Low Rank Plus Sparse Decomposition of Synthetic Aperture Radar Data for Target Imaging

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.