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Proper mathematical modeling of inverse scattering problem is of utmost importance in applications such as optical imaging and microscopy, radar, acoustic, seismic and medical imaging. However, the problem is non-linear and ill-posed due to the diffusive nature of wave propagation through the scattering medium. Born and Rytov approximation are two widely used techniques to linearize the inverse scattering problem that simpli es the mathematics and modeling of wave propagation through scattering medium in special cases. The linear inverse scattering problem is still severely ill-posed and hence, in general, the solution is not stable and unique, unless a priori knowledge about the solution is used to regularize the inverse problem. In many of the inverse scattering problem it is known a priori that the object to be imaged is sparse in spatial domain or in some transform domain. In such cases, regularization techniques that impose sparsity of the solution should be used. The focus of this dissertation is sparsity regularization of the linear inverse scattering problem. The major contributions can be divided in two segments: (i) Investigate the condition of uniqueness for the sparsity regularized linear inverse scattering problem and (ii) Propose a dimensionality reduction based optimization method for rapid and high resolution sparse image reconstruction for the inverse scattering problem in optical imaging. After studying the scattering wave measurement process and the nature of the inverse problem, the condition for obtaining a unique sparsest solution of the linear inverse scattering problem is derived. The condition is based on the degree of sparsity of the image for a xed source-detector geometry. This result will be useful to determine when one can use Born/Rytov approximation reliably for inverse scattering problem. Computer simulations and laboratory phantom experiments are performed and state-of-the-art sparse signal reconstruction scheme is used to reconstruct the solution. The results show that the quality of reconstruction is satisfactory within the derived sparsity limit. In the second part of this dissertation, a novel optimization scheme is proposed to solve a particular instance of inverse scattering problem, namely, di use optical tomography (DOT), which is a promising low cost and portable imaging modality. Conventional sparse optimization approaches to solve DOT are computationally expensive and have no selection criteria to optimize the regularization parameter. A novel algorithm, Dimensionality Reduction based Optimization for DOT (DRODOT), is proposed in this research. It reduces the dimensionality of the inverse DOT problem by reducing the number of unknowns in two steps and thereby makes the overall process fast. First, it constructs a low resolution voxel basis based on the sensing-matrix properties to nd an image support. Second, it reconstructs the sparse image inside this support. To compensate for the reduced sensitivity with increasing depth, depth compensation is incorporated in DRO-DOT. An e cient method to optimally select the regularization parameter is developed for obtaining a high-quality DOT image. DRO-DOT is also able to reconstruct high-resolution image even with a limited number of optodes in a spatially limited imaging set-up which leads towards further application in in-vivo prostate DOT imaging
Inverse Problems in Scattering and Imaging is a collection of lectures from a NATO Advanced Research Workshop that integrates the expertise of physicists and mathematicians in different areas with a common interest in inverse problems. Covering a range of subjects from new developments on the applied mathematics/mathematical physics side to many areas of application, the book achieves a blend of research, review, and tutorial contributions. It is of interest to researchers in the areas of applied mathematics and mathematical physics as well as those working in areas where inverse problems can be applied.
The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg–Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg’s S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev–Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis.
This book presents papers given at a Conference on Inverse Scattering on the Line, held in June 1990 at the University of Massachusetts, Amherst. A wide variety of topics in inverse problems were covered: inverse scattering problems on the line; inverse problems in higher dimensions; inverse conductivity problems; and numerical methods. In addition, problems from statistical physics were covered, including monodromy problems, quantum inverse scattering, and the Bethe ansatz. One of the aims of the conference was to bring together researchers in a variety of areas of inverse problems which have seen intensive activity in recent years. scattering
This book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.