A Regularised Total Least Squares Approach for 1D Inverse Scattering

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense.

Inverse scattering for three-dimensional quasi-linear biharmonic operator

We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito’s formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.

A Non-Iterative Method Combined with Neural Network Embedded in Physical Model to Solve the Imaging of Electromagnetic Inverse Scattering Problem

The main purpose of this paper is to solve the electromagnetic inverse scattering problem (ISP). Compared with conventional tomography technology, it considers the interaction between the internal structure of the scene and the electromagnetic wave in a more realistic manner. However, due to the nonlinearity of ISP, the conventional calculation scheme usually has some problems, such as the unsatisfactory imaging effect and high computational cost. To solve these problems and improve the imaging quality, this paper presents a simple method named the diagonal matrix inversion method (DMI) to estimate the distribution of scatterer contrast (DSC) and a Generative Adversarial Network (GAN) which could optimize the DSC obtained by DMI and make it closer to the real distribution of scatterer contrast. In order to make the distribution of scatterer contrast generated by GAN more accurate, the forward model is embedded in the GAN. Moreover, because of the existence of the forward model, not only is the DSC generated by the generator similar to the original distribution of the scatterer contrast in the numerical distribution, but the numerical of each point is also approximate to the original.

Target signatures for thin surfaces

We investigate an inverse scattering problem for a thin inhomogeneous scatterer in ${\mathbb R}^m$, $m=2,3$, which we model as a $m-1$ dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results.

Down-Looking Airborne Radar Imaging Performance: The Multi-Line and Multi-Frequency

The paper proposes an analytical study regarding airborne radar imaging performances and accounts for a down-looking radar system moving along parallel lines far, in terms of probing wavelength, from the investigated domain and collecting multi-frequency and multi-monostatic data. The imaging problem is formulated in a constant depth plane by exploiting the Born approximation. Hence, a linear inverse scattering problem is faced by considering both the Adjoint and the Truncated Singular Value Decomposition reconstruction schemes. Analytical and simulated results are provided to state how the achievable performances depend on the measurement configuration. These results are of practical usefulness because, in operative conditions, it is unfeasible to plan a flight grid made up by a high number of closely (in terms of probing wavelength) spaced lines. Hence, the understanding of how the availability of under-sampled data affects the radar imaging allows for a trade-off between operative data collection constrains and reliable reconstructions of the scenario under test.

One remark on the inverse scattering problem for the perturbed Stark operator on the semiaxis

In the present paper, it is indicated that the proof of the main lemma is not valid, which relates to the inverse scattering problem for the perturbed Stark operator on the semiaxis. A correct proof of the mentioned lemma is given.

Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

Numerical solution of the system of Marchenko integral equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

NDF and PSF Analysis in Inverse Source and Scattering Problems for Circumference Geometries

This paper aims at discussing the resolution achievable in the reconstruction of both circumference sources from their radiated far-field and circumference scatterers from their scattered far-field observed for the 2D scalar case. The investigation is based on an inverse problem approach, requiring the analysis of the spectral decomposition of the pertinent linear operator by the Singular Value Decomposition (SVD). The attention is focused upon the evaluation of the Number of Degrees of Freedom (NDF), connected to singular values behavior, and of the Point Spread Function (PSF), which accounts for the reconstruction of a point-like unknown and depends on both the NDF and on the singular functions. A closed-form evaluation of the PSF relevant to the inverse source problem is first provided. In addition, an approximated closed-form evaluation is introduced and compared with the exact one. This is important for the subsequent evaluation of the PSF relevant to the inverse scattering problem, which is based on a similar approximation. In this case, the approximation accuracy of the PSF is verified at least in its main lobe region by numerical simulation since it is the most critical one as far as the resolution discussion is concerned. The main result of the analysis is the space invariance of the PSF when the observation is the full angle in the far-zone region, showing that resolution remains unchanged over the entire source/investigation domain in the considered geometries. The paper also poses the problem of identifying the minimum number and the optimal directions of the impinging plane waves in the inverse scattering problem to achieve the full NDF; some numerical results about it are presented. Finally, a numerical application of the PSF concept is performed in inverse scattering, and its relevance in the presence of noisy data is outlined.