Issue 
Natl Sci Open
Volume 3, Number 5, 2024
Special Topic: Microwave Vision and SAR 3D Imaging



Article Number  20230085  
Number of page(s)  17  
Section  Information Sciences  
DOI  https://doi.org/10.1360/nso/20230085  
Published online  20 August 2024 
RESEARCH ARTICLE
RMCSTV: An effective highresolution method of nonlineofsight millimeterwave radar 3D imaging
School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
^{*} Corresponding author (email: weishunjun@uestc.edu.cn)
Received:
15
December
2023
Revised:
26
January
2024
Accepted:
28
May
2024
Nonlineofsight (NLOS) imaging is a novel radar sensing technology that enables the reconstruction of hidden targets. However, it may suffer from synthetic aperture length reduction caused by ambient occlusion. In this study, a complex total variation (CTV) regularizationbased sparse reconstruction method for NLOS threedimensional (3D) imaging by millimeterwave (mmW) radar, named RMCSTV method, is proposed to improve imaging quality and speed. In this scheme, the NLOS imaging model is first introduced, and associated geometric constraints for NLOS objects are established. Second, an effective highresolution NLOS imaging method based on the range migration (RM) kernel and complex sparse joint total variation constraint, dubbed as modified RMCSTV, is proposed for 3D highresolution imaging with edge information. The experiments with multitype NLOS targets show that the proposed RMCSTV method can provide effective and highresolution NLOS targets 3D imaging.
Key words: NLOS imaging / 3DSAR / 3D imaging / sparse reconstruction
© The Author(s) 2024. Published by Science Press and EDP Sciences.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
Nonlineofsight (NLOS) imaging is a novel method used to reconstruct hidden targets located behind corners. The NLOS imaging technique leverages the characteristics of multipath electromagnetic (EM) propagation. As EM and light waves bounce off the surface of the lineofsight (LOS) medium, they illuminate the hidden target and reflect back to the detector. Through the process, valuable information such as the range, azimuth, and geometry of the NLOS target can be detected from the multipath echo. With the capacity of allday and allweather working, the NLOS radar imaging has received widespread attention. Combined with the threedimensional (3D) synthetic aperture radar (SAR) imaging algorithm, it becomes possible to reconstruct and identify highprecision stereo images of hidden targets located behind obstacles.
The NLOS imaging technology originated from the study of optical methods. In recent years, there have been many studies on NLOS scene imaging in the optical field [15]. In ref. [6], based on the timeofflight (TOF) theory, the scholars utilize a femtosecond laser and an ultrafast photodetector array to reconstruct the 3D images of the hidden objects. In ref. [7], Rapp et al. exploited an arc scanning light source to control the illuminated portion of the hidden area and achieved 2.5 dimensional veiled scene reconstruction. For the extraction of hidden target details, Liu et al. [8] proposed an NLOS imaging algorithm based on a Bayesian framework to achieve the recovery of fine details. However, the optical NLOS sensing requires expensive optical equipment and is heavily affected by meteorology. Other NLOS imaging technologies, such as acoustic [9] and WiFi [10, 11], also have similar limitations. Compared to optical devices, radar is inexpensive and stable. Meanwhile, there are several 3D imaging achievements in the radar field. Hence, NLOS radar imaging has great potential.
Tshaped hidden scene is a research hotspot of existing radar NLOS sensing. The target and the radar are blocked by the obstacle, and the EM waves cannot illuminate the target in the LOS direction[1219]. In ref. [20], Li et al. achieved multihidden target localization by measuring and eliminating the ghosts caused by multipath. For LOS 3D radar imaging[2127], Wang et al. [28] proposed a perceptual learning framework by unfolding the fast iterative shrinkagethresholding algorithm (FISTA) and exploring the sparse prior. For LOS 3D microwave vision[2931], several SAR 3D imaging systems and methods are discussed. In refs. [32, 33], the scholars focus on tomographic synthetic aperture radar (TomoSAR) and proposed a method dubbed ATASINet, which overcomes the drawbacks of the traditional algorithm, such as weak noise resistance and high computational complexity.
For radar NLOS sensing, the main research direction is to focus on 3D positioning[19, 20, 3436]. In order to realize NLOS 3D imaging, we have carried out a series of studies[3741]. We proposed the MSBP algorithm for the looking around corner (LAC) case and verified the method by the experiment data collected by MIMO millimeter wave array antennas[42]. Although highresolution 3D reconstruction of hidden targets was achieved, the signal undersampling problem caused by complex NLOS scenes exists, which leads to the failure of proposed algorithm. Thus, a no prior information required joint compressed sensing algorithm with total variation (TV) regularization, L_{1} norm and range migration (RM) kernel called RMCSTV is proposed to achieve 3D highresolution imaging.
In this study, for the echo model of NLOS imaging is different from that of LOS region due to the multipath effect in electromagnetic propagation, the matrix model and wavenumber domain model of NLOS echo are first analyzed theoretically, and the detection and correction methods without prior information without prior information of NLOS targets are given according to NLOS geometric constraints. Additionally, aiming at the sparse characteristics of NLOS echoes, we propose a 3D sparse reconstruction method of NLOS based on L_{1} norm and complex TV regular, and an effective 3D imaging method based on RM kernel function is proposed to solve the problem of huge computation. Furthermore, we build up NLOS 3D imaging experimental system to verify the effectiveness of the proposed method. Finally, we summarize the content of this study.
The main contribution of this article are as follow: (1) a geometric constraint based NLOS target detecting method for array SAR 3D imaging is proposed, which provides a theoretical basis for 3D NLOS target detection; (2) compared with present algorithm and our previous[42], CSTV method avoids the loss of contour information, and exploits the sparse characteristic of NLOS echo to achieve high resolution imaging; (3) by introducing the RM kernel, the calculation time of RMCSTV method reduced two orders of magnitude compared with CSTV method, which provides a possibility for practical application of NLOS 3D imaging. The rest of this study is organized as follows. Section 2 introduces the problem formulation related to NLOS 3D imaging is analyzed. In section 3, the theories and algorithms flows of NLOS targets 3D effective reconstruction is proposed. The experimental scene and 3D imaging results are shown and discussed in section 4. Finally, we give a conclusion in section 5.
PROBLEM FORMULATION
In this section, we present the basic theory for mmW NLOS 3D imaging in the “T" type LAC scene. First, the geometric constraints of LAC scene is discussed for detecting the reflecting walls. Then, to obtain highresolution imaging, the CS theory based model of NLOS radar echo is proposed. Finally, the theoretical resolution for mmW NLOS imaging system is analyzed.
A typical NLOS scene is shown in Figure 1. The millimeter wave undergoes reflection by the reflecting surface to illuminate the NLOS target, and is subsequently received by the array after undergoing three scatterings. The Considering distance of representation in linear array scanning mmW radar NLOS 3D imaging, the single trip of signal R can be described as$R={R}_{\text{los}}+{R}_{\text{nlos}}\mathrm{.}$(1)
Figure 1 The layout of a typical LAC scene. The wall is used as an intermediary mmW propagating, and the echoes are received from the threebounces path. 
Specifically, the scattering process can be seen as ideallyreflection when the reflecting surface roughness height Δh satisfied$\Delta h\le \frac{\lambda}{8\mathrm{cos}{\theta}_{\text{in}}}\mathrm{,}$(2)where λ is signal wave length, θ_{in} is the angle of incidence. In this scenario, deterministic reflection path exhibit, the transmitting and reflecting distances of given array element for the designated target are equivalent.
Through the threebounce path, the single point NLOS target echo can be described as$\begin{array}{c}s\left(t\mathrm{,}{\tau}_{\text{nlos}}\right)=\sigma \ast {\rho}_{r}\left({\theta}_{i}\mathrm{,}{\theta}_{o}\right)\\ \ast \text{exp}\left\{\text{j}2{f}_{0}\left(t{\tau}_{\text{nlos}}\right)+\text{j\pi}K{\left(t{\tau}_{\text{nlos}}\right)}^{2}\right\}\mathrm{,}\end{array}$(3)where ${\tau}_{\text{nlos}}=2\left({R}_{\text{los}}+{R}_{\text{nlos}}\right)/c$, f_{0} denotes carrier frequency, σ is the scattering coefficient of target, ${\rho}_{r}\left({\theta}_{i}\mathrm{,}{\theta}_{o}\right)$ denotes bidirectional reflection distribution function (BRDF), influenced by angle of incidence θ_{i} and angle of emergence θ_{o}. K is frequency slope.
Expanding to 3D targets imaging, the echo can be described by exploiting the CS method:$\text{S}=\Psi {x}^{\prime}+n\mathrm{,}$(4)where $S\in {\u2102}^{M\times 1}$ is signal echo vector, $\Psi \in {\u2102}^{M\times {N}_{x}{N}_{y}}$ is the measure matrix. $x\text{'}\in {\u2102}^{{N}_{x}{N}_{y}\times 1}$ is the scattering coefficient of virtual NLOS targets, n is the noise.
Additionally, expressing eq. (3) in wavenumber domain for reducing the computational complexity, we can obtain the scattering coefficient x(P_{s}, r) of range plane r as follows:$x\left({P}_{s}\mathrm{,}r\right)={\displaystyle \int}IF{T}_{2D}\left(F{T}_{2D}\left(S\left({P}_{a}\mathrm{,}k\right)\right)\times {k}_{y}\mathrm{exp}\left\{\text{j}{k}_{y}r\right\}\right)\text{d}k\mathrm{,}$(5)where IFT_{2D}(·) is the twodimensional inverse Fourier transform operator, P_{s} is the position of scattering coefficient, P_{a} is the antenna phase center, $k=\left(2\left(f+{f}_{0}\right)\right)/c$, and k_{y} can be denoted as${k}_{y}=\sqrt{4{k}^{2}{k}_{x}{}^{2}{k}_{z}{}^{2}}\mathrm{,}{k}_{x}{}^{2}+{k}_{z}{}^{2}\le 4{k}^{2}\mathrm{.}$(6)
In this study, we exploit geometrical relationships in the NLOS scene to reconstruct the hidden target in a real position. As shown in eq. (4), the NLOS targets are reconstructed in a virtual position with normal LOS objects.
Assuming that the reflective surface is similar to vertical and flat, which can be seen as the mirror reflector for mmW radar wave. Based on the 2D image, the position of the reflective wall can be determined. Considering the xy axis in z=z_{0} slice, the reflective surface is described as Ax+By+C=0.
Hence, the relationship between position of virtual target ${{P}^{\prime}}_{T}$ and real target P_{T} is satisfied:$\begin{array}{c}{P}_{T}=\left[\begin{array}{ccc}\frac{{B}^{2}{A}^{2}}{{A}^{2}+{B}^{2}}& \frac{2AB}{{A}^{2}+{B}^{2}}& 0\\ \frac{2AB}{{A}^{2}+{B}^{2}}& \frac{{A}^{2}{B}^{2}}{{A}^{2}+{B}^{2}}& 0\\ 0& 0& 1\end{array}\right]{P}_{{T}^{\prime}}\left[\begin{array}{c}\frac{2BC}{{A}^{2}+{B}^{2}}\\ \frac{2AC}{{A}^{2}+{B}^{2}}\\ 0\end{array}\right]\\ ={M}_{S}{P}_{T}{}^{\prime}E\mathrm{.}\end{array}$(7)
The equalheight slice of NLOS scene is shown in Figure 2. The target T is located at ${P}_{T}=\left({x}_{0}\mathrm{,}{y}_{0}\mathrm{,}{z}_{0}\right)$, and the reflecting wall W_{1} is the intermediary for multipath echo. The radar antenna array A is located between the points A_{1} and A_{2}. As shown in Figure 2, for the impact of W_{2} , which might be buildings or cars in the urban environment, the direct path between antenna and target is blocked. The imaging space that can provide a direct echo for radar is called LOS region Ω_{los}. And the hidden space caused by obstacles is called NLOS region Ω_{nlos}. The mainly propagation of NLOS echo is constituted by three parts of bounces: $A\to {W}_{1}\to T\to {W}_{1}\to A$[42], where A is on the line segment $\overline{{A}_{1}{A}_{2}}$.
Figure 2 The Geometric relationship between radar array, reflecting wall and NLOS target. 
For the relationship shown in Figure 2, the virtual NLOS targets can be detected by the following constraints:$\{\begin{array}{c}{n}_{w}\cdot \left({P}_{T}{}^{\prime}{W}_{11}\right)\ge \mathrm{0,}\\ \Vert {P}_{WT1}{P}_{wa1}\Vert \le \Vert {P}_{wa1}{P}_{wa2}\Vert \mathrm{,}\\ \Vert {P}_{WT1}{P}_{wa2}\Vert \le \Vert {P}_{wa1}{P}_{wa2}\Vert \mathrm{,}\\ \Vert {P}_{WT2}{P}_{wa1}\Vert \le \Vert {P}_{wa1}{P}_{wa2}\Vert \mathrm{,}\\ \Vert {P}_{WT2}{P}_{wa2}\Vert \le \Vert {P}_{wa1}{P}_{wa2}\Vert \mathrm{,}\end{array}$(8)where A_{1} and A_{2} are the two endpoint coordinate vectors of the antenna array, and W_{11} is the point on the reflecting wall W_{1}. The incident point of the echo corresponding to the end of the radar array on the reflecting wall are P_{WT1} and P_{WT2}, which can be calculated by${P}_{WT1}={A}_{1}+\frac{\left({W}_{11}{A}_{1}\right)\times {W}_{1}}{\left({P}_{T}{}^{\prime}{A}_{1}\right)\times {W}_{1}}\left({P}_{T}{}^{\prime}{A}_{1}\right)\mathrm{,}$(9)${P}_{WT2}={A}_{2}+\frac{\left({W}_{11}{A}_{2}\right)\times {W}_{1}}{\left({P}_{T}{}^{\prime}{A}_{2}\right)\times {W}_{1}}\left({P}_{T}{}^{\prime}{A}_{2}\right)\mathrm{.}$(10)
METHODOLOGY
In this section, the CS theory based NLOS 3D imaging method is proposed. The 3D reconstruction is divided into three steps: highresolution imaging, reflecting surface judgment, and virtual targets adjusting. Meanwhile, the CS algorithms working in timedomain with high dimension matrixector operators employed in the optimization iterations generate largescale storage of the measurement matrix and huge computational complexity. We proposed a novel RMA kernelbased STV 3D imaging method to overcome these shortcoming. In this section, the sparse and total variation regularization operator combined (STV) highresolution 3D targets reconstructing method is introduced firstly. And then by exploiting the geometric constraints, we correct the virtual target to real position without any previous knowledge. At last, the RMA kernel is embedded into STV to improve the calculation speed.
The sparseTV algorithm
Considering the CS theory, the echo in eq. (11) can be transfered into the following function:${S}_{\text{NLOS}}=\Psi {x}_{v}+n\mathrm{,}$(11)where ${S}_{\text{NLOS}}\in {\u2102}^{MN\times 1}$ is NLOS echo vector, $\Psi \in {\u2102}^{MN\times {N}_{x}{N}_{z}}$ is the measure matrix proposed in ref. [43]. ${x}_{v}\in {\u2102}^{{N}_{x}{N}_{z}\times 1}$ is the scattering coefficient of virtual NLOS targets, n is the noise.
Generally, to obtain highresolution imaging, eq. (11) is solved by the following optimization method:$\begin{array}{c}\mathrm{min}{\alpha}_{1}{\Vert \text{CTV}\left(x\right)\Vert}_{1}+{\alpha}_{2}{\Vert x\Vert}_{1}\mathrm{,}\\ \text{s.t.}{\Vert S\Psi x\Vert}_{2}^{2}\mathrm{,}\end{array}$(12)where ${\Vert x\Vert}_{1}$ is used to characterize sparsity, ${\Vert \cdot \Vert}_{TV}$ denotes the 2D TV norm, which is using for preserving the contour information. ${\alpha}_{1}$ and ${\alpha}_{2}$ are the parameters for balancing the influence of these regular terms. Considering that the NLOS targets are isotropy, CTV operator can be described as$\text{CTV}\left(x\right)=\left({{\displaystyle \text{Re}}}_{}\left({\nabla}_{x}\right)\mathrm{,}{{\displaystyle \text{Im}}}_{}\left({\nabla}_{x}\right)\mathrm{,}{{\displaystyle \text{Re}}}_{}\left({\nabla}_{y}\right)\mathrm{,}{{\displaystyle \text{Im}}}_{}\left({\nabla}_{y}\right)\right)\mathrm{,}$(13)with$\begin{array}{l}{{\displaystyle \text{Re}}}_{}{\left\{{\nabla}_{x}\right\}}_{m\mathrm{,}n}={{\displaystyle \text{Re}}}_{}{\left\{x\right\}}_{m\mathrm{,}n}{{\displaystyle \text{Re}}}_{}{\left\{x\right\}}_{m+\mathrm{1,}n}\mathrm{,}{{\displaystyle \text{Re}}}_{}{\left\{{\nabla}_{x}\right\}}_{N\mathrm{,}n}=\mathrm{0,}\hfill \\ {{\displaystyle \text{Re}}}_{}{\left\{{\nabla}_{y}\right\}}_{m\mathrm{,}n}={{\displaystyle \text{Re}}}_{}{\left\{x\right\}}_{m\mathrm{,}n}{{\displaystyle \text{Re}}}_{}{\left\{x\right\}}_{m\mathrm{,}n+1}\mathrm{,}{{\displaystyle \text{Re}}}_{}{\left\{{\nabla}_{x}\right\}}_{m\mathrm{,}N}=\mathrm{0,}\hfill \\ {{\displaystyle \text{Im}}}_{}{\left\{{\nabla}_{x}\right\}}_{m\mathrm{,}n}={{\displaystyle \text{Im}}}_{}{\left\{x\right\}}_{m\mathrm{,}n}{{\displaystyle \text{Im}}}_{}{\left\{x\right\}}_{m+\mathrm{1,}n}\mathrm{,}{{\displaystyle \text{Im}}}_{}{\left\{{\nabla}_{x}\right\}}_{N\mathrm{,}n}=\mathrm{0,}\hfill \\ {{\displaystyle \text{Im}}}_{}{\left\{{\nabla}_{y}\right\}}_{m\mathrm{,}n}={{\displaystyle \text{Im}}}_{}{\left\{x\right\}}_{m\mathrm{,}n}{{\displaystyle \text{Im}}}_{}{\left\{x\right\}}_{m\mathrm{,}n+1}\mathrm{,}{{\displaystyle \text{Im}}}_{}{\left\{{\nabla}_{y}\right\}}_{m\mathrm{,}N}=\mathrm{0,}\hfill \end{array}$(14)where $I\in {\mathbb{R}}^{{\text{N}}_{x}\times {\text{N}}_{x}}$ is the unit matrix.
Hence, the estimation of scattering coefficient $\widehat{x}$ can be transformed into the following unconstrained optimization problem:$\widehat{x}=\underset{x}{{\displaystyle \mathrm{min}}}\text{}{\alpha}_{1}{\Vert \text{CTV}\left(x\right)\Vert}_{1}+{\alpha}_{2}{\Vert x\Vert}_{1}+\frac{1}{2}{\Vert S\Psi x\Vert}_{2}^{2}\mathrm{.}$(15)
Convert eq. (15) to the following two variable representations:$\begin{array}{c}\mathrm{min}\frac{1}{2}{\Vert \Psi xS\Vert}_{2}^{2}+{\alpha}_{1}{\Vert {z}_{1}\Vert}_{1}+{\alpha}_{2}{\Vert {z}_{2}\Vert}_{1}\mathrm{,}\\ s\mathrm{.}t\mathrm{.}{z}_{1}=\text{CTV}\left(x\right)\mathrm{,}{z}_{2}=x\mathrm{.}\\ \end{array}$(16)
By introducing Lagrangian multiplier $u={\left[{u}_{1}^{\text{T}}\mathrm{,}\text{}{u}_{2}^{\text{T}}\right]}^{\text{T}}$, the argument Lagrangian function can be described as$\begin{array}{c}L\left(x\mathrm{,}z\mathrm{,}u\right)=\frac{1}{2}{\Vert \Psi xS\Vert}_{2}^{2}+{\alpha}_{1}{\Vert {z}_{1}\Vert}_{1}+{\alpha}_{2}{\Vert {z}_{2}\Vert}_{1}\\ +{u}_{1}{}^{\text{T}}\left(\text{CTV}\left(x\right){z}_{1}\right)+{u}_{2}{}^{\text{T}}\left(x{z}_{2}\right)\\ +\frac{\lambda}{2}{\Vert \text{CTV}\left(x\right){z}_{1}\Vert}_{2}^{2}+\frac{\lambda}{2}{\Vert x{z}_{2}\Vert}_{2}^{2}\mathrm{.}\end{array}$(17)
Hence, the scattering coefficient x in eq. (17) can be solved by the following three steps[44].
Step 1: Minimized the scattering coefficient x:${x}^{k+1}=\underset{x}{{\displaystyle \mathrm{min}}}\frac{1}{2}{\Vert \Psi xS\Vert}_{2}^{2}+\frac{\lambda}{2}{\Vert \text{CTV}\left({x}^{k}\right){z}_{1}{}^{k}+{u}_{1}{}^{k}\Vert}_{2}^{2}+\frac{\lambda}{2}{\Vert {x}^{k}{z}_{2}{}^{k}+{u}_{2}{}^{k}\Vert}_{2}^{2}\mathrm{,}$(18)which can be solved by finding the gradient,$\begin{array}{l}{x}^{k+1}={\left({\Psi}^{\text{T}}\Psi +\lambda {\alpha}_{1}+\lambda {\alpha}_{2}\right)}^{1}\left\{{\Psi}^{\text{T}}s+\lambda {\alpha}_{1}\left({z}_{1}{}^{k}{u}_{1}{}^{k}\right)+\lambda {\alpha}_{2}\left({z}_{2}{}^{k}{u}_{2}{}^{k}\right)\right\}\mathrm{.}\hfill \end{array}$(19)
Step 2: The problem of optimizing z_{1} and z_{2} can be solved by divide it into the following two subproblems,${z}_{1}^{k+1}=\underset{{z}_{2}}{{\displaystyle \mathrm{min}}}{\Vert {z}_{1}^{k+1}\Vert}_{1}+\frac{\lambda}{2}{\Vert {\alpha}_{2}\text{CTV}\left({x}^{k+1}\right){z}_{1}^{k}+{u}_{1}{}^{k}\Vert}_{2}^{2}\mathrm{,}$(20)${z}_{2}^{k+1}=\underset{{z}_{1}}{{\displaystyle \mathrm{min}}}{\Vert {z}_{2}^{k+1}\Vert}_{1}+\frac{\lambda}{2}{\Vert {x}^{k+1}{z}_{2}^{k}+{u}_{2}{}^{k}\Vert}_{2}^{2}\mathrm{.}$(21)
The subproblems can be solved by exploiting the iterative shrinkage threshold (ISTA),${z}_{1}{}^{k+1}=\text{IST}\left(\text{CTV}\left({x}^{k+1}\right)+{u}_{1}{}^{k}\mathrm{,1}/{\alpha}_{1}\lambda \right)\mathrm{,}$(22)${z}_{2}{}^{k+1}=\text{IST}\left({x}^{k+1}+{u}_{2}{}^{k}\mathrm{,1}/{\alpha}_{2}\lambda \right)\mathrm{,}$(23)where $\text{IST}\left(x\mathrm{,}\rho \right)=\text{sign}\left(x\right)\cdot \text{max}\left(\leftx\right\rho \mathrm{,0}\right)$ is the iterative shrinkage threshold function[45].
Step 3: Solving the Lagrange multipliers u${u}_{i}{}^{k+1}\text{=}{u}_{i}{}^{k}+{x}_{i}{}^{k+1}{z}_{i}^{k+1}\mathrm{,}\text{\hspace{0.17em}}i=\mathrm{1,}\text{}2.$(24)
SparseTV algorithm
RMA kernel based effective sparseTV imaging
For solving the shortcoming that CS based algorithms relies heavily computing resource, the RMA kernel is proposed to replace the measurement matrix Ψ. The RM kernel solve the iteration problem in the Fourier domain by exploiting the 2D fast Fourier transformation.
Consequently, we pad and transform the NLOS echo ${S}_{f}\in {\u2102}^{{\text{N}}_{x}\times {N}_{z}}$ from $S\in {\u2102}^{\text{MN}\times 1}$ by the function $F(\cdot ):{\u2102}^{\text{M}N\times 1}\to {\u2102}^{{\text{N}}_{x}\times {N}_{z}}$ to perform the 2D Fourier transform of the corresponding size. And we suppose that ${x}^{\prime}\in {\u2102}^{{\text{N}}_{x}\times {N}_{z}}$ is the matrix representing virtual imaging.
For the echo expressed in eq. (5), eq. (4) and its inverse operation can be written with RM kernel function as follow:${S}_{f}=\text{RM}\left\{{x}^{\prime}\right\}=IF{T}_{2D}\left\{F{T}_{2D}\left\{{x}^{\prime}\right\}\odot {\Phi}_{r}\right\}\mathrm{,}$(25)${x}^{\prime}{\text{=RM}}^{\u2020}\left\{{S}_{f}\right\}=IF{T}_{2D}\left\{F{T}_{2D}\left\{{S}_{f}\right\}\odot {\Phi}_{r}{}^{\u2020}\right\}\mathrm{,}$(26)where $\odot $ denotes the Hadamard product, RM(·) and RM^{†}(·) are RM kernel and its inverse operation. Notably, eq. (25) is a classical nearfield imaging assumption, and it suffers from decline when facing the downsampling performed.
The phase propagation matrix and its inverse which shown in eqs. (25) and (26) are defined as$\begin{array}{l}{\Phi}_{r}={k}_{y}{}^{1}{\text{e}}^{\text{j}{k}_{y}r}\mathrm{,}\\ {\Phi}_{r}{}^{\u2020}={k}_{y}{\text{e}}^{\text{j}{k}_{y}r}\mathrm{,}\end{array}$(27)where ${k}_{\text{y}}\in {\u2102}^{{\text{N}}_{\text{x}}\times {\text{Nz}}_{}}$ is described as the frequency wavenumber variable matrix.
We introduce the RM kernel and its inverse operation into the optimization problem described in eq. (15). The problem can be transformed into$\widehat{x}=\underset{x}{{\displaystyle \mathrm{min}}}\text{}{\alpha}_{1}{\Vert \text{CTV}\left(x\right)\Vert}_{1}+{\alpha}_{2}{\Vert x\Vert}_{1}+\frac{1}{2}{\Vert S\text{RM}\left\{x\right\}\Vert}_{2}^{2}\mathrm{.}$(28)
Since the RM kernel and its inverse kernel satisfy$\begin{array}{c}\text{RM}\left\{{\text{RM}}^{\u2020}\left\{x\right\}\right\}=IF{T}_{2D}\left\{F{T}_{2D}\left\{{\text{RM}}^{\u2020}\left\{x\right\}\right\}\odot {\Phi}_{r}\right\}\\ =IF{T}_{2D}\left\{F{T}_{2D}\left\{IFT\left\{FT\left\{x\right\}\odot {\Phi}_{r}^{\u2020}\right\}\right\}\odot {\Phi}_{r}\right\}\mathrm{,}\end{array}$(29)the RM kernel based sparseTV algorithm optimization is concluded as$\{\begin{array}{l}{x}^{k+1}={\left({\Psi}^{\text{T}}\Psi +\lambda {\alpha}_{1}+\lambda {\alpha}_{2}\right)}^{1}\left\{{\Psi}^{\text{T}}s+\lambda {\alpha}_{1}\left({z}_{1}{}^{k}{u}_{1}{}^{k}\right)+\lambda {\alpha}_{2}\left({z}_{2}{}^{k}{u}_{2}{}^{k}\right)\right\},\hfill \\ {z}_{1}{}^{k+1}=\text{IST}\left(\text{CTV}\left({x}^{k+1}\right)+{u}_{1}{}^{k}\mathrm{,}\text{}1/{\alpha}_{1}\lambda \right)\mathrm{,}\hfill \\ {z}_{2}{}^{k+1}=\text{IST}\left({x}^{k+1}+{u}_{2}{}^{k}\mathrm{,}\text{}1/{\alpha}_{2}\lambda \right)\mathrm{,}\hfill \\ {u}_{i}{}^{k+1}\text{=}{u}_{i}{}^{k}+{x}_{i}{}^{k+1}{z}_{i}^{k+1}\mathrm{,}\text{}i=\mathrm{1,}\text{}2.\hfill \end{array}$(30)
RMCSTV Algorithm
NLOS region extraction and correction
In order to avoid the dependence on the prior information, a fast imaging of the xy plane is first performed to perceive the horizontal wall shaped object in the NLOS scene. By Hough transform and NLOS geometric constraint which given in eq. (8), reflective surfaces in 2D slices are detected. Then, the NLOS imaging region is separated and corrected by eq. (7).
EXPERIMENTS AND RESULTS
In this section, the performance of proposed imaging method on NLOS 3D sparse reconstruction is validated by NLOS highresolution imaging experiment system, and the comparison of NLOS imaging by RMCSTV, MSBP and mirrorsymmetry based traditional compressed sensing (MSCS) method has been given, qualitative results are given with different sampling rates.
Considering the complexity of electromagnetic propagation in NLOS systems, it is difficult to simulate NLOS echoes accurately in a simulated system. Therefore, we use measured data to validate the imaging system and imaging algorithms.
To verify the imaging principle and algorithm performance of NLOS radar threedimensional imaging. The imaging system consists of three parts: the data processing module, the radio frequency (RF) module, and the highprecision antenna motion rail module. The NLOS 3D SAR imaging experimental system mainly uses the “TI AWR2243" millimeterwave radar sensor and the “DCA1000" data acquisition card as the RF module, operating in a 1T1R singleinput singleoutput (SISO) mode. The equivalent 2D virtual array is achieved using the antenna motion module.
For the antenna motion module, the synthesized array element has a length of 400 mm in both the xaxis and zaxis, i.e., D_{x}=D_{z}=400 mm. To satisfy the condition of array without grating lobes, Array element spacing in the xaxis is set to d_{x}=1 mm, and Array element spacing in the zaxis is set to d_{z}=2 mm. The original data dimension size is 200×400×256.
To validate the accuracy of the RMCSTV algorithm for target localization and the quality of 3D imaging, two sets of nearfield NLOS imaging experiments were conducted using two metal tools and a metal flower stand as nonlineofsight targets. Additionally, to achieve the experimental scene under ideal conditions, four pieces of the same mmW absorbing material were used to absorb possible clutter, and aluminum foil was pasted on the reflecting surface to achieve ideal reflection of electromagnetic waves on the reflecting surface.
To evaluate the performance of the proposed algorithm, this chapter mainly uses image entropy (ENT), image contrast (IC), image sharpness (SHA), and computation time for algorithm evaluation.
Image entropy is a statistical measure that reflects the average amount of information in an image. For the same imaging scene, a lower image entropy indicates a better focus effect and higher image quality, without causing any loss of image information. Image entropy can be defined as follows:$\text{ENT}={\displaystyle \sum _{i}}{\displaystyle \sum _{j}}{\displaystyle \sum _{k}}{\left{I}_{S}\left(i\mathrm{,}j\mathrm{,}k\right)\right}^{2}\mathrm{log}{\left{I}_{S}\left(i\mathrm{,}j\mathrm{,}k\right)\right}^{2}\mathrm{,}$(31)where I_{S}(i, j, k) represents the complex value corresponding to the (i, j, k)th pixel in the 3D imaging result.
Image contrast is an estimation of the distinguishable brightness levels, and higher contrast indicates better image quality. It can be represented as follows:$IC=\sqrt{\frac{I\times J\times K}{{\Vert {I}_{S}\Vert}_{2}^{2}}{\displaystyle \sum _{i}}{\displaystyle \sum _{j}}{\displaystyle \sum _{k}}{\left{I}_{S}\left(i\mathrm{,}j\mathrm{,}k\right)\right}^{4}1}\mathrm{,}$(32)where I, J, K denote the lengthes of 3D result.
Image sharpness (SHA) is an indicator that reflects the clarity of the image plane and the sharpness of the image edges. The higher the image sharpness, the better the quality of test image.$\text{SHA}={\displaystyle \sum _{i}}{\displaystyle \sum _{j}}{\displaystyle \sum _{k}}{\left{I}_{S}\left(i\mathrm{,}j\mathrm{,}k\right)\right}^{4}\mathrm{.}$(33)
The NLOS virtual target correction
As shown in Figure 3, the virtual antenna array is represented by the blue lines, which indicate the length and position of the array in the horizontal dimension. The gap width between the occluding surface and the reflecting wall is denoted by d_{w}=300 mm, and the angle between the scanning array and the wall is denoted by θ_{s}=50°. The dimensions of the metal knives are shown in Figure 3B. The length and width of the kitchen knife are L_{1}=290 mm and W_{1}=60 mm, and the length and width of the small knife are L_{2}=226 mm and W_{2}=23 mm. The two knives are inserted into square foam blocks with side lengths of L_{3}=200 mm and L_{4}=150 mm.
Figure 3 NLOS 3D imaging experiment scene. (A) Metal knives experiment scenario; (B) optical images of metal knives. 
Figure 4 represents the rangecompressed echoes of the metal tools in the distance domain. The range echoes from the reflecting wall mainly concentrate around the 200th distance cell, while the echoes from the occluding surface are weaker due to the presence of absorbing material, and the targets echo is focused around the 100th distance cell. Around the 250th distance cell, clear pulse compression results provided by the two virtual knives can be observed, with the echo from target T_{2} being significantly stronger than the target T_{1} with a smaller reflecting cross section (RCS).
Figure 4 Echo data of knives. (A) Rangepulse compression result; (B) twodimensional imaging results of the original NLOS echo; (C) threedimensional imaging results of the original NLOS echo. 
Significantly, the detection of NLOS target exists reflecting error. By exploiting mirror correction denoted at eq. (7), the maximum projection along the xz plane of NLOS 3D result is shown in Figure 5. The sampling rate of the NLOS echo is 70%. BP, MSCS and RMCSTV methods are exploited for 3D reconstruction. As show in Figure 5C, the edge information is retained, and the reflecting error is corrected.
Figure 5 2D maximum projection results of NLOS 3D imaging. (A) MSBP method; (B) MSCS method; (C) RMCSTV method. 
The 3D imaging result
To verify the robustness of proposed method, ornaments are 3D reconstructed as another NLOS targets. The experiment scene is built as shown in Figure 6. And the size of the ornaments are shown in Figure 6B.
Figure 6 NLOS 3D imaging experiment scene. (A) Metal knives experiment scenario; (B) optical images of metal knives. 
For the NLOS experiments, we compare the performance of MSBP, RMA, CSTV, and RMCSTV (Figures 7 and 8). As reported in Tables 1 and 2, CSTV method has the longest operation time because of relying on largescale matrix operations and multiple iterations, which is far from meeting the realtime requirements in radar imaging, but it has high image contrast at each sampling rate. The RMA method has the lowest operation time, but its imaging quality is poor, and its imaging results deteriorate sharply as the sampling rate decreases. The various imaging metrics of the MSBP method are between the RMA method and the compressed sensing method.
Figure 7 NLOS knives targets 3D imaging result. 
Figure 8 NLOS ornament targets 3D imaging result. 
Quantitative comparisons and speed evaluations of knives targets
Quantitative comparisons and speed evaluations of ornament targets
Even in full sampling conditions, the BP and RMA imaging results are significantly affected by side lobes and clutter interference, with the edges of the objects being heavily polluted by noise. In contrast, the proposed algorithm effectively suppresses image side lobes and filters out a large amount of background clutter, resulting in clearer images and significantly improving imaging quality. Furthermore, the proposed algorithm successfully recovers partial detailed contours of the target even at low sampling rates, further proving the effectiveness and high performance of the algorithm.
Compared with the other three methods, the RMCSTV method can achieve lower image entropy and higher image sharpness at the sampling rates of 70%, 50%, and 30%, which indicates that the proposed method has better image focusing effect. At the same time, the imaging time required by the proposed method is reduced by at least two orders of magnitude.
CONCLUSION AND DISCUSSION
In this article, we proposed an accelerated mmW NLOS 3D sparse reconstruction method, i.e., RMCSTV. The algorithm has the capability of not only reconstructing the NLOS targets with high resolution from multipath echoes in the LAC situation but also preserving the edge information. Using only a single radar array and single imaging, reflective surfaces are extracted from the LOS and NLOS mixed echoes and used to correct the position of the NLOS target. Considering the TV and sparse constraints of the target, the highresolution problem of NLOS radar imaging is transformed into the optimization problem with multiconstraint. By solving the multiconstraint problem, defocuses of NLOS images are suppressed, and the contour information is preserved. Also, an RM kernel based accelerated algorithm is proposed to improve computational efficiency. With the RMCSTV method, a wellfocus and edge sharpness 3D image can be obtained by the ADMM method to solve the optimization problem with sparse and TV constraints. Extensive NLOS 3D mmW imaging experiments, including multitarget with 3D characteristics, the weakly sparse surface target and weapons, validate the superiority of the RMCSTV algorithm over the existing NLOS imaging methods in terms of both quality and efficiency. Except visually verifying that the NLOS imaging results of the proposed method have a precise contour, the indexes of ENT, Contrast, SHA and TBR are used to quantitatively ascertain the performance of our approach. Meanwhile, the acceleration method introduced RM kernel is two orders of magnitude times faster than the original method.
Funding
This work was supported by the National Natural Science Foundation of China (62271108).
Author contributions
X.L. and S.W. designed the research and analyzed the data. S.W. supervised the project. W.P., L.K. and X.Z. conducted the system demonstration and the experimental scheme designing. X.L. and X.C. collected the NLOS 3D radar data. Y.W. and X.C preprocessed the data. Y.W. and S.G. analyzed the results. X.L. and S.W. cowrote the manuscript. All authors contributed to the discussions.
Conflict of interest
The authors declare no conflict of interest.
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All Tables
All Figures
Figure 1 The layout of a typical LAC scene. The wall is used as an intermediary mmW propagating, and the echoes are received from the threebounces path. 

In the text 
Figure 2 The Geometric relationship between radar array, reflecting wall and NLOS target. 

In the text 
Figure 3 NLOS 3D imaging experiment scene. (A) Metal knives experiment scenario; (B) optical images of metal knives. 

In the text 
Figure 4 Echo data of knives. (A) Rangepulse compression result; (B) twodimensional imaging results of the original NLOS echo; (C) threedimensional imaging results of the original NLOS echo. 

In the text 
Figure 5 2D maximum projection results of NLOS 3D imaging. (A) MSBP method; (B) MSCS method; (C) RMCSTV method. 

In the text 
Figure 6 NLOS 3D imaging experiment scene. (A) Metal knives experiment scenario; (B) optical images of metal knives. 

In the text 
Figure 7 NLOS knives targets 3D imaging result. 

In the text 
Figure 8 NLOS ornament targets 3D imaging result. 

In the text 
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