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Examples of linear inverse problems
Published in Mario Bertero, Patrizia Boccacci, Christine De Mol, Introduction to Inverse Problems in Imaging, 2021
Mario Bertero, Patrizia Boccacci, Christine De Mol
We assume that the scatterer is contained in a bounded domain D of the 3D space. Then in general two different kinds of inverse scattering problems are considered: the inverse medium problem and the inverse obstacle problem.
Examples of linear inverse problems
Published in Bertero Mario, Boccacci Patrizia, Introduction to Inverse Problems in Imaging, 2020
Bertero Mario, Boccacci Patrizia
We assume that the scatterer is contained in a bounded domain V of the 3D space. Then two different kinds of inverse scattering problems are, in general, considered: the inverse medium problem and the inverse obstacle problem.
Uniqueness to inverse acoustic and electromagnetic scattering from locally perturbed rough surfaces
Published in Applicable Analysis, 2021
Yu Zhao, Guanghui Hu, Baoqiang Yan
Applications of these two inverse scattering problems occur in such diverse areas as medical imaging, nondestructive testing, radar, remote sensing, and geophysical exploration. As in the acoustic case, one can easily see that these inverse problems are formally determined with all , a single electric dipole located at and a fixed polarization , since the measurements depend on the same number of variables as the boundary to be reconstructed. The aim of this paper is to prove the unique determination of polyhedral-type perfect conductors by a single electric dipole with a fixed wave number, a fixed dipole point and a fixed polarization. The main tools we have used are the reflection principle for the Maxwell equations [24] and the path arguments developed in [23].
An adaptive quadrature-based factorization method for inverse acoustic scattering problems
Published in Inverse Problems in Science and Engineering, 2019
Koung Hee Leem, Jun Liu, George Pelekanos
where the abstract operator maps the boundary of the obstacle D to the corresponding far-field pattern for all pair of directions . This operator equation turns out to be highly nonlinear and severely ill-posed, which has been solved by Newton’s method [11,12], with the Fréchet derivative of being inverted using Tikhonov regularization at each iteration. It is well-known, that such nonlinear iterative methods are costly in practical computations and their effectiveness often depends on the faithfulness of the initial guess (i.e. a priori information) of the scatterer . In a typical formulation of an inverse scattering problem, however, it is assumed that we have no or very limited information about the number of components and the physical properties (i.e. sound-soft or penetrable) of the obstacle. With the advantage of requiring no a priori information of D, the above-mentioned linear sampling method [29,30] and its variants have been widely used in practice. Such linear sampling methods need to solve the following linear integral equation (defined for all )
Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data
Published in Inverse Problems in Science and Engineering, 2021
Trung Truong, Dinh-Liem Nguyen, Michael V. Klibanov
This inverse problem belongs to a wider class of coefficient inverse scattering problems which in general aim to recover information about the coefficient (e.g. its support and/or its values) from the knowledge of the scattered wave generated by a number of incident waves. Inverse scattering problems occur in many applications, including non-destructive testing, explosive detection, medical imaging, radar imaging and geophysical exploration. There is a vast literature about theoretical results and numerical solution to inverse scattering problems, see for instance [4] and references therein. Due to the interest of this paper, we discuss only some numerical methods. The conventional approach is based on the optimization-based methods, see, e.g. [9–13]. However, it is well known that these methods may suffer from multiple local minima and ravines and their convergence analysis is also unknown in many situations. An important attempt in overcoming the drawbacks of the optimization-based methods is the qualitative approach which aims to compute the geometry of the scattering object or the support of the coefficient . We refer to [4,14–19] and references therein for the development of qualitative methods in solving inverse scattering problems. Although one may be able to avoid local minima or the use of advanced a priori information of the solution, still only geometrical information of the scatterer can be reconstructed with qualitative methods. Furthermore, these methods typically require muti-static data which are sometimes not available in practical applications.