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Super-resolution Imaging for Vortex Electromagnetic Wave Radar Based on Mode Correlation Weighting and Adaptive Regularization
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摘要: 涡旋电磁波雷达(VEWR)利用轨道角动量(OAM)模态的正交性，理论上为突破传统雷达的方位向分辨率限制提供了新的物理维度，从而也为目标微动感知与前视成像开辟了新途径。然而，实际应用中有限可用模态与复杂电磁噪声导致严重的模态混叠和分辨率退化，现有稀疏成像方法普遍存在精度-效率失衡、噪声鲁棒性不足等问题。该文提出一种融合模态相关性加权与自适应正则化(MCW-AR)的超分辨成像框架。首先构建VEWR前视成像几何与波前调制信号模型；进而设计OAM模态相关矩阵量化模态间辐射能量的非均匀分布特性，通过贝塞尔函数幅值加权调制强化主导模态的低秩约束；最终建立联合稀疏性与低秩性的复合优化模型，引入自适应权重机制动态平衡结构保持与噪声抑制，并设计基于交替方向乘子法(ADMM)与增广拉格朗日乘子法(ALM)的联合优化框架，其中核心图像更新子问题采用动量加速的二维共轭梯度最小二乘(2D-CGLS)法高效求解。数值仿真与电磁仿真实验表明：该方法在有限模态和强噪声下仍能保持目标结构完整性，计算效率与成像质量得到显著提升。Abstract: Vortex Electromagnetic Wave Radar (VEWR) leverages the orthogonality of Orbital Angular Momentum (OAM) modes, introducing a new physical dimension that theoretically overcomes the azimuth resolution limitations of conventional radar systems and enables enhanced micro-motion perception and forward-looking imaging. However, in practical engineering applications, the limited number of available OAM modes and the presence of complex electromagnetic noise often cause severe mode aliasing and resolution degradation. Existing sparse imaging methods face inherent trade-offs between accuracy and computational efficiency and exhibit limited robustness to noise. To address these issues, this paper proposes a super-resolution imaging framework that integrates Mode Correlation Weighting and Adaptive Regularization (MCW-AR). First, a forward-looking imaging geometry and a wavefront-modulated signal model for VEWR are established. Subsequently, an OAM mode correlation matrix is designed to characterize the nonuniform distribution of radiation energy among modes, where Bessel-function-modulated weights reinforce the low-rank constraints of dominant radiation components. Finally, a compound optimization model combining sparsity and low-rankness priors is developed, incorporating an adaptive weighting mechanism that dynamically balances structural preservation and noise suppression. A joint optimization framework based on the Alternating Direction Method of Multipliers (ADMM) and Augmented Lagrangian Method (ALM) algorithms is constructed, in which the core image-updating subproblem is efficiently solved using a momentum-accelerated Two-Dimensional-Conjugate Gradient Least Squares (2D-CGLS) method. Numerical simulations and electromagnetic experiments verify that the proposed method preserves target structural integrity under limited modes and strong noise, while effectively improving both computational efficiency and imaging quality.
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图 1 涡旋电磁波雷达成像几何模型
Figure 1. Imaging geometry model of vortex electromagnetic wave radar
图 2 图像矩阵的奇异值分布与能量累积特性
Figure 2. Singular value distribution and energy accumulation characteristics of image matrix
图 5 不同带宽参数$ \eta $取值对成像性能的影响
Figure 5. The impact of different bandwidth parameter $ \eta $ values on imaging performance
图 7 传统CG与所提2D-CGLS的相对残差收敛曲线对比
Figure 7. Comparison of relative residual convergence between conventional CG and the proposed 2D-CGLS
图 8 传统CG与所提2D-CGLS的成像结果对比
Figure 8. Comparison of imaging results between conventional CG and the proposed 2D-CGLS
图 11 无噪环境情况下利用不同OAM模态范围获得的VEWR成像结果
Figure 11. VEWR imaging results with different OAM mode ranges in a noise-free environment
图 12 不同信噪比条件下通过OAM模态范围$ [ - 40,40] $获得的VEWR成像结果
Figure 12. VEWR imaging results with OAM modes $ [ - 40,40] $ under different SNRs
图 14 仿真点目标的图像相关系数曲线图
Figure 14. The image correlation value curve of simulated point target
图 15 各种算法的运行时间与方位网格数的关系
Figure 15. The running time of various algorithms vs. the number of azimuth grids
图 16 贝塞尔函数调制下各方法的成像对比
Figure 16. Imaging comparison of various methods under Bessel function modulation
图 18 SNR=0 dB情况下利用不同OAM模态范围获得的电磁数据VEWR成像结果
Figure 18. VEWR imaging results of electromagnetic data with different OAM mode ranges at SNR=0 dB
图 19 电磁仿真数据的熵值对比
Figure 19. Comparison of entropy values for electromagnetic simulation data
1 所提MCW-AR算法的成像求解流程
1. The imaging procedure of the proposed MCW-AR algorithm
输入：观测矩阵$ {\boldsymbol{S}} $，字典$ {\boldsymbol{\varPhi}} $，模态相关矩阵$ {\boldsymbol{C}} $，最大迭代次数$ {K_{{\max}}} $，容差$ \varepsilon $，参数$ \lambda _1 $, $ \lambda _2 $, $ \mu_1^0 $, $ \mu_2^0$, $ \mu_3^0 $, $ \rho $, $ \eta $ 初始化： 计算$ {\boldsymbol{G}} = {{\boldsymbol{\varPhi}} ^{{\mathrm{H}}}}{\boldsymbol{S}} $ 对G进行SVD：$ {\boldsymbol{G}} = {{\boldsymbol{U}}_G}{{\boldsymbol{\varSigma}} _G}{\boldsymbol{V}}_G^{{\mathrm{H}}} $, $ {{\boldsymbol{\varSigma}} _G} = {\mathrm{diag}}({\sigma _{G,1}},{\sigma _{G,2}}, \cdots ,{\sigma _{G,{r_G}}}) $, $ {\sigma _{G,1}} \ge {\sigma _{G,2}} \ge \cdots \ge 0 $, $ {r_G} = \min ({N_{{\mathrm{r}}}},{N_{{\mathrm{a}}}}) $ 估计有效秩$ {K_{{{\mathrm{eff}}}}} $：$ {K_{{{\mathrm{eff}}}}} = \min \left\{ {q \in {\mathbb{Z}^ + }\left| {\displaystyle\sum\limits_{y = 1}^q {\sigma _y^2({\boldsymbol{X}}) \ge 0.95\displaystyle\sum\limits_{z = 1}^r {\sigma _z^2({\boldsymbol{X}})} } } \right.} \right\} $ 初始化$ {{\boldsymbol{X}}^0} $：$ {{\boldsymbol{X}}^0} = {{\boldsymbol{U}}_{G(:,1:{K_{{{\mathrm{eff}}}}})}}{{\boldsymbol{\varSigma}} _{G(1:{K_{{{\mathrm{eff}}}}},1:{K_{{{\mathrm{eff}}}}})}}{\boldsymbol{V}}_{G(:,1:{K_{{{\mathrm{eff}}}}})}^{{\mathrm{H}}} $，$ {{\boldsymbol{U}}_{G(:,1:{K_{{{\mathrm{eff}}}}})}} $表示$ {{\boldsymbol{U}}_G} $矩阵的前$ {K_{{{\mathrm{eff}}}}} $列，$ {{\boldsymbol{\varSigma}} _{G(1:{K_{{{\mathrm{eff}}}}},1:{K_{{{\mathrm{eff}}}}})}} $表示$ {{\boldsymbol{\varSigma}} _G} $左上角的
$ {K_{{{\mathrm{eff}}}}} \times {K_{{{\mathrm{eff}}}}} $子矩阵，$ {{\boldsymbol{V}}_{G(:,1:{K_{{{\mathrm{eff}}}}})}} $表示$ {{\boldsymbol{V}}_G} $矩阵的前$ {K_{{{\mathrm{eff}}}}} $列，$ {{\boldsymbol{Z}}^0} = {{\boldsymbol{C}}^{ - 1/2}}{{\boldsymbol{X}}^0} $, $ {{\boldsymbol{J}}^0} = {{\boldsymbol{X}}^0} $, $ {\boldsymbol{E}}^0 = 0 $, $ {\boldsymbol{Q}}^0_1 = 0 $, $ {\boldsymbol{Q}}_2^0 = 0 $, ${\boldsymbol{Q}}_3^0 = 0 $, $ k = 0 $迭代：While $ k \lt {K_{{\max}}} $ and $ \Vert {{\boldsymbol{X}}}^{k}-{{\boldsymbol{X}}}^{k-1}{\Vert }_{\text{F}}/\Vert {{\boldsymbol{X}}}^{k-1}{\Vert }_{\text{F}} \gt \varepsilon $ do: 1. 更新自适应权重： 计算$ {{\boldsymbol{U}}^k} = {{\boldsymbol{C}}^{ - 1/2}}{{\boldsymbol{X}}^k} + {\boldsymbol{Q}}_2^k/\mu _2^k $ 对$ {{\boldsymbol{U}}^k} $进行SVD：$ {{\boldsymbol{U}}^k} = {\boldsymbol{U\varSigma}} {{\boldsymbol{V}}^{\mathrm{H}}} $，计算奇异值权重$ {{w_y} = {\sigma _y}({{\boldsymbol{U}}^k})/\displaystyle\sum\nolimits_{z = 1}^{{\mathrm{rank}}({{\boldsymbol{U}}^k})} {{\sigma _z}} ({{\boldsymbol{U}}^k})} $，其中，${y = 1,2, \cdots ,{\mathrm{rank}}({{\boldsymbol{U}}^k})} $ 更新稀疏性参数：更新稀疏阈值$ \lambda _1^k = \lambda _1^0 \cdot \dfrac{{{{\left\| {{{\boldsymbol{J}}^k}} \right\|}_0}}}{{{N_{{\mathrm{r}}}}{N_{{\mathrm{a}}}}}} $ 2. 更新$ {\boldsymbol{Z}}^{k+1} $： $ {{\boldsymbol{Z}}^{k + 1}} = \mathcal{U} \cdot {\mathcal{D}_\alpha }({\boldsymbol{\varSigma}} ) \cdot {{\boldsymbol{\mathcal{V}}}^{\mathrm{H}}} $，其中，$ {\mathcal{D}_\alpha }({\boldsymbol{\varSigma}} ) = {\mathrm{diag}}\left( {\max\left( {{\sigma _y} - {\alpha _y},0} \right)} \right) $ 3. 更新$ {\boldsymbol{J}}^{k+1} $： ${\boldsymbol{J}}^{k+1}={\mathrm{sign}}({\boldsymbol{V}}^k)\odot\max\left(|{\boldsymbol{V}}^k|-\dfrac{ \lambda_1^k}{\mu_3},0 \right) $，其中，$ {\boldsymbol{V}}^k={\boldsymbol{X}}^k+{\boldsymbol{Q}}_3^k/\mu_3^k\ $ 4. 更新$ {\boldsymbol{X}}^{k+1} $： 构造线性算子$ \mathcal{H}({\boldsymbol{X}}) = \mu _1^k{\boldsymbol{\varPhi}} _{{\mathrm{r}}}^{{\mathrm{H}}}({{\boldsymbol{\varPhi}} _{{\mathrm{r}}}}{\boldsymbol{X\varPhi}} _{{\mathrm{a}}}^{{\mathrm{H}}}){{\boldsymbol{\varPhi}} _{{\mathrm{a}}}} + \mu _2^k{{\boldsymbol{C}}^{ - 1}}{\boldsymbol{X}} + \mu _3^k{\boldsymbol{X}} $及右端项$ {{\boldsymbol{B}}^k} $以$ {{\boldsymbol{X}}^k} $为初值，应用2D-CGLS算法求解$ \mathcal{H}({\boldsymbol{X}}) = {{\boldsymbol{B}}^k} $ 5. 更新${\boldsymbol{E}}^{k + 1} $： $ {\boldsymbol{E}}^{k + 1} = \dfrac{\mu_1}{ \lambda_2+\mu_1}{\boldsymbol{W}}^k $，其中，$ {\boldsymbol{W}}^k={\boldsymbol{S}}-{\boldsymbol{\varPhi}}{\boldsymbol{X}}^{k+1}+{\boldsymbol{Q}}_1^k/\mu_1^k $ 6. 更新乘子： $ {\boldsymbol{Q}}_1^{k + 1} = {\boldsymbol{Q}}_1^k + {\mu _1}({\boldsymbol{S}} - {\boldsymbol{\varPhi }}{{\boldsymbol{X}}^{k + 1}} - {{\boldsymbol{E}}^{k + 1}}) $, $ {\boldsymbol{Q}}_2^{k + 1} = {\boldsymbol{Q}}_2^k + {\mu _2}({{\boldsymbol{C}}^{ - 1/2}}{{\boldsymbol{X}}^{k + 1}} - {{\boldsymbol{Z}}^{k + 1}}) $, $ {\boldsymbol{Q}}_3^{k + 1} = {\boldsymbol{Q}}_3^k + {\mu _3}({{\boldsymbol{X}}^{k + 1}} - {{\boldsymbol{J}}^{k + 1}}) $ 7. 更新惩罚系数：$ \mu_w^{k+1}=\rho\mu_w^k $ for $ w = 1,2,3 $ 8. $ k = k + 1 $ End While 输出：超分辨图像$ {\boldsymbol{X}} $ 
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表 1 VEWR仿真参数
Table 1. Simulation parameters of VEWR
参数 数值 载频f0 10 GHz 带宽B 180 MHz 脉冲周期PR 50 μs 子脉冲数D 100 阵列半径a 0.5 m 目标俯仰角$ {\theta _0} $ $0.3 \text{π} $ 目标距离范围 (880, 920) m 目标方位角范围 ($0 $, $0.5\text{π} $) rad 距离单元数$ {N_{{\mathrm{r}}}} $ 120 方位网格数$ {N_{{\mathrm{a}}}} $ 120
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