A Three-Dimensional Imaging Algorithm of Downward-looking Sparse Linear Array SAR Based on Low-rank Tensor
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摘要: 为了解决3维稀疏数据处理中向量化或矩阵化带来的原始空间结构破坏与计算复杂度高的问题，该文针对下视稀疏线阵3维SAR成像几何模型和回波信号特点，构建了张量空间信号模型，提出了一种基于低秩张量补全的3维SAR稀疏成像算法。该算法首先利用回波张量的低秩性，通过张量补全重构稀疏回波中的丢失元素，再对补全后的全采样信号张量进行3维成像，从而获得高效率、低旁瓣、高分辨率3维图像。基于X波段下视稀疏线阵3维SAR点目标回波进行了3维成像仿真实验，比较了在不同信噪比和采样率条件下的成像性能，并基于实测数据进一步验证了该算法的有效性和优势。Abstract: In order to solving the problems of the inner structure damage and the high computation load brought by the vectorizing or matrixing of 3-D sparse data, the 3-D signal model is established in tensor space for downward-looking sparse linear array three-dimensional SAR. Based on this signal model, a three-dimensional SAR sparse imaging algorithm is proposed in this paper. The missing data firstly can be recovered by tensor completion on the assumption that the echo tensor is essentially low rank. Then, the resulting 3-D images can be well focused by any Fourier transform-based 3-D imaging algorithms with the recovered full-sampled data tensor. The proposed algorithm achieves not only high resolution and low-level side-lobes but also the ideal computational cost and memory consumption, which verified by several numerical simulations and multiple comparative studies on real data.
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表 1 基于低秩张量补全的下视稀疏线阵3维SAR成像算法流程
输入：稀疏回波信号张量${{{\cal S}}}$，采样集$\varOmega $，正则参数$\rho $，最大迭代次数$J$ 输出：3维图像${{{\cal I}}}$ 初始化：${ {{{\cal X}}}_\varOmega } = { {{{\cal S}}}_\Omega }$, ${ {{{\cal Y}}}_i} = 0$, ${ {{{\cal M}}}_i} = {{{\cal X}}}$, ${\rho ^0} \ge 1$ //步骤 1 张量补全稀疏回波信号 (1) for j = 0 to J do (2) for i = 1 to 3 do (3) 更新${{ {\cal M} } }_i^{j + 1} \!=\! {\rm{fol} }{ {\rm{d} }_i}\left( { {{ {\cal M} } }_{i(i)}^{j + 1} } \right) \!=\! {\rm{fol} }{ {\rm{d} }_i}\left[ { {D_{ { { {\alpha _i} } / { {\rho ^j} } } } }({ {{ {\cal X} } }_{(i)} } + 1/{\rho ^j}{ {{ {\cal Y} } }_{i(i)} })} \right]$，其中fold表示将矩阵表示为对应阶的张量，${D_{{{{\alpha _i}} / {{\rho ^j}}}}}$表示
$\tau = {{{\alpha _i}} / {{\rho ^j}}}$时的软阈值因子${D_\tau }$。(4) end for
(5) 更新${{ {\cal X} } }_\varOmega ^{j + 1} = 1/3{\left(\displaystyle\sum\limits_{i = 1}^3 { {{ {\cal M} } }_i^{j + 1} } - 1/{\rho ^j}{{ {\cal Y} } }_i^{j + 1}\right)_\varOmega }$(6) 更新拉格朗日算子${{ {\cal Y} } }_i^{j + 1} = {{ {\cal Y} } }_i^j + {\rho ^j}({ {{ {\cal X} } }^{j + 1} } - {{ {\cal M} } }_i^{j + 1})$ (7) 更新${\rho ^{j{\rm{ + }}1}} = {t}{\rho ^j},{t} \in [1.1,1.2]$ (8) end for //步骤 2 3维RD处理 (9) ${{ {\cal I} } } = {\rm{3D {\text{-}} RD(} }{{ {\cal X} } })$，其中3D-RD表示3维距离徙动校正(Range Doppler, RD)处理，见参考文献[16]。 
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表 2 仿真系统参数
参数 数值 中心频率 10 GHz 信号带宽 150 MHz 飞行高度 2000 m 飞行速度 200 m/s 脉冲重复频率 1000 Hz 线阵长度 6 m 方位向全采样数 200 切航向全采样数 120
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表 3 基于80%采样率稀疏数据的点目标的3维成像性能
成像算法 峰值旁瓣比 积分旁瓣比 80%采样 60%采样 80%采样 60%采样 方位向 算法1 –0.94 –0.57 –0.21 –0.09 算法2 –10.69 –10.90 –1.79 –1.44 算法3 –10.03 –2.77 –0.94 –0.13 切航向 算法1 –0.16 –0.16 –0.10 –0.18 算法2 –8.33 –12.03 –3.59 –3.19 算法3 –9.19 –1.44 –2.45 –0.21 高度向 算法1 –0.19 –0.11 –0.31 –0.38 算法2 –16.69 –13.41 –7.11 –5.97 算法3 –11.57 –1.33 –1.96 –0.49
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表 4 下视线阵3维SAR系统参数
参数 数值 中心载频 10 GHz 带宽 4 GHz 频率步进 1 MHz 高度 2.2 m 切航向全采样扫描点数 50 切航向全采样扫描间隔 0.02 m 方位向全采样扫描点数 160 方位向全采样扫描间隔 0.01 m
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