A Generalized Phase Gradient Autofocus Algorithm

The phase gradient autofocus (PGA) algorithm has seen widespread use and success within the synthetic aperture radar (SAR) imaging community. However, its use and success has largely been limited to collection geometries where either the polar format algorithm (PFA) or range migration algorithm is suitable for SAR image formation. In this work, a generalized phase gradient autofocus (GPGA) algorithm is developed which is applicable with both the PFA and backprojection algorithm (BPA), thereby directly supporting a wide range of collection geometries and SAR imaging modalities. The GPGA algorithm preserves the four crucial signal processing steps comprising the PGA algorithm, while alleviating the constraint of using a single scatterer per range cut for phase error estimation which exists with the PGA algorithm. Moreover, the GPGA algorithm, whether using the PFA or BPA, yields an approximate maxi- mum marginal likelihood estimate (MMLE) of phase errors having marginalized over unknown complex-valued reflectivities of selected scatterers. Also, in this work a new approximate MMLE, termed the max-semidefinite relaxation (Max-SDR) phase estimator, is proposed for use with the GPGA algorithm. 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., solution set to the non-deterministic polynomial- time hard (NP-hard) GPGA phase estimation problem). Moreover, in this work a specialized interior-point method is presented for more efficiently performing Max- SDR phase estimation by exploiting low-rank structure typically associated with the GPGA phase estimation problem. Lastly, simulation and experimental results produced by applying the GPGA algorithm with the PFA and BPA are presented

A sparsity-driven approach for joint SAR imaging and phase error correction

Image formation algorithms in a variety of applications have explicit or implicit dependence on a mathematical model of the observation process. Inaccuracies in the observation model may cause various degradations and artifacts in the reconstructed images. The application of interest in this paper is synthetic aperture radar (SAR) imaging, which particularly suffers from motion-induced model errors. These types of errors result in phase errors in SAR data which cause defocusing of the reconstructed images. Particularly focusing on imaging of fields that admit a sparse representation, we propose a sparsity-driven method for joint SAR imaging and phase error correction. Phase error correction is performed during the image formation process. The problem is set up as an optimization problem in a nonquadratic regularization-based framework. The method involves an iterative algorithm each iteration of which consists of consecutive steps of image formation and model error correction. Experimental results show the effectiveness of the approach for various types of phase errors, as well as the improvements it provides over existing techniques for model error compensation in SAR

A sparsity-driven approach for joint SAR imaging and phase error correction

Image formation algorithms in a variety of applications have explicit or implicit dependence on a mathematical model of the observation process. Inaccuracies in the observation model may cause various degradations and artifacts in the reconstructed images. The application of interest in this paper is synthetic aperture radar (SAR) imaging, which particularly suffers from motion-induced model errors. These types of errors result in phase errors in SAR data which cause defocusing of the reconstructed images. Particularly focusing on imaging of fields that admit a sparse representation, we propose a sparsity-driven method for joint SAR imaging and phase error correction. Phase error correction is performed during the image formation process. The problem is set up as an optimization problem in a nonquadratic regularization-based framework. The method involves an iterative algorithm each iteration of which consists of consecutive steps of image formation and model error correction. Experimental results show the effectiveness of the approach for various types of phase errors, as well as the improvements it provides over existing techniques for model error compensation in SAR

A Linear Algebraic Framework for Autofocus in Synthetic Aperture Radar

Synthetic aperture radar (SAR) provides a means of producing high-resolution microwave images using an antenna of small size. SAR images have wide applications in surveillance, remote sensing, and mapping of the surfaces of both the Earth and other planets. The defining characteristic of SAR is its coherent processing of data collected by an antenna at locations along a trajectory in space. In principle, we can produce an image of extraordinary resolution. However, imprecise position measurements associated with data collected at each location cause phase errors that, in turn, cause the reconstructed image to suffer distortion, sometimes so severe that the image is completely unrecognizable. Autofocus algorithms apply signal processing techniques to restore the focused image. This thesis focuses on the study of the SAR autofocus problem from a linear algebraic perspective. We first propose a general autofocus algorithm, called Fourier-domain Multichannel Autofocus (FMCA), that is developed based on an image support constraint. FMCA can accommodate nearly any SAR imaging scenario, whether it be wide-angle or bistatic (transmit and receive antennas at separate locations). The performance of FMCA is shown to be superior compared to current state-of-the-art autofocus techniques. Next, we recognize that at the heart of many autofocus algorithms is an optimization problem, referred to as a constant modulus quadratic program (CMQP). Currently, CMQP generally is solved by using an eigenvalue relaxation approach. We propose an alternative relaxation approach based on semidefinite programming, which has recently attracted considerable attention in other signal processing applications. Preliminary results show that the new method provides promising performance advantages at the expense of increasing computational cost. Lastly, we propose a novel autofocus algorithm based on maximum likelihood estimation, called maximum likelihood autofocus (MLA). The main advantage of MLA is its reliance on a rigorous statistical model rather than on somewhat heuristic reverse engineering arguments. We show both the analytical and experimental advantages of MLA over existing autofocus methods.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86443/1/khliu_1.pd

Autofocused compressive SAR imaging based on the alternating direction method of multipliers

We present an alternating direction method of multipliers (ADMM) based autofocused Synthetic Aperture Radar (SAR) imaging method in the presence of unknown 1-D phase errors in the phase history domain, with undersampled measurements. We formulate the problem as one of joint image formation and phase error estimation. We assume sparsity of strong scatterers in the image domain, and as such use sparsity priors for reconstruction. The algorithm uses l(p)-norm minimization (p <= 1) [8] with an improvement by integrating the phase error updates within the alternating direction method of multipliers (ADMM) steps to correct the unknown 1-D phase error. We present experimental results comparing our proposed algorithm with a coordinate descent based algorithm in terms of convergence speed and reconstruction quality

Joint sparsity-driven inversion and model error correction for SAR imaging

Image formation algorithms in a variety of applications have explicit or implicit dependence on a mathematical model of the observation process. Inaccuracies in the observation model may cause various degradations and artifacts in the reconstructed images. The application of interest in this thesis is synthetic aperture radar (SAR) imaging, which particularly suffers from motion-induced model errors. These types of errors result in phase errors in SAR data which cause defocusing of the reconstructed images. Particularly focusing on imaging of fields that admit a sparse representation, we propose a sparsity-driven method for joint SAR imaging and phase error correction. In this technique, phase error correction is performed during the image formation process. The problem is set up as an optimization problem in a nonquadratic regularization-based framework. The method involves an iterative algorithm each iteration of which consists of consecutive steps of image formation and model error correction. Experimental results show the effectiveness of the proposed method for various types of phase errors, as well as the improvements it provides over existing techniques for model error compensation in SAR

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

Recent Techniques for Regularization in Partial Differential Equations and Imaging

abstract: Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain. This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges. Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.Dissertation/ThesisDoctoral Dissertation Mathematics 201