# Motion Detection in Synthetic Aperture Radar using Robust Principal Component Analysis: Matrix and Tensor Methods

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 a signal that is needed for target imaging and tracking, respectively. The approach we use is to decompose the data matrix into a low-rank part 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 provide 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 gives also 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.

We further consider an extension of the algorithm to a tensor case. We introduce a representation of the data as a third-order tensor formed from data coming from partially overlapping sub-apertures. We then apply a tensor robust principal component analysis (TRPCA) to the tensor data which separates them into the parts coming from the stationary and moving reflectors. Our analysis shows a distinctly improved performance of TRPCA, compared to the usual matrix case, in some cases. In particular, the tensor decomposition can identify motion features that are undetectable when using conventional motion estimation methods, including matrix RPCA. We illustrate the performance of the methods with numerical simulations in the X-band radar regime.

*Matan Leibovich is a 5th year Ph.D. student in Computational and Mathematical Engineering, at Stanford University, advised by Prof. George Papanicolaou. His research focuses on computational imaging. Prior to coming to Stanford, Matan was a researcher with the Israeli Navy, focusing on propagation models for underwater acoustic waves. *