Thorough Understanding and 3D Super-Resolution Imaging for Forward-Looking Missile-Borne SAR via a Maneuvering Trajectory

Abstract

For missile-borne platforms, traditional SAR technology consistently encounters two significant shortcomings: geometric distortion of 2D images and the inability to achieve forward-looking imaging. To address these issues, this paper explores the feasibility of using a maneuvering trajectory to enable forward-looking and three-dimensional imaging by analyzing the maneuvering characteristics of an actual missile-borne platform. Additionally, it derives the corresponding resolution characterization model, which lays a theoretical foundation for future applications. Building on this, the paper proposes a three-dimensional super-resolution imaging algorithm that combines axis rotation with compressed sensing. The axis rotation not only realizes the dimensionality reduction of data, but also can expand the observation scenario in the cross-track dimension. The proposed algorithm first focuses on the track-vertical plane to extract 2D position parameters. Then, a compressed sensing-based process is applied to extract reflection coefficients and super-resolution cross-track position parameters, thereby achieving precise 3D imaging reconstruction. Finally, numerical simulation results confirm the effectiveness and accuracy of the proposed algorithm.

1. Introduction

Thanks to the ability to provide high-resolution microwave imagery of the observed area regardless of weather conditions, synthetic aperture radar (SAR) [1,2,3,4] has become one of the most attractive radar techniques. Meanwhile, with the rapid development of electronic technology and the miniaturization of components in recent years, missile-borne SAR has become possible. However, different from traditional side-looking SAR, missile-borne SAR [5,6,7] often requires the antenna to present a large squint observation angle in order to detect longer distances and a forward target. Further, various algorithms have been proposed to solve the above requirements, such as time domain algorithms [8,9] (back-projection algorithm (BPA) and the fast factorized back-projection algorithm (FFBPA), etc.) and frequency domain algorithms [10,11,12,13,14,15,16,17,18] (Range–Doppler algorithm (RDA), Chirp Scaling algorithm, Nonlinear Chirp Scaling algorithm (NCSA) and Frequency Scaling algorithm (FSA), etc.). Nevertheless, both the geometric model and the imaging algorithm of the above missile-borne SAR entail 2D imaging detection through traditional linear trajectory, the limitations of which are twofold, as follows: the geometric distortion of 3D object on 2D images; the large squint cannot detect the front object. For solving these issues, the concept of 3D forward-looking imaging has gained the attention of scholars.

For achieving 3D forward-looking imaging detection, the primary task is to solve the issue of a missing aperture compared to traditional SAR. To date, there are two main ways to complete the aperture: Firstly, multi-channel interference [19,20,21,22,23,24,25,26], such as TomoSAR, HoloSAR and Linear array SAR, which has no requirements for the flight trajectory of the platform, but needs to be able to form a large aperture antenna. However, due to the limited volume of the missile platform, the above technical approaches are difficult to apply. Secondly, trajectory deviation [27,28,29,30,31,32,33], such as CSAR, CLSAR. Their principle is to form the third dimensional aperture through the trajectory, thereby achieving 3D object detection. From the technical point of view, it is more suitable for the missile platform, but the trajectory deviation of the above technical approach is concentrated in the plane and does not involve the research and analysis of 3D deviation similar to the missile platform. Following the second technical idea and combining it with an actual ballistic study, a novel 3D forward-looking super-resolution imaging method for missile-borne SAR via maneuvering trajectory is proposed, in which we first validate the feasibility of using maneuvering trajectory for 3D forward-looking through multiple actual ballistic data, and then the 3D resolution is derived, including an analysis of the influence of different parameters on the 3D resolution. Based on this, an effective 3D forward-looking imaging algorithm is proposed, which consist of two steps: axis rotation and super-resolution extraction through compressed sensing [34,35,36,37,38,39,40]. The operation of axis rotation is used to widen the observation scenario, thereby extracting the position parameter of the vertical and track dimension. Then, compressed sensing is applied to extract the reflection coefficient and cross-track position. Finally, the 3D image will be reconstructed effectively.

This paper is organized as follows: Section 2 assesses the feasibility of using maneuvering trajectory for the 3D forward-looking approach through multiple actual ballistics, and then analyzes its 3D resolution in detail; Section 3 describes a 3D forward-looking super-resolution imaging algorithm, combining axis rotation and compressed sensing; Section 4 further tests the proposed algorithm through several simulation experiments. Finally, summations of the whole paper are provided in Section 5.

2. The Understanding of Maneuvering Trajectory for Three-Dimensional and Forward-Looking Missile-Borne SAR

As illustrated in Figure 1, we here simulate multiple actual ballistic trajectories (note that the data source cannot be disclosed). Each trajectory exhibits a significant deviation arc caused by ballistic maneuvers, and these maneuvering paths are typically underutilized during the guidance process. Given this, we are exploring the potential value of these trajectories. From Figure 1b,c, it is evident that the maneuvering trajectories include pull deviations in the X-cross-track, Y-track, and Z-vertical dimensions. From a detection standpoint, this implies three-dimensional degrees of freedom/apertures, which could enable the possibility of three-dimensional target detection. To further clarify this, we also provide the aperture size of the maneuvering trajectories in the standard coordinate system, as shown in

Table 1. The 3D trajectory drop of multiple simulated actual ballistics.

Ballistic	Trajectory Drop-X	Trajectory Drop-Y	Trajectory Drop-Z
trajectory 1	32,910 m	54,860 m	22,370 m
trajectory 2	9060 m	79,400 m	2440 m
trajectory 3	16,490 m	93,000 m	15,710 m
trajectory 4	34,272 m	82,530 m	34,070 m

. It is clear from these data that utilizing these maneuvering trajectories for three-dimensional imaging detection is indeed feasible.

Using the parameters in Table 1, we analyze the changes in the view angle between the observation angle and the target along the maneuvering trajectory segment. Typically, an angle within a range of plus or minus 5 degrees is considered to fall within the forward-looking category. As shown in Figure 2, the overall angle change caused by the maneuvering trajectory remains within 3 degrees, with a slight reduction in the later stages due to the trajectory maneuvering back. In summary, it can be concluded that this maneuvering trajectory enables both forward-looking and 3D imaging simultaneously, offering a distinct advantage over the traditional large squint SAR mode.

After completing the feasibility analysis of three-dimensional forward-looking imaging, another crucial factor to consider is the resolution capability afforded by the maneuvering trajectory, as this directly determines its practical application potential. For simplicity, we assume the target is located at the origin, and the position coordinates of the missile platform at the $i-\mathrm{th}$ slow time are $\left(x_i,y_i,z_i\right)$. The pitch and azimuth angles from the missile platform relative to the target are denoted as $\phi \left(t_i\right)$ and $\theta \left(t_i\right)$, respectively. Thus, the vector representation of the line between the target and the platform can be expressed as

$$r\left(\phi \left({\tau }_i\right),\theta \left({\tau }_i\right)\right)=r_i\left[\begin{array}{c}\cos \left(\phi \left({\tau }_i\right)\right)\cos \left(\theta \left({\tau }_i\right)\right)\\ \cos \left(\phi \left({\tau }_i\right)\right)\sin \left(\theta \left({\tau }_i\right)\right)\\ \sin \left(\phi \left({\tau }_i\right)\right)\end{array}\right]$$

(1)

in which $r_i=\sqrt{x_i^2+y_i^2+z_i^2}$. Assuming that the radar system sends broadband linear frequency modulation signals and processes them with dechirp, then the echo signal can be written as

$$S\left({\tau }_i,{\theta }_i,{\phi }_i\right)=rect\left({\displaystyle \frac{t-2r_i/c}{T_p}}\right)\exp \left(-jk{r}_i\right)$$

(2)

where $k=4\pi \left(f_c+\gamma t\right)/c$ ($f_c$ is the center frequency, $\gamma$ is the frequency modulation rate and $c$ is the light speed). Define $\Omega \left({\tau }_i,{\phi }_{i,}{\theta }_i\right)=k{r}_i$ as the phase history of signal echo propagation; then, its spatial frequencies at the standard wavenumber spectrum geometry—k_x-k_y-k_z, as shown in Figure 3—are as follows:

$$\begin{array}{l}{k}_x={\displaystyle \frac{\partial \Omega }{\partial x}}=k\cos \left(\phi \left({\tau }_i\right)\right)\cos \left(\theta \left({\tau }_i\right)\right)\\ k_y={\displaystyle \frac{\partial \Omega }{\partial y}}=k\cos \left(\phi \left({\tau }_i\right)\right)\sin \left(\theta \left({\tau }_i\right)\right)\\ k_z={\displaystyle \frac{\partial \Omega }{\partial z}}=k\sin \left(\phi \left({\tau }_i\right)\right)\end{array}$$

(3)

In theory, its 3D resolution corresponds to the wavenumber spectrum widths projected on the k_x, k_y and k_z axes, that is,

$$\begin{array}{l}{\delta }_{kx}=c/2/\left[\max \left(k_x\right)-\min \left(k_x\right)\right]\\ {\delta }_{ky}=c/2/\left[\max \left(k_y\right)-\min \left(k_y\right)\right]\\ {\delta }_{kz}=c/2/\left[\max \left(k_z\right)-\min \left(k_z\right)\right]\end{array}$$

(4)

However, it is relatively difficult to solve Formula (4) directly due to the irregular, asymmetric and non-straight maneuvering trajectory. Therefore, we give an alternative—Kh-Kv-Ku, as shown in Figure 3, which is established based on the maneuvering trajectory. The advantage of this coordinate system design is that it allows for the detailed calculation of resolution based on the actual ballistic trajectory. Here, we assume that the azimuth angle and pitch angle at the initial time of maneuvering trajectory are $\theta \left({\tau }_0\right)$ and $\phi \left({\tau }_0\right)$. Then, the Kh-Kv-Ku coordinate system can be realized by rotating $\theta \left({\tau }_0\right)$ along the k_z axis and $\pi /2-\phi \left({\tau }_0\right)$ along the k_x axis, respectively. The red grid area denotes the 3D wavenumber spectrum, which begins at the initial time of the maneuvering trajectory and ends at the end of the maneuvering trajectory on the horizontal plane. The width is determined by the bandwidth $B=\gamma \cdot T_p$ of the transmitted signal. Based on the above coordinate system design, we define the range dimension by the line-of-sight at initial time, and the normalized Ku axis can be written as ($\mathsf{\Lambda }$ notes the normalized operation)

$$\stackrel{\mathsf{\Lambda }}{Ku}=\left[\cos \left(\phi \left({\tau }_0\right)\right)\cos \left(\theta \left({\tau }_0\right)\right),\cos \left(\phi \left({\tau }_0\right)\right)\sin \left(\theta \left({\tau }_0\right)\right),\sin \left(\phi \left({\tau }_0\right)\right)\right]$$

(5)

At this moment, the projection of the wavenumber spectrum along the Ku axis can be easily calculated, as follows:

$$B_{Ku}=\max \left\{{\displaystyle \frac{4\pi \left(f_c+B\right)}{c}}\stackrel{\mathsf{\Lambda }}{Ku}\times \left[\begin{array}{c}\cos \left(\phi \left({\tau }_k\right)\right)\cos \left(\theta \left({\tau }_k\right)\right)\\ \cos \left(\phi \left({\tau }_k\right)\right)\sin \left(\theta \left({\tau }_k\right)\right)\\ \sin \left(\phi \left({\tau }_k\right)\right)\end{array}\right]\right\}-\min \left\{{\displaystyle \frac{4\pi f_c}{c}}\stackrel{\mathsf{\Lambda }}{Ku}\times \left[\begin{array}{c}\cos \left(\phi \left({\tau }_l\right)\right)\cos \left(\theta \left({\tau }_l\right)\right)\\ \cos \left(\phi \left({\tau }_l\right)\right)\sin \left(\theta \left({\tau }_l\right)\right)\\ \sin \left(\phi \left({\tau }_l\right)\right)\end{array}\right]\right\}$$

(6)

where $k, l\left(k,l=1,2,\cdots ,N\right)$ is the $k/l-\mathrm{th}$ sampling position of the maneuvering trajectory. Form Formula (6), we can infer the value of $B_{Ku}$ in advance, while the maneuvering trajectory is determined. Then, the resolution can be written as

$${\delta }_{Ku}={\displaystyle \frac{c}{2B_{Ku}}}$$

(7)

Furthermore, Figure 4 gives the projection of the wavenumber spectrum along the Kh and Kv axis, and the definition of normalized Kh axis can be written as

$$\stackrel{\mathsf{\Lambda }}{Kh}=\left[\sin \left(\theta \left({\tau }_0\right)\right),-\cos \left(\theta \left({\tau }_0\right)\right),0\right]$$

(8)

As shown in Figure 4a, we first project the wavenumber spectrum onto the k_x-k_y plane with the range of value being [$\left[4\pi f_c\cos \left(\phi \left({\tau }_i\right)\right)/c,4\pi \left(f_c+B\right)\cos \left(\phi \left({\tau }_j\right)\right)/c\right],i,j=1,2,\cdots ,N$], and then project it vertically onto the Kh axis. So, the bandwidth on the Kh axis can be written as

$$B_{Kh}=\max \left\{{\displaystyle \frac{4\pi \left(f_c+B\right)}{c}}\cos \left(\phi \left({\tau }_k\right)\right)\sin \left(\theta \left({\tau }_k\right)-{\theta }_0\right)\right\}-\min \left\{{\displaystyle \frac{4\pi f_c}{c}}\cos \left(\phi \left({\tau }_l\right)\right)\sin \left(\theta \left({\tau }_l\right)-{\theta }_0\right)\right\}$$

(9)

where ${\theta }_0$ is the azimuth angle at the initial time. Similarly, Figure 4b gives the process of projection onto the Kv axis, and its bandwidth can be written as

$$\begin{array}{c}{B}_{Kv}=\max \left\{{\displaystyle \frac{4\pi \left(f_c+B\right)}{c}}\sqrt{1-\cos {\left(\phi \left({\tau }_k\right)\right)}^2\sin {\left(\theta \left({\tau }_k\right)-{\theta }_0\right)}^2}\cdot \sin \left({\phi }_0-{\phi }_{new\_k}\right)\right\}\\ -\min \left\{{\displaystyle \frac{4\pi f_c}{c}}\sqrt{1-\cos {\left(\phi \left({\tau }_l\right)\right)}^2\sin {\left(\theta \left({\tau }_l\right)-{\theta }_0\right)}^2}\cdot \sin \left({\phi }_0-{\phi }_{new\_l}\right)\right\}\end{array}$$

(10)

where ${\phi }_0$ is the pitch angle at the initial time and the definition of ${\phi }_{new}$ is

$${\phi }_{new}=a\sin {\displaystyle \frac{\sin \left(\phi \left(\tau \right)\right)}{\sqrt{1-\cos {\left(\phi \left(\tau \right)\right)}^2\sin \left(\theta \left(\tau \right)-{\theta }_0\right)}}}$$

(11)

Based on the above analysis, we can determine the 3D resolution of the rotated Ku-Kh-Kv coordinate system. This resolution can also be used to calculate the 3D resolution in the standard coordinate system, as follows:

$$\left[\begin{array}{c}{B}_{kx}\\ B_{ky}\\ B_{kz}\end{array}\right]=\left[\begin{array}{ccc}\cos {\phi }_0\cos {\theta }_0& \sin {\theta }_0& \sin {\phi }_0\cos {\theta }_0\\ \cos {\phi }_0\sin {\theta }_0& -\cos {\theta }_0& \sin {\phi }_0\sin {\theta }_0\\ \sin {\phi }_0& 0& -\cos {\phi }_0\end{array}\right]\left[\begin{array}{c}{B}_{Ku}\\ B_{Kh}\\ B_{Kv}\end{array}\right]$$

(12)

and

$$\begin{array}{l}{\delta }_{kx}\approx {\displaystyle \frac{c}{2B_{kx}}}={\displaystyle \frac{c}{2\left(B_{Ku}\cos {\phi }_0\cos {\theta }_0+B_{Kv}\sin {\phi }_0\cos {\theta }_0+B_{Kh}\sin {\theta }_0\right)}}\\ {\delta }_{ky}\approx {\displaystyle \frac{c}{2B_{ky}}}={\displaystyle \frac{c}{2\left(B_{Ku}\cos {\phi }_0\sin {\theta }_0+B_{Kv}\sin {\phi }_0\sin {\theta }_0-B_{Kh}\cos {\theta }_0\right)}}\\ {\delta }_{kz}\approx {\displaystyle \frac{c}{2B_{kz}}}={\displaystyle \frac{c}{2\left(B_{Ku}\sin {\phi }_0-B_{Kv}\cos {\phi }_0\right)}}\end{array}$$

(13)

To validate the accuracy of the 3D resolution derived above, we use the simulation parameters listed in Table 1. Figure 5 illustrates the 3D envelope at the scenario center, demonstrating the feasibility of 3D imaging using the maneuvering trajectory. For this simulation, we selected maneuvering trajectory 2, with a center frequency of 40 GHz and a transmitting bandwidth of 400 MHz. A comparison between the estimated and actual values of 3D resolution is presented in

Table 2. Detailed comparison of 3D resolution.

Resolution	kx/m	ky/m	kz/m
True value	16.25	0.588	0.425
Estimate value	16.23	0.574	0.440

, showing that the estimated values align closely with the true values. However, since the 3D resolution is fundamentally a multi-dimensional variable function influenced by factors such as center frequency, transmitting bandwidth, pitch angle, and azimuth angle, we further investigate the impacts of these parameters. Here, we define the ranges of center frequency, bandwidth, pitch angle, and azimuth angle as $\left[20 \mathrm{GHz}~40 \mathrm{GHz}\right]$, $\left[200 \mathrm{MHz}~800 \mathrm{MHz}\right]$, $[{50}^{\circ },{80}^{\circ }]$ and $\left[{30}^{\circ },{60}^{\circ }\right]$, respectively (note that while the angle range is broad, it will only apply to a small portion of this range in practice). Figure 6 illustrates the variation in 3D resolution. It can be observed that the 3D resolution is significantly influenced by bandwidth, increasing as both bandwidth and center frequency rise. From Figure 6a,d,g,j, we see that the resolution in the k_z dimension is less affected by azimuth angle, though its upper bound increases with a rising pitch angle. In the k_x dimension, as shown in Figure 6b,e,h,k, the upper bound of the resolution increases rapidly with the azimuth angle. Finally, Figure 6c,f,i,l indicate that the upper bound of the resolution in the k_y dimension is inversely proportional to the azimuth angle but directly proportional to the pitch angle.

In summary, we have demonstrated the feasibility of 3D forward-looking imaging using the maneuvering trajectory and conducted a detailed analysis of its resolution and influencing factors. The subsequent section will focus on the image processing methodology.

3. Three-Dimensional Super-Resolution Imaging Combining Axis Rotation and Compressed Sensing

Before proceeding with imaging processing, it is essential to first analyze the representation of the echo signal. Assuming the 3D scenario is discretized into uniform 3D grids-$\Omega \in M\times N\times L$ and a linear frequency modulation signal is transmitted, the received echo signal, denoted as Formula (2), can be rewritten as:

$$\begin{array}{ll}S\left({\tau }_i,{\theta }_i,{\phi }_i\right)& ={\displaystyle \underset{\Omega }{\int }{\delta }_{\Omega }\cdot rect\left(\frac{t-2r_i/c}{T_p}\right)\exp \left(-jk{r}_i\right)}\\ & ={\displaystyle \underset{\Omega }{\int }{\delta }_{\Omega }\cdot rect\left(\frac{t-2r_i/c}{T_p}\right)\exp \left\{-jk\left(\begin{array}{l}{x}_{\Omega }\cos \left(\phi \left({\tau }_i\right)\right)\cos \left(\theta \left({\tau }_i\right)\right)\\ +y_{\Omega }\cos \left(\phi \left({\tau }_i\right)\right)\sin \left(\theta \left({\tau }_i\right)\right)+z_{\Omega }\sin \left(\phi \left({\tau }_i\right)\right)\end{array}\right)\right\}}\end{array}$$

(14)

where ${\delta }_{\Omega }$ denotes the reflection coefficient of $\Omega$. In order to get a more intuitive understanding of Formula (14), we perform matrix processing on it, as follows:

$$\begin{array}{l}Define:\\ {\alpha }_i=\left[\begin{array}{c}\exp \left\{-jk\left(x_1\cos \left(\phi \left({\tau }_i\right)\right)\cos \left(\theta \left({\tau }_i\right)\right)\right)\right\}\\ \exp \left\{-jk\left(x_2\cos \left(\phi \left({\tau }_i\right)\right)\cos \left(\theta \left({\tau }_i\right)\right)\right)\right\}\\ ⋮\\ \exp \left\{-jk\left(x_M\cos \left(\phi \left({\tau }_i\right)\right)\cos \left(\theta \left({\tau }_i\right)\right)\right)\right\}\end{array}\right]{\beta }_i=\left[\begin{array}{c}\exp \left\{-jk\left(y_1\cos \left(\phi \left({\tau }_i\right)\right)\sin \left(\theta \left({\tau }_i\right)\right)\right)\right\}\\ \exp \left\{-jk\left(y_2\cos \left(\phi \left({\tau }_i\right)\right)\sin \left(\theta \left({\tau }_i\right)\right)\right)\right\}\\ ⋮\\ \exp \left\{-jk\left(y_N\cos \left(\phi \left({\tau }_i\right)\right)\sin \left(\theta \left({\tau }_i\right)\right)\right)\right\}\end{array}\right]\\ {\chi }_i=\left[\begin{array}{c}\exp \left\{-jk\left(z_1\sin \left(\theta \left({\tau }_i\right)\right)\right)\right\}\\ \exp \left\{-jk\left(z_2\sin \left(\theta \left({\tau }_i\right)\right)\right)\right\}\\ ⋮\\ \exp \left\{-jk\left(z_L\sin \left(\theta \left({\tau }_i\right)\right)\right)\right\}\end{array}\right] {\Omega }_{\delta }=\left[\begin{array}{l}\delta \left(x_1,y_1,z_1\right),\delta \left(x_2,y_1,z_1\right),\cdots ,\delta \left(x_M,y_1,z_1\right),\cdots ,\\ \delta \left(x_1,y_N,z_L\right),\cdots ,\delta \left(x_M,y_N,z_L\right)\end{array}\right]\\ \Downarrow \\ S=A_{\left(K\right)\times \left(M\cdot N\cdot L\right)}=\left[\begin{array}{c}{\alpha }_1\otimes {\beta }_1\otimes {\chi }_1\\ {\alpha }_2\otimes {\beta }_2\otimes {\chi }_2\\ \cdots \\ {\alpha }_K\otimes {\beta }_K\otimes {\chi }_K\end{array}\right]\cdot {\Omega }_{\delta } \end{array}$$

(15)

where $\otimes$ denotes the Kronecker product [41]. For Formula (15), there are two ways to solve it. One way is the time domain algorithm, such as BP, FBP or FFBP, which performs imaging through point by point compensation. However, it is time-consuming and inefficient. The other way involves solving a linear programming problem efficiently. However, the computational complexity of this method is primarily determined by the sampling ratio of $\Omega$, which corresponds to the measurement matrix $A$. If the size of $A$ is too large, even traditional toolboxes become inefficient. So, can we combine these two approaches to reduce the size of $A$ while also minimizing the 3D compensation required by the time domain algorithm?

To address this, we start by examining the wavenumber spectrum and its corresponding resolution. Figure 5 shows that the resolutions along k_y and k_z axes are at the sub-meter level, while that along the k_x axis is on the order of ten meters. This discrepancy is also evident from the wavenumber spectrum projection results in Figure 7. Based on these characteristics, we propose an efficient super-resolution imaging algorithm that utilizes FFBP to image the k_y-k_z plane and compress sensing to achieve a super-resolution reconstructed along the k_x axis. However, this algorithm still faces a significant issue: the imaging area range along the k_x axis is equal to its theoretical resolution (i.e., the imaging area range is constrained by the k_x axis resolution, and the scatters with different k_y/k_z positions within one k_x axis resolution unit will be focused at the same position). Consequently, once the spacing of any two scatters exceeds the k_x axis resolution, the false scatters may appear in the k_y-k_z imaging plane. So, what can be done to alleviate this issue? In other words, how does one reduce the resolution of the k_x axis?

In order to solve this, we further analyze the wavenumber spectrum projection results. Upon closer inspection of Figure 7b, it becomes apparent that the projection bandwidth along the k_x axis is not minimal in the current k_x-k_y axes definition. Therefore, we propose rotating the coordinate system to reduce the k_x axis resolution and broaden the range of the observation area. As illustrated in the inset of Figure 7b, we define new k_x′-k_y′ axes by rotating the k_y axis to align with the initial azimuth position (i.e., by rotating ${\theta }_0$ with the k_z axis as the rotation axis). The projection bandwidth of the new coordinate system can then be expressed as follows:

$$\left[\begin{array}{c}{B}_{k{x}^{\prime }}\\ B_{k{y}^{\prime }}\\ B_{k{z}^{\prime }}\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ \cos {\phi }_0& 0& \sin {\phi }_0\\ \sin {\phi }_0& 0& -\cos {\phi }_0\end{array}\right]\left[\begin{array}{c}{B}_{Ku}\\ B_{Kh}\\ B_{Kv}\end{array}\right]$$

(16)

For verification, we select two scatters separated by 18 m along the k_x axis for verification. Figure 8a,b present a comparison of the 3D resolution, where the solid line corresponds to the standard coordinate system and the dashed line corresponds to the rotated co-ordinate system. Figure 8a indicates that the resolutions along the (k_y, k_z) and (k_y′, k_z′) axes are approximately the same, which aligns with the findings from the previous theoretical analysis in Formula (16). In contrast, Figure 8b illustrates that the resolution along the k_x axis is significantly smaller than that of the k_x′ axis. Based on these conclusions, Figure 8c shows the k_y-k_z and k_y′-k_z′ reconstructed plane. The k_y-k_z reconstructed plane shows a false scatter due to the separation between the two scatters exceeding the resolution along the k_x aixs. However, the k_y′-k_z′ reconstructed plane eliminates the false scatter, indicating that the k_x′ axis resolution is greater than the separation between the two scatters. Therefore, we can conclude that the coordinate system rotation approach is indeed feasible.

After the k_y′ and k_z′ positions are extracted from the k_y′-k_z′ reconstructed plane, the next task is to get the k_x′ positions of all scatters. As the k_y′ and k_z′ positions are known, the set $\Omega$ will reduce to $M\times J$, and $J\ll N,L$ denotes the number of extracted positions. Therefore, both the measurement matrix $A$ and reflection coefficient vector ${\Omega }_{\delta }$ will achieve dimensionality reduction, and can be rewritten as

$$\begin{array}{l}{A}_{K\times \left(M\times J\times J\right)}^{\prime }={\left[{\alpha }_1\otimes \left({\beta }_1\cdot {\chi }_1\right),{\alpha }_2\otimes \left({\beta }_2\cdot {\chi }_2\right),\cdots ,{\alpha }_K\otimes \left({\beta }_K\cdot {\chi }_K\right)\right]}^T\\ {\Omega }_{\delta }^{\prime }=\left[\delta \left(x_1,y_1,z_1\right),\delta \left(x_2,y_1,z_1\right),\cdots ,\delta \left(x_M,y_1,z_1\right),\cdots ,\delta \left(x_1,y_J,z_J\right),\cdots ,\delta \left(x_M,y_J,z_J\right)\right]\\ \Downarrow \\ S=A^{\prime }\cdot {\Omega }_{\delta }^{\prime } \end{array}$$

(17)

For Formula (17), we can convert it into the following typical linear programming problem [32]:

$$minimize{‖A^{\prime }\cdot {\Omega }_{\delta }^{\prime }-S‖}_2^2+\lambda {‖{\Omega }_{\delta }^{\prime }‖}_1$$

(18)

where ${‖\cdot ‖}_1$ denotes the $L_1$ norm and $\lambda >0$ is the regularization parameter. To solve the Formula (18), high-quality implementations of the interior-point method including l1-magic [42] and PDCO [43] can be used, which utilize iterative algorithms, such as the conjugate gradients (CG) [44] or LSQR algorithm [45], to compute the search step. After applying compressed sensing, we obtain the reflection coefficients and positions of all scatters. Summarizing the entire processing procedure, the flowchart of the proposed dimension-reduction super-resolution 3D imaging algorithm is shown in Figure 9.

4. Simulation and Results

Here, we conduct the simulation verification in three steps. First, we verify the feasibility of the proposed 3D imaging algorithm using multiple scatters with both identical and varying reflection coefficients. Next, we explore the boundaries of the algorithm and examine the effects of signal-to-noise ratio and sampling rate on imaging performance. Finally, we validate the algorithm using point cloud models of actual complex target objects. It is important to note that the maneuvering trajectory II is used in all the simulations below.

4.1. 3D Imaging for Multi-Scatters with the Same and Varying Reflection Coefficients

To verify the feasibility of the proposed imaging algorithm, we not only present the final imaging results, but also demonstrate the corresponding parameter extraction process. First, consider Figure 10, where five scatters with varying reflection coefficients are depicted. Figure 10a,b show the k_y′-k_z′ positions and the k_x′ positions, respectively. It is also evident that the vertical axis in Figure 10b corresponds to the reflection coefficients. In Figure 10a, it can be observed that the number of scatters is reduced to three. This reduction occurs because the three scatters distributed along the k_x′ axis fall within a single k_x′ axis resolution unit, causing them to overlap in the k_y′-k_z′ focused plane. To separate these three scatters, we apply compressed sensing to achieve super-resolution extraction, as shown in Figure 10b, where the five scatters reappear, and their k_x′ positions and reflection coefficients are effectively extracted. Finally, the complete 3D imaging result and detailed comparisons are presented in Figure 10c and

Table 3. Detailed parameter comparison of five scatters with varying reflection coefficients.

Parameters	Reflection Coefficients					3D Positions
True	1.00	2.40	3.60	1.60	3.00	(0, 0, 0)	(0, 4, 4)	(0, 4, −4)	(4, 0, 0)	(−4, 0, 0)
Reconstructed	0.948	2.40	3.578	1.558	2.973	(−0.02, 0, 0)	(0.10, 4, 4)	(−0.13, 4, −4)	(4.07, 0, 0)	(−4.05, 0, 0)

. Similarly, Figure 11 presents the 3D imaging process for seven scatters with identical reflection coefficients, with detailed comparisons shown in

Table 4. Detailed parameter comparisons of seven scatters with same reflection coefficients.

Parameters	Reflection Coefficients							3D Positions
True	10.00	10.00	10.00	10.00	10.00	10.00	10.00	(0, 0, 0)	(0, 4, 4)	(0, 4, −4)	(4, 0, 0)	(−4, 0, 0)	(0, 4, 0)	(0, −4, 0)
Reconstructed	9.83	9.94	9.81	9.93	9.93	9.90	9.86	(−0.02, 0, 0)	(0.10, 4, 4)	(−0.13, 4, −4)	(4.07, 0, 0)	(−4.05, 0, 0)	(0.05, 4, 0)	(−0.08, −4, 0)

. Overall, both the 3D imaging and parameter comparison results demonstrate that the proposed algorithm is a feasible and high-performance 3D imaging processing method for the maneuvering trajectory.

4.2. Effects of the Signal-to-Noise Ratio and Sampling Rate on 3D Imaging Processing

Figure 12 illustrates the effects of signal-to-noise ratio (SNR) and sampling rate on 3D imaging processing, where real scatters are denoted by blue pentagrams and reconstructed scatters by red hexagons. The positions of the multi-scatters are referenced in Table 4. As shown in Figure 12a,d,g, the shapes of the multi-scatters are well reconstructed at a high sampling rate of M = 12,000, even with an SNR of −10 dB. In the middle row, where M = 8000, the performance slightly declines compared to M = 12,000, with a small offset appearing along the k_x′ axis. Meanwhile, Figure 12c,f show similar performances to M = 8000, but in Figure 12i, there is a phenomenon of missing scatters, and the shapes of the multi-scatters are not properly reconstructed. To further evaluate the resolution capability in detail, we here analyze it by reconstructing the position error of a multi-point target with two perspectives: the number of k_x/cross-track dimension grids and the signal-to-noise ratio of echoes. Here, the definition of reconstruction error is

$$P_{error}={\displaystyle \sum _{i=1}^K\frac{\left(x_i-{\tilde{x}}_i\right)}{K}}$$

(19)

where $K$ denotes the number of scatters, and $x_i$ and ${\tilde{x}}_i$ represent the true k_x position and estimated k_x position, respectively. Then, we conduct 100 Monte Carlo experiments to provide a relatively clear demonstration of resolution ability, as shown in Figure 13. Regarding doubts as to why this approach is used to evaluate resolution capability, here, we will provide an explanation. The three-dimensional imaging process proposed in this paper is essentially achieved by extracting and reconstructing the target scattering point position and reflection coefficient. Therefore, we evaluate the resolution capability of the proposed algorithm by using the positioning accuracy of scattered points in three-dimensional space.

Figure 13 shows three curves, each corresponding to a different number of sampling grids. Overall, when the SNR is negative, the positioning error across all curves is relatively high. However, at an SNR of 0 dB, there is a noticeable drop in positioning error, which continues to decrease as the SNR increases. A further comparison of the curves for different sampling grids reveals that the more sampling grids there are, the smaller the positioning error becomes. This can be explained by the fact that increasing the number of sampling grids results in a finer division of the k_x axis space, thereby increasing the likelihood of accurately sampling the true target position. However, this also leads to a corresponding increase in computational load.

4.3. 3D Imaging Verification with the Point Cloud Models of Actual Complex Tank Object

To further validate the practical applicability of the proposed algorithm, we selected a complex tank object for verification. The 3D imaging results are presented in Figure 14, with the sampling rate set at M = 50,000 and SNR = 10 dB. In Figure 14b, the 2D position parameters of the tank object in the k_y′-k_z′ focused plane are shown, where the shape of the tank is clearly discernible. The image appears clear, with sidelobes effectively eliminated, as indicated by the ample minimum value on the color bar. Figure 14c,d display the projections of the 3D imaging result in the k_x′-k_y′ and k_x′-k_z′ planes, respectively. Although a slight offset is observed, the overall shape of the tank remains identifiable. Overall, the tank object is accurately and effectively reconstructed, demonstrating the feasibility and practicality of the proposed algorithm for the 3D imaging of complex real-world objects under a maneuvering trajectory.

5. Conclusions

Building on a comprehensive understanding of actual ballistic trajectories, this paper delves into the maneuvering trajectory characteristics of missile platforms, verifying the feasibility of 3D forward-looking imaging. This research represents a significant innovation in the field of missile technology, offering a viable technical solution for achieving forward detection in three-dimensional space.

The main contributions of this study are as follows. First, it introduces and validates the use of maneuvering trajectories for 3D forward-looking imaging—a concept not explored in the existing literature. Second, the paper provides an in-depth analysis of 3D resolution and its influencing factors, offering valuable theoretical insights that can inform future ballistic design. Finally, the study proposes a novel 3D super-resolution imaging algorithm that combines axis rotation with compressed sensing. The effectiveness and accuracy of this algorithm are validated through several rigorous experiments.