Generalized Phase Gradient Autofocus Using Semidefinite Relaxation Phase Estimation

Abstract

Recently, we developed a generalized phase gradient autofocus (GPGA) algorithm for performing synthetic aperture radar (SAR) autofocus over arbitrary flight paths, including both near-field and bistatic collection geometries. A key step of the GPGA algorithm is solving or finding an approximate solution to a non-deterministic polynomial-time hard (NP-hard) optimization problem, whose solution set consists of maximum marginal likelihood estimates (MMLEs) of the phase errors having marginalized over unknown complex-valued reflectivities of selected scatterers. In this work, a new approximate MMLE, termed the max-semidefinite relaxation (Max-SDR) phase estimator, is proposed for use with the GPGA algorithm. Leveraging recent work on SDR, the Max-SDR phase estimator provides a phase error estimate with a worst-case approximation bound compared to the solution set of MMLEs (i.e., worst-case suboptimality for a feasible point to the NP-hard GPGA phase estimation problem). Additionally, in this work a specialized interior-point method (IPM) is presented for efficiently performing Max-SDR phase estimation by exploiting low-rank structure typically associated with the GPGA phase estimation problem. The presented specialized IPM is shown to have reduced computational complexity and yield a significant run-time performance improvement over a generic IPM for large-scale problems commonly encountered with SAR imaging applications. Simulation and experimental results produced using Max-SDR phase estimation are presented.