Exploiting signal sparsity has recently received considerable attention in a variety of areas including signal and image processing, compressive sensing, machine learning and so on. Many of these applications involve optimization models that are regularized by certain sparsity-promoting metrics. Two most popular regularizers are based on the l1 norm that approximates sparsity of vectorized signals and the total variation (TV) norm that serves as a measure of gradient sparsity of an image. Nevertheless, the l1 and TV terms are merely two representative measures of sparsity. To explore the matter of sparsity further, in this thesis we investigate relaxations of the regularizers to nonconvex terms such as lp and TVp "norms" with 0 <= p < 1. The contributions of the thesis are two-fold. First, several methods to approach globally optimal solutions of related nonconvex problems for improved signal/image reconstruction quality have been proposed. Most algorithms studied in the thesis fall into the category of iterative reweighting schemes for which nonconvex problems are reduced to a series of convex sub-problems. In this regard, the second main contribution of this thesis has to do with complexity improvement of the l1/TV-regularized methodology for which accelerated algorithms are developed. Along with these investigations, new techniques are proposed to address practical implementation issues. These include the development of an lp-related solver that is easily parallelizable, and a matrix-based analysis that facilitates implementation for TV-related optimizations. Computer simulations are presented to demonstrate merits of the proposed models and algorithms as well as their applications for solving general linear inverse problems in the area of signal and image denoising, signal sparse representation, compressive sensing, and compressive imaging. / Graduate

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.