We analyze synthetic aperture radar (SAR) imaging of complex ground scenes that contain both stationary and moving targets. In the usual SAR acquisition scheme, we consider ways to preprocess the data so as to separate the contributions of the moving targets from those due to stationary background reflectors. Both components of the data, that is, reflections from stationary and moving targets, are considered as signal and are needed for target imaging and tracking, respectively. The approach we use is to decompose the data matrix into a low rank and a sparse part. This decomposition enables us to capture the reflections from moving targets into the sparse part and those from stationary targets into the low rank part of the data. The computational tool for this is robust principal component analysis (RPCA) applied to the SAR data matrix. We also introduce a lossless baseband transformation of the data, which simplifies the analysis and improves the performance of the RPCA algorithm. A modified version of RPCA, the stable principal component pursuit (PCP), is robust to additive noise. Our main contribution is a theoretical analysis that determines an optimal choice of parameters for the RPCA algorithm so as to have an effective and stable separation of SAR data coming from moving and stationary targets. This analysis also gives a lower bound for detectable target velocities. We show in particular that the rank of the sparse matrix is proportional to the square root of the target's speed in the direction that connects the SAR platform trajectory to the imaging region. The robustness of the approach is illustrated with numerical simulations in the X-band SAR regime.

中文翻译：

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用于目标成像的合成孔径雷达数据的低秩稀疏分解

我们分析了包含静止和移动目标的复杂地面场景的合成孔径雷达 (SAR) 成像。在通常的 SAR 获取方案中，我们考虑预处理数据的方法，以便将移动目标的贡献与静止背景反射器的贡献分开。数据的两个组成部分，即来自静止目标和移动目标的反射，都被视为信号，分别用于目标成像和跟踪。我们使用的方法是将数据矩阵分解为低秩和稀疏部分。这种分解使我们能够捕获从移动目标到稀疏部分的反射以及从静止目标到数据的低秩部分的反射。用于此的计算工具是应用于 SAR 数据矩阵的稳健主成分分析 (RPCA)。我们还引入了数据的无损基带变换，简化了分析并提高了 RPCA 算法的性能。RPCA 的修改版本，稳定的主成分追踪 (PCP)，对加性噪声具有鲁棒性。我们的主要贡献是理论分析，确定 RPCA 算法参数的最佳选择，以便有效和稳定地分离来自移动和静止目标的 SAR 数据。该分析还给出了可检测目标速度的下限。我们特别表明，稀疏矩阵的秩与目标在将 SAR 平台轨迹连接到成像区域的方向上的速度的平方根成正比。该方法的稳健性通过 X 波段 SAR 机制中的数值模拟来说明。