About Phase: Synthetic Aperture Radar and the Phase Retrieval Problem

Synthetic aperture radar (SAR) uses relative motion to produce fine resolution images from microwave frequencies and is a useful tool for regular monitoring and mapping applications. Unfortunately, if target distance is estimated poorly, then phase errors are incurred in the data, producing a blurry reconstruction of the image. In this thesis, we introduce a new multistatic methodology for determining these phase errors from interferometry-inspired combinations of signals. To motivate this, we first consider a more general problem called phase retrieval, in which a signal is reconstructed from linear measurements whose phases are either unreliable or unavailable. We make significant theoretical progress on the phase retrieval problem, to include characterizing injectivity in the complex case, devising the theory of almost injectivity, and performing a stability analysis. We then apply certain ideas from phase retrieval to resolve phase errors in SAR. Specifically, we use bistatic techniques to measure relative phases, and then we apply a graph-theoretic phase retrieval algorithm to recover the phase errors. We conclude by devising an image reconstruction procedure based on this algorithm, and we provide simulations that demonstrate stability to noise.