ is the sensing matrix based on the URRF; $R_{\Delta p}^q=R_p^q-R_{ref}$, and $R_p^q$ is the range between the q-th range slice and radar phase center; $E^q$ is the phase matrix corresponding to different imaging planes. $n$ is the system noise. ${\left(\cdot \right)}^T$ represents the transpose operation.

2.2.2. 3D Imaging Model Based on the URRF

The principle of 3D imaging based on microwave correlation is shown in Figure 6. A series of two-dimensional images on the equidistant plane can be obtained by reconstructing the wavefront plane at different distances using broadband signals. These images are stitched along the radar line of sight direction, and the 3D image can be obtained.

The 3D observation scene and the URRF are discretized, which can be divided into $N_Q$ range slices. There are $K=N_{\theta }\times N_{\beta }$ imaging grid units on each range slice. Under the URRF, the 3D parametric imaging model based on microwave correlation can be expressed as

$$\begin{array}{l}{S}_{dc}={\left[\begin{array}{c}{S}_{dc}^1\\ S_{dc}^2\\ ⋮\\ S_{dc}^{N_Q}\end{array}\right]}_{\left(L\times N_Q\right)\times 1}=F_{U\_3D}Ea+n\\ S_{dc}^q={\left(vec{\left(S_{dc}^q\left(t_1\right),S_{dc}^q\left(t_2\right),\cdots S_{dc}^q\left(t_L\right)\right)}^T\right)}_{L\times 1}\\ F_{U\_3D}={\left[\begin{array}{cccc}{F}_U^1& 0& \cdots & 0\\ 0& F_U^2& 0& ⋮\\ ⋮& 0& \ddots & 0\\ 0& \cdots & 0& F_U^{N_{\mathrm{Q}}}\end{array}\right]}_{\left(L\times N_Q\right)\times \left(K\times N_Q\right)}\\ F_U^q={\left(\left[\begin{array}{c}{F}_U^q\left(\theta ,\beta ;t_1\right)\\ F_U^q\left(\theta ,\beta ;t_2\right)\\ ⋮\\ F_U^q\left(\theta ,\beta ;t_L\right)\end{array}\right]\right)}_{L\times K}\\ F_U^q\left(\theta ,\beta ;t_l\right)={\left[F_U^q\left({\theta }_1,{\beta }_1;t_l\right),F_U^q\left({\theta }_2,{\beta }_2;t_l\right),\cdots ,F_U^q\left({\theta }_K,{\beta }_K;t_l\right)\right]}_{1\times K} \left(\begin{array}{l}1\le l\le L,\\ t_1\le t_l\le t_L\end{array}\right)\\ F_U^q\left({\theta }_k,{\beta }_k;t_l\right)={\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}{I}_{mn}\left(t_l\right)\times \exp \left(\begin{array}{l}j\frac{2\pi }{c}\left(f_c+\Delta f_{mn}\left(t_l\right)\right)\\ \times \left(\begin{array}{l}\left(\begin{array}{l}\left(m-1\right)d_x\sin \theta \cos \beta \\ -\left(m-1\right)d_x\sin {\theta }_0\cos {\beta }_0\end{array}\right)\\ +\left(\begin{array}{l}\left(n-1\right)d_y\sin \beta \\ -\left(n-1\right)d_y\sin {\beta }_0\end{array}\right)\end{array}\right)\\ -j\Delta {\varphi }_{mn}\left(t_l\right)\end{array}\right)}}\left(\begin{array}{l}1\le k\le K,\\ {\theta }_1\le {\theta }_k\le {\theta }_K,\\ {\beta }_1\le {\beta }_k\le {\beta }_K\end{array}\right)\\ E={\left[\begin{array}{cccc}{E}^1& 0& \cdots & 0\\ 0& E^2& 0& ⋮\\ ⋮& 0& \ddots & 0\\ 0& \cdots & 0& E^{N_Q}\end{array}\right]}_{\left(K\times N_Q\right)\times \left(K\times N_Q\right)}\\ E^q=\mathrm{diag}{\left(\exp \left(-j2\pi f_c{\displaystyle \frac{2R_{\Delta p}^q}{c}}\right)\right)}_{K\times K}\end{array}$$

(25)

where $S_{dc}$ is the echo vector obtained by multiple range slices after the dechirping process; $a$ represents the three-dimensional image that needs to be reconstructed; $F_{U\_3D}$ represents the sensing matrix corresponding to the three-dimensional scene; $E^q$ is the phase matrix corresponding to different imaging planes; $n$ is the system noise.

2.2.3. MC-AAMPE Algorithm

From the parametric imaging model based on the URRF established in Section 2.2.1 and Section 2.2.2, the echo after the dechirping process includes the information of the scattering points in the scene, and the URRF meets the requirements of space–time 2D incoherence. The imaging scene has universal sparsity, especially the limited penetration ability of microwaves, so that the target on each range slice only occupies a small part of the scene, so that the microwave correlation imaging based on the URRF can be converted into a multi-parameter optimization problem with sparse constraints. However, due to the inherent limitations of echo signal accumulation and SNR gain, the influence of environmental noise becomes more obvious. Therefore, effectively suppressing the influence of noise is very important for obtaining accurate imaging results.

Based on the obtained parametric imaging model, this section proposes the MC-AAMPE algorithm, which decomposes the high-dimensional multivariate optimization problem into several sub-optimization problems. In each alternating estimation process, the non-estimated parameters are fixed and the estimated parameter is optimized to gradually approach the optimal solution of the overall optimization problem. This method enhances the ability to distinguish the influence of signal and noise in the process of alternating estimation optimization. Therefore, this method ensures the robust performance of microwave correlation imaging based on URRF under different SNR conditions. This method is more adaptable in noisy scenes and can obtain high-resolution imaging results while reducing the influence of SNR.

• Prior Model

The complex Gaussian model is used to represent this noise distribution [27,50]. The microwave-correlated 3D imaging model based on the URRF is obtained by splicing the 2D imaging models of multiple range slices along the radar line of sight. To facilitate the analysis, the following 2D imaging model is used for method derivation. The posterior probability density functions of the observed data can be written as

$$p\left(\left.S_{dc}^q\right|a,{\sigma }^2\right)={\left(\pi {\sigma }^2\right)}^{-L}\exp \left(-{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2\right)$$

(26)

where ${\sigma }^2$ denotes the variance of $n$. ${‖·‖}_2$ is the $l_2$ norm.

The strong scattering points of the image only account for a few distribution units in the imaging domain, which has strong sparsity. The sparse prior of the image can be characterized by the Laplace probability distribution [51,52]. Each element is independent of each other, and the probability density function of the high-resolution image $a$ that needs to be reconstructed can be expressed as

$$p\left(\left.a\right|\gamma \right)={\displaystyle \prod _{k=1}^K\left(\frac{{\gamma }_k}{2}\right)\exp \left(-{\gamma }_k\left|a_k\right|\right)}$$

(27)

where $a_k$ represents the k-th $\left(1\le k\le K\right)$ component of $a$; ${\gamma }_k$ is the Laplace scale coefficient corresponding to $a_k$.

• Statistical Modeling and Solution of the Optimization Problem

Based on the prior model, according to the Bayesian criterion, the posterior probability density function of the high-resolution image $a$ to be reconstructed can be written as

$$p\left(\left.a\right|S_{dc}^q,\gamma ,{\sigma }^2\right)={\displaystyle \frac{p\left(\left.S_{dc}^q\right|a,{\sigma }^2\right)p\left(\left.a\right|\gamma \right)}{p\left(\left.S_{dc}^q\right|\gamma ,{\sigma }^2\right)}}$$

(28)

The maximum a posteriori (MAP) [27,53] estimation is used to estimate the high-resolution image $a$, which can be expressed as

$$\begin{array}{cc}\hfill a_{MAP}=& \underset{a}{\arg \max }\left[p\left(\left.a\right|S_{dc}^q,\gamma ,{\sigma }^2\right)\right]\hfill \\ & =\underset{a}{\arg \max }\left[{\displaystyle \frac{p\left(\left.S_{dc}^q\right|a,{\sigma }^2\right)p\left(\left.a\right|\gamma \right)}{p\left(\left.S_{dc}^q\right|\gamma ,{\sigma }^2\right)}}\right]\hfill \end{array}$$

(29)

$p\left(\left.S_{dc}^q\right|\gamma ,{\sigma }^2\right)$ and the high-resolution image $a$ are independent. Equation (29) can be rewritten as

$$a_{MAP}=\underset{a}{\arg \max }\left[p\left(\left.S_{dc}^q\right|a,{\sigma }^2\right)p\left(\left.a\right|\gamma \right)\right]$$

(30)

Substituting Equations (26) and (27) into Equation (30), we can obtain

$$p\left(\left.S_{dc}^q\right|a,{\sigma }^2\right)p\left(\left.a\right|\gamma \right)={\left(\pi {\sigma }^2\right)}^{-L}\exp \left(-{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2\right){\displaystyle \prod _{k=1}^K\left(\left(\frac{{\gamma }_k}{2}\right)\exp \left(-{\gamma }_k\left|a_k\right|\right)\right)}$$

(31)

Taking the natural logarithm of both sides of Equation (31)

$$\begin{array}{cc}\hfill In\left(p\left(\left.S_{dc}^q\right|a,{\sigma }^2\right)p\left(\left.a\right|\gamma \right)\right)& =In\left({\left(\pi {\sigma }^2\right)}^{-L}\exp \left(-{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2\right){\displaystyle \prod _{k=1}^K\left(\frac{{\gamma }_k}{2}\right)\exp \left(-{\gamma }_k\left|a_k\right|\right)}\right)\hfill \\ & =-L\times In\left(\pi \right)-L\times In\left({\sigma }^2\right)-{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2+{\displaystyle \sum _{k=1}^{K}In\left(\frac{{\gamma }_k}{2}\right)}-{\displaystyle \sum _{k=1}^K{\gamma }_k\left|a_k\right|}\hfill \end{array}$$

(32)

Make the following definition:

$$L\left(a,{\sigma }^2,\gamma \right)=-L\times In\left(\pi \right)-L\times In\left({\sigma }^2\right)-{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2+{\displaystyle \sum _{k=1}^{K}In\left(\frac{{\gamma }_k}{2}\right)}-{\displaystyle \sum _{k=1}^K{\gamma }_k\left|a_k\right|}$$

(33)

we can obtain

$$\begin{array}{cc}\hfill \left(a,{\sigma }^2,\gamma \right)& =\arg \max L\left(a,{\sigma }^2,\gamma \right)\hfill \\ & =\arg \min \left(L\times In\left({\sigma }^2\right)+{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2-{\displaystyle \sum _{k=1}^{K}In\left(\frac{{\gamma }_k}{2}\right)}+{\displaystyle \sum _{k=1}^K{\gamma }_k\left|a_k\right|}\right)\hfill \end{array}$$

(34)

The complete objective function can be written in the following form:

$$\begin{array}{cc}\hfill J\left(a,{\sigma }^2,\gamma \right)& =L\times In\left({\sigma }^2\right)+{\displaystyle \frac{1}{{\sigma }^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2-{\displaystyle \sum _{k=1}^{K}In\left(\frac{{\gamma }_k}{2}\right)}+{\displaystyle \sum _{k=1}^K{\gamma }_k\left|a_k\right|}\hfill \\ & ={\sigma }^2\times L\times In\left({\sigma }^2\right)+{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2-{\sigma }^2{\displaystyle \sum _{k=1}^{K}In\left(\frac{{\gamma }_k}{2}\right)}+{\sigma }^2{\displaystyle \sum _{k=1}^K{\gamma }_k\left|a_k\right|}\hfill \end{array}$$

(35)

High-resolution image reconstruction based on the URRF can be transformed into the following multi-parameter optimization problem with sparse constraints.

$$\left(\widehat{a},{\widehat{\sigma }}^2,\widehat{\gamma }\right)=\underset{a,{\sigma }^2,\gamma }{\arg \min }J\left(a,{\sigma }^2,\gamma \right)$$

(36)

The proposed method transforms microwave correlation imaging based on the URRF into a multi-parameter optimization problem with sparse constraints. The optimization problem involves complex multivariable coupling, resulting in extremely high computational complexity of direct solutions. In addition, due to common constraints, it is difficult for traditional methods to perform effective variable decomposition, which further leads to the difficulty of direct solutions.

To address this challenge, we propose a MC-AAMPE algorithm. Under the parametric model of microwave correlation imaging, this algorithm decomposes the high-dimensional multivariate optimization problem into several sub-optimization problems. In each alternating estimation process, the non-estimated parameters are fixed while the current estimated parameter is optimized. By gradually estimating noise variance, the Laplace scale coefficient, and the reconstructed image, the method approaches the optimal solution of the overall optimization problem. This process accurately estimates the sparse description parameters of the target signal and the noise characteristic parameters of the scene. Consequently, the method better distinguishes the influence of signal and noise, improving microwave correlation imaging resolution and reducing noise interference during optimization. The proposed method adds some penalty terms. Therefore, the proposed method enhances the noise suppression performance and improves the high-resolution capabilities.

In Equation (36), the high-resolution image $a$, noise variance ${\sigma }^2$, and Laplace scale coefficient $\gamma$ are all unknown. The noise variance ${\sigma }^2$ and Laplace scale coefficient $\gamma$ can be estimated, as detailed below:

$${\displaystyle \frac{\partial J\left(a,{\sigma }^2,\gamma \right)}{\partial \left({\sigma }^2\right)}}={\displaystyle \frac{L}{{\sigma }^2}}-{\displaystyle \frac{1}{{\left({\sigma }^2\right)}^2}}{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2$$

(37)

$${\displaystyle \frac{\partial J\left(a,{\sigma }^2,{\gamma }_k\right)}{\partial \left({\gamma }_k\right)}}=-\left({\displaystyle \frac{1}{{\gamma }_k}}\right)+\left|a_k\right|$$

(38)

The estimated parameters can be expressed as

$${\sigma }^2={\displaystyle \frac{{‖S_{dc}^q-F_U{E}^{q}a‖}_2^2}{L}}$$

(39)

$${\gamma }_k={\displaystyle \frac{1}{\left|a_k\right|+\epsilon }}$$

(40)

where $\epsilon$ denotes a small constant.

The MC-AAMPE algorithm solves the multi-parameter optimization problem with sparse constraints by the following equations:

$${\Lambda }_1\left({\gamma }^g\right)=diag\left({\gamma }_k^g\right)$$

(41)

$$H\left({\widehat{a}}^g\right)={\left({\sigma }^2\right)}^g{\Lambda }_1\left({\gamma }^g\right){\Lambda }_2\left(a^g\right)+2{\left(F_U{E}^q\right)}^H{F}_U{E}^q$$

(42)

$$H\left({\widehat{a}}^g\right){\widehat{a}}^{g+1}-2{\left(F_U{E}^q\right)}^H{S}_{dc}=0$$

(43)

$${\widehat{\gamma }}_k^{g+1}={\displaystyle \frac{1}{\left|{\widehat{a}}_k^{g+1}\right|+\epsilon }}$$

(44)

$${\left({\widehat{\sigma }}^2\right)}^{g+1}={\displaystyle \frac{{‖S_{dc}^q-F_U{E}^q{\widehat{a}}^{g+1}‖}_2^2}{L}}$$

(45)

where ${\widehat{a}}^g$ and ${\widehat{a}}^{g+1}$ represent the estimates of $a$ in the g-th and (g + 1)-th iterations, respectively; ${\left({\widehat{\sigma }}^2\right)}^g$ and ${\left({\widehat{\sigma }}^2\right)}^{g+1}$ represent the estimates of ${\sigma }^2$ in the g-th and (g + 1)-th iterations, respectively. ${\left({\widehat{\sigma }}^2\right)}^g$ can be computed using the estimated ${\widehat{a}}^g$; $H\left({\widehat{a}}^g\right)$ is an approximate Hessian matrix; ${\Lambda }_1\left({\gamma }^g\right)$ is the diagonal matrix, and its k-th element is ${\widehat{\gamma }}_k^g$. ${\widehat{\gamma }}_k^g$ and ${\widehat{\gamma }}_k^{g+1}$ represent the estimates of ${\gamma }_k$ in the g-th and (g + 1)-th iterations, respectively; ${\Lambda }_2\left(a^g\right)$ is the diagonal matrix, and its k-th element is ${\left({\left|a_k^g\right|}^2+\tau \right)}^{-1/2}$. $a_k^g$ is the k-th component of ${\widehat{a}}^g$. $\tau$ is a small non-negative constant to avoid the non-differentiable case of $\left|a_k\right|$.

The reconstructed high-resolution image ${\widehat{a}}^{g+1}$ at (g + 1)-th iteration can be obtained by solving Equation (43) with the conjugate gradient method, which can effectively avoid the high computational cost associated with matrix inversion during the optimization iterations. Combining the ${\widehat{a}}^{g+1}$, Equation (44), and Equation (45), the estimated parameter for the noise variance ${\left({\widehat{\sigma }}^2\right)}^{g+1}$ and the Laplace scale coefficient ${\widehat{\gamma }}_k^{g+1}$ at the (g + 1)-th iteration can be obtained. Through adaptive alternating estimation, until the convergence condition ${{‖{\widehat{a}}^{g+1}-{\widehat{a}}^g‖}_2/‖{\widehat{a}}^g‖}_2\le \rho$ is achieved, where $\rho$ represents a predefined threshold.

Based on the idea of sub-problem decomposition, this manuscript proposes the MC-AAMPE method. Compared with the common sparse Bayesian method [4], the proposed method not only continuously updates the reconstructed target image during the iteration process, but also alternately estimates the sparsity description parameters of the target and noise parameters of the scene. The proposed method enhances the ability to distinguish the influence of signal and noise on the optimization function, thus improving the performance of high-resolution image reconstruction and noise suppression.

The proposed method decomposes the high-dimensional multi-parameter optimization problem into several simple sub-optimization problems, and uses the adaptive alternating estimation method to approximate the optimal solution of the overall optimization problem. When dealing with several simple sub-optimization problems, the direct matrix inversion operation is avoided as much as possible, and the matrix inversion is replaced by matrix element operation or other algorithms with lower computational complexity, thus greatly reducing the computational complexity. Specifically, when using Equations (42) and (43) to obtain the high-resolution image, this manuscript proposes using the conjugate gradient algorithm to avoid the inversion of the Hessian matrix, thereby improving computation speed. Direct matrix inversion results in a computational complexity of $O\left(N_Q{K}^3\right)$ for Equation (43), but this is reduced to $O\left(N_Q{N}_{CG}{K}^2\right)$ with the conjugate gradient method. $N_Q$ represents the number of iterations of the whole algorithm to solve the Equation (42), $N_{CG}$ is the number of iterations of each conjugate gradient solution, which is generally dozens of times, and $N_{CG}\ll K$. Therefore, it can be seen that the proposed method in this manuscript can effectively reduce the computational complexity.

The MC-AAMPE algorithm is shown in Figure 7. The noise variance ${\sigma }^2$ and the Laplace scale coefficient $\gamma$ help suppress the influence of noise on the imaging results during the high-resolution image reconstruction. The Laplace scale coefficient $\gamma$ adds a small value reduction constraint to the imaging region, and imposes a large value increase constraint on the noise region. The constraint enhances the sparsity of the reconstructed signal, better distinguishes the influence of signal and noise on solving the optimization function, and reduces the influence of noise in the process of high-resolution image reconstruction. Therefore, the proposed algorithm can more accurately distinguish the influence of signal and noise and obtain high-resolution images while reducing the influence of noise, and has good noise robustness.

3. Results and Discussion

In this section, we conduct experiments on the different situations of data to verify the effectiveness and robustness of the proposed method. For further comparison, several reference methods [4,50,54] are also employed.

3.1. Performance Analysis of URRF

In this section, we perform experiments using radar echo data based on the URRF. These experiments aim to validate the effectiveness of the proposed method. Assuming that the radar transmits an LFM signal, the parameters are listed in

Table 1. Parameters used in the simulation.

Parameter	Value	Parameter	Value
Center frequency	17 GHz	Sampling frequency	900 MHz
Wavelength	0.0176 m	Slant range of scene center	500 m
Pulse width	5 µs	Antenna aperture	0.42 m × 0.42 m
Bandwidth	3 GHz	Angle resolution	Pitch resolution: 2.4°
			Azimuth resolution: 2.4°

. In the URRF integrating phase, amplitude, and frequency modulation, there is at least one random modulation mode, and the modulation mode satisfies $\Delta {\varphi }_{mn}\left(t\right)\sim U\left(0,2\pi \right)$,$I_{mn}\left(t\right)\sim U\left(-1,1\right)$, or $\Delta f_{mn}\left(t\right)\sim U\left(-B/10,B/10\right)$.

The far-field radiation patterns for different radiation fields are shown in Figure 8. Figure 8a,b display the 2D and 3D radiation patterns for the conventional radiation field. Figure 8c–h present the 2D and 3D radiation patterns of the URRF corresponding to three different moments in the beam irradiation time, which are three randomly selected moments. It can be seen from Figure 8 that under the URRF, the far fields of each array element are no longer coherently combined in space. The far-field radiation pattern of the phased array radar exhibits random fluctuations, and the main lobe also shows random variations rather than the sinc-function-like pattern typical of conventional beams. Additionally, the URRF varies at different times. It demonstrates that the proposed URRF model can achieve a far-field radiation pattern that is incoherent in both spatial and temporal dimensions.

The condition number of the random radiation field is used to measure the correlation and randomness of the random radiation field. The condition number of the random radiation field depends on the time and space randomness of the radiation field. The larger the condition number of the random radiation field, the stronger the coherence of the radiation field and the weaker the randomness. The condition number of the random radiation field is defined as follows [39]:

$$cond\left(F\right)={\displaystyle \frac{{\delta }_1}{{\delta }_{er}}}$$

(46)

where $F$ represents the random radiation field, ${\delta }_1$ denotes the maximum singular value of $F$, and ${\delta }_{er}$ is the minimum singular value of $F$ under the condition of effective rank.

Figure 9 shows the condition number of the random radiation field under different modulations. Observing Figure 9, it can be clearly seen that under different effective ranks, the condition number of the unified random radiation field is the smallest, the correlation of the radiation field is the smallest, and the randomness is the strongest.

To further validate the effectiveness of the URRF method, imaging simulation experiments were conducted on the imaging scenario shown in Figure 10a. Figure 10b,c are the real aperture scanning imaging results and the microwave correlation imaging results based on the URRF, respectively. The real aperture scanning imaging uses the radiation pattern in Figure 8a. Observing Figure 10, it can be seen that when the target separation is 3°, which is greater than the angular resolution, real aperture imaging can effectively distinguish the two targets. However, when the target separation is 1.5°, which is less than the ideal angular resolution, real aperture imaging fails to distinguish the two targets. In contrast, the proposed method can effectively separate targets with a separation smaller than the ideal angular resolution and can effectively reconstruct the target scene information. Therefore, the following experiments are based on the echo data obtained under the URRF.

3.2. Performance Analysis of 2D Imaging Results

To verify the effectiveness of the microwave-correlated 2D imaging method based on the MC-AAMPE algorithm under different SNRs, white Gaussian noise is added to the echo to generate echo data with varying SNRs.

In the experimental process, qualitative and quantitative metrics are employed to objectively evaluate the performance of the proposed algorithm. The Mean Square Error (MSE) [55,56] and Correlation Coefficient (CORR) [27,50] are used to evaluate the performance of the microwave correlation imaging based on the MC-AAMPE algorithm under the URRF model. MSE is defined as follows:

$$\mathrm{MSE}={\displaystyle \frac{1}{N_{\theta }\times N_{\beta }}}{\displaystyle \sum _{k=1}^{N_{\theta }\times N_{\beta }}{\left({\widehat{a}}_k-a_{0\_k}\right)}^2}$$

(47)

where $a_{0\_k}$ and ${\widehat{a}}_k$ denote the k-th $\left(1\le k\le N_{\theta }\times N_{\beta }\right)$ component of the amplitude-normalized reference image and the high-resolution image reconstructed by the proposed method, respectively. The smaller MSE indicates that the calculated target scattering coefficient is closer to the real value and can reflect superior high-resolution performance.

CORR reflects the similarity between the high-resolution image reconstructed by the proposed method and the ideal reference image, which is expressed as

$$\mathrm{CORR}={\displaystyle \frac{〈\left|a_0\right|,\left|\widehat{a}\right|〉}{\sqrt{{\left|a_0\right|}^2}·\sqrt{{\left|\widehat{a}\right|}^2}}}$$

(48)

where $a_0$ and $\widehat{a}$ are the ideal reference image and the high-resolution image reconstructed by the proposed method, respectively. $〈·〉$ represents the scalar product. CORR reflects the similarity between the target scattering coefficient and the ideal target scattering coefficient. The larger the value, the better the high-resolution reconstruction performance.

Figure 11, Figure 12, Figure 13 and Figure 14 are the imaging results of various algorithms in different scenarios with SNR of 20 dB, 15 dB, and 10 dB respectively. The first row of Figure 11, Figure 12, Figure 13 and Figure 14 shows the image scenes with the following target intervals: 2°, 1°, 0.75°, and 0.5°, respectively. The imaging results obtained using the correlation method (CM) [54] are shown in the first column of the second to fourth rows in Figure 11, Figure 12, Figure 13 and Figure 14. The imaging results obtained using the spectral projected gradient for L1 minimization (SPGL1) [50] method are displayed in the second column of the second to fourth rows in Figure 11, Figure 12, Figure 13 and Figure 14. The imaging results obtained using the Bayesian compressed sensing (BCS) [4,50] method are presented in the third column of the second to fourth rows in Figure 11, Figure 12, Figure 13 and Figure 14. The imaging results obtained by the proposed method in this manuscript are shown in the fourth column of the second to fourth rows in Figure 11, Figure 12, Figure 13 and Figure 14.

It can be clearly seen that the high-resolution performance obtained by the CM is very limited, the target interval becomes smaller, and the CM will not be able to effectively distinguish the target. The SPGL1 method can effectively separate the targets in the scene with a target interval of 2°, but when the target interval becomes smaller, the method cannot effectively distinguish all the targets in the scene. With the decrease of SNR and the decrease of target interval, the BCS method will produce strong false points. The proposed method achieves high-resolution imaging results even under conditions of small target interval and low SNR, without introducing significant false points.

To validate the effectiveness and robustness of the proposed method, we conducted Monte Carlo tests to assess imaging performance under different scenarios and different SNRs. For each scenario and each SNR, we conducted 100 random experiments. As shown in Figure 15, the average of MSE of the four methods under different SNRs for various scenarios. Figure 16 displays the average of the correlation coefficient of the four methods under different SNRs for various scenarios. Observing Figure 15 and Figure 16, it can be clearly seen that the reconstruction error of the CM is very large, and it is difficult to distinguish the target with small interval; the SPGL1 method can effectively distinguish the target and obtain good imaging results when the target interval is 2°. However, when the target interval continues to become smaller, the algorithm cannot obtain good imaging results, and the high-resolution performance of the algorithm is very limited. The imaging performance of the BCS method will decrease rapidly when the target interval becomes smaller and the SNR decreases. The microwave correlation imaging method based on MC-AAMPE algorithm can obtain accurate target information and high-performance imaging results. In addition, compared with other reference methods, the proposed method has the lowest MSE. Meanwhile, the proposed method also achieves higher correlation coefficients than other reference methods. The proposed algorithm has better performance under small target intervals and low SNR.

3.3. Performance Analysis of 3D Imaging Results

In order to verify the effectiveness of the microwave-correlated 3D imaging method based on the MC-AAMPE algorithm, we carried out 3D imaging experiments with the impact of SNR. The imaging scene is shown in Figure 17, and 10 dB white Gaussian noise is added to the echo.

The real aperture scanning imaging results are shown in Figure 18. In the ideal target distribution scene, the minimum target interval in the X and Y directions is 4.5 m, and the maximum is 13.5 m. However, the real aperture scanning radar has a resolution of 20.9 m in these directions. The target interval is less than the resolution of the real aperture scanning radar, so the target cannot be effectively distinguished, resulting in poor imaging results.

The imaging results obtained using the microwave-correlated 3D imaging method based on the MC-AAMPE algorithm are shown in Figure 19. It is evident that scattering points, which could not be separated using the real aperture scanning imaging method, are effectively distinguished. The positions of the scattering points are accurately estimated, with both noise and sidelobes effectively suppressed, demonstrating the effectiveness of the proposed method in high-resolution 3D imaging.

4. Conclusions

In this manuscript, we propose a novel high-resolution and robust microwave correlation imaging framework based on the URRF and utilizing the MC-AAMPE algorithm. The URRF is designed to achieve a more diverse random radiation field distribution, enhancing the flexibility and adaptability of the random radiation field. We establish the parametric imaging model based on the URRF, which clearly reveals the relationship between the image to be reconstructed and the URRF and provides a model support for image reconstruction. High-resolution and robust imaging results are achieved by solving a multi-parameter optimization problem with the MC-AAMPE algorithm. It more precisely distinguishes between the influences of signal and noise, enhances the imaging resolution, and reduces noise interference in the optimization process. Experimental results also demonstrate that the proposed algorithm delivers high-resolution performance and good noise robustness.