Research report | Convex Relaxation Methods: A Review and Application to Sparse Radar Imaging of Rotating Targets

Abstract

In this report we explore the use of sparse signal representation methods in the radar imaging problem of rotating targets and compare their results. The ultimate goal is to estimate the spatial locations and corresponding refectivities of the scatterers constituting a target, based on a signal scattered from it. We pay particular attention to the so-called convex relaxation methods, which presumably can give the sparsest possible solutions and are computationally tractable while providing provable theoretical performance guarantees. We provide a comprehensive survey on various convex relaxation problem formulations known to date, as well as known computational algorithms for solving the optimization problems. By using extensive numerical simulations with simple rotating point targets, we show that, while many of these methods perform satisfactorily for 'on-grid' cases, performance for 'o_-grid' cases is mostly unsatisfactory, warranting much further research before they can be efficiently applied to the inverse problem of radar imaging.

Executive Summary

In radar imaging, sparsity and compressed sensing have been widely exploited in various problems such as sparse phase coherent imaging, wide-angle synthetic aperture radar (SAR) imaging for anisotropic scattering, multichannel SAR imaging and moving target indication, to name but a few. In this study, we focus our attention on the problem of sparsity-based radar imaging of a rotating target.

Signal analysis and radar imaging of fast-rotating objects are of particular research interest. The radar signals returned from such objects are commonly described under the general category of micro-Doppler signals which cannot be processed by conventional range-Doppler techniques and should be separated from other non-rotating scattering components prior to further processing. Specific attention is paid to rotating point-scatterer targets. The point- scatterer target model has been studied in the literature based on compressive sensing but with the restriction to the case of small angles of rotation and signals of `moderate' bandwidths, and much less attention has been given to narrowband signals where wider angles of rotation are required.

We focus the investigation on convex relaxation methods and explore their possible applications to the radar imaging problem of a rotating target within the point-scatterer approximation, which expands on the research theme initiated in a previous report. We present a literature survey on various convex relaxation problem formulations known to date, as well as computational algorithms for solving the optimization problems and extensive numerical simulations and results. For simplicity, the current study is mostly restricted to simulated data examples where the true scatterers are `on-grid', i.e. corresponding exactly to some of the `atoms' in the defined dictionary. The objective is to explore how some of the best known convex relaxation methods perform when applied to the problem of estimating the spatial parameters of the scatterers, and gain insight into the performance of each relevant technique. We also find that the methods, in their current forms, do not perform satisfactorily for `off-grid', which warrants further research.