Approximation and bounding of distortion errors in polar format SAR imaging for squinted geometries

Abstract

Synthetic aperture radar (SAR) imaging is a powerful tool that can be utilized where other conventional surveillance methods fail. It has a variety of applications including reconnaissance and surveillance for defense purposes, natural resource exploration, and environmental monitoring, among others. SAR systems generally create large datasets that need to be processed to form a final image. Processing this data can be computationally intensive, and applications may demand algorithms that can form images quickly. The goal and motivation of this research is to analyze algorithms that permit a large SAR dataset to be efficiently processed into a high-resolution image of a large scene. The backprojection algorithm (BPA)1 can serve as a baseline for performance relative to other SAR imaging algorithms. It results in accurately formed images for a vast variety of imaging scenarios. The tradeoff comes in its computational complexity which is O(N³) for an N × N pixel image. The polar format algorithm (PFA⁾² is a long-standing and popular alternative to the BPA. The PFA allows the use of fast Fourier Transforms (FFTs), leading to a computational complexity of O(N2 logN) for an N × N pixel image. However, the PFA relies on a far-field approximation, wherein the curved wavefront of the transmitted pulses is approximated as a planar wavefront, thereby introducing spatially variant phase errors and hence distortion and defocus in the PFA formed image. The defocus and distortion errors can be corrected, but this is a non-trivial process.3 It can be shown that first-order Taylor expansion of a differential range expression yields the assumed received signal phase used to generate images from SAR phase history data with the PFA.4 This work focuses on error terms introduced by the PFA assumption that introduce geometric distortion in the resulting image. This distortion causes a point scatterer located at a true (x, y) coordinate to appear at some (x, y) in the formed image, i.e, unwanted translation of point target locations is introduced. Complicating matters, the distortion is a function of a pixel's coordinates in the scene, thus making the distortion spatially-variant such that each pixel will be distorted differently. This is often referred to as an image warping. Previously, it has been assumed that the second-order Taylor series of the differential range defines the dominant error,2, 4, 5 due to the factorial decay of the Taylor series. This assumption is tested here by performing a Taylor expansion on a differential range error expression. Instead of assuming the second-order differential range expansion term to be the sole source of error, the true error term is used to approximate the distortion. The results of this comparison are presented. The differential range error approach will be referred to as the DRE approach and the dominant polynomial approach as the DPE. Additionally, with an accurate distortion approximation, it has been shown that the distortion can be removed in post-processing.3 With this in mind, bounds on scene size are derived limiting the visible distortion to within an arbitary number of resolution cells, both before and after the second-order distortion correction. These bounds are also verified in simulation. The paper is outlined as follows. In Section 2, we will first introduce the differential range term and demonstrate its relationship to the PFA imaging kernel and the source of the phase error terms. Next in Section 3, the distortion functions will be derived from these error terms using both the DRE and DPE approaches before and after applying the second-corrections. Then in Section 4, these results will be bounded such that the worst-case distortion at a specific pixel in the scene is within an arbitrary number of resolution cells, giving an approximated distortion-free scene size. Finally in Section 5, the results and comparison of the approaches will be presented.