The inverse scattering problem for a cylinder

1979 ◽  
Vol 84 (1-2) ◽  
pp. 135-143 ◽  
Author(s):  
David Colton
Keyword(s):  
Plane Wave ◽  
Moment Problem ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Wave Number ◽  
Field Pattern ◽  
Far Field ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Far Field Pattern

SynopsisLet D be a bounded simply connected domain in the plane and Ω the unit disk. Let F(Θ;k) be the far field pattern arising from the scattering of an incoming plane wave by the obstacle D and let an(k) denote the nth Fourier coefficient of F. Then if f conformally maps ℝ2\D onto ℝ2\Ω, a “moment” problem is derived which expresses an(k) in terms of f−1 for small values of the wave number k. The solution of this moment problem then gives the Laurent coefficients of f−1 and hence ∂D.

scholarly journals An injective far-field pattern operator and inverse scattering problem in a finite depth ocean

1991 ◽  
Vol 34 (2) ◽  
pp. 295-311 ◽  
Author(s):  
Yongzhi Xu
Keyword(s):  
Inverse Scattering ◽  
Acoustic Waves ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Finite Depth ◽  
Field Pattern ◽  
Far Field ◽  
Optimal Scheme ◽  
Loss Of Information ◽  
Far Field Pattern

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

Hybrid Newton-type methods for reconstructing sound-soft obstacles from a single far field

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Zewen Wang ◽  
Xiaoxia Li ◽  
Yun Xia
Keyword(s):  
Newton Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Pattern ◽  
Far Field ◽  
Newton Iterations ◽  
The One ◽  
Far Field Pattern ◽  
Quasi Newton ◽  
Multiple Obstacles

AbstractThe inverse scattering problem considered in this paper is to reconstruct multiple sound-soft obstacles from one incident wave and its far field pattern. Based on the ideas of the Kirsch–Kress method and the hybrid Newton method, three variant Newton-type methods are developed for reconstructing the shape of multiple obstacles. The proposed hybrid Newton-type methods I and II can choose auxiliary curves adaptively, and do not require them to be contained in the unknown multiple obstacles. The proposed hybrid Newton-type method III is simpler than the hybrid Newton method developed by Kress in terms of computational complexity since it adopts quasi-Newton iterations in numerical reconstructions. Results of numerical examples show that the proposed methods, especially the one with both adaptively choosing auxiliary curves and quasi-Newton iterations, are more efficient and stable for reconstructing multiple obstacles.

scholarly journals Fast Localization of Small Inhomogeneities from Far-Field Pattern Data in the Limited-Aperture Inverse Scattering Problem

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2087
Author(s):  
Won-Kwang Park
Keyword(s):  
Inverse Scattering ◽  
Random Noise ◽  
Scattering Problem ◽  
Synthetic Data ◽  
Inverse Scattering Problem ◽  
Mathematical Structure ◽  
Indicator Function ◽  
Field Pattern ◽  
Far Field ◽  
Far Field Pattern

In this study, we consider a sampling-type algorithm for the fast localization of small electromagnetic inhomogeneities from measured far-field pattern data in the limited-aperture inverse scattering problem. For this purpose, we designed an indicator function based on the structure of left- and right-singular vectors of a multistatic response matrix, the elements of which were measured far-field pattern data. We then rigorously investigated the mathematical structure of the indicator function in terms of purely dielectric permittivity and magnetic permeability contrast cases by establishing a relationship with an infinite series of Bessel functions of an integer order of the first kind and a range of incident and observation directions before exploring various intrinsic properties of the algorithm, including its feasibility and limitations. Simulation results with synthetic data corrupted by random noise are presented to support the theoretical results.

Simultaneous Scatterer Shape Estimation and Partial Aperture Far-Field Pattern Denoising

2012 ◽  
Vol 11 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Yaakov Olshansky ◽  
Eli Turkel
Keyword(s):  
Scattering Problem ◽  
Least Square ◽  
Field Pattern ◽  
Error Estimator ◽  
Far Field ◽  
The Finite Element Method ◽  
Shape Estimation ◽  
Ill Posed ◽  
Filtering Technique ◽  
Far Field Pattern

AbstractWe study the inverse problem of recovering the scatterer shape from the far-field pattern(FFP) in the presence of noise. Furthermore, only a discrete partial aperture is usually known. This problem is ill-posed and is frequently addressed using regularization. Instead, we propose to use a direct approach denoising the FFP using a filtering technique. The effectiveness of the technique is studied on a scatterer with the shape of the ellipse with a tower. The forward scattering problem is solved using the finite element method (FEM). The numerical FFP is additionally corrupted by Gaussian noise. The shape parameters are found based on a least-square error estimator. If ũ∞ is a perturbation of the FFP then we attempt to find Γ, the scatterer shape, which minimizes ∣∣ũ∞ − ũ∞∣∣ using the conjugate gradient method for the denoised FFP

scholarly journals Uniqueness in inverse electromagnetic scattering problem with phaseless far-field data at a fixed frequency

2020 ◽  
Author(s):  
Xiaoxu Xu ◽  
Bo Zhang ◽  
Haiwen Zhang
Keyword(s):  
Electromagnetic Scattering ◽  
Field Data ◽  
Scattering Problem ◽  
Plane Waves ◽  
Index Of Refraction ◽  
Field Pattern ◽  
Far Field ◽  
Fixed Frequency ◽  
Uniqueness Results ◽  
Far Field Pattern

Abstract This paper is concerned with uniqueness in inverse electromagnetic scattering with phaseless far-field pattern at a fixed frequency. In our previous work (2018,SIAM J. Appl. Math. 78, 3024–3039), by adding a known reference ball into the acoustic scattering system, it was proved that the impenetrable obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the acoustic phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency. In this paper, we extend these uniqueness results to the inverse electromagnetic scattering case. The phaseless far-field data are the modulus of the tangential component in the orientations ${\boldsymbol{e}}_\phi $ and ${\boldsymbol{e}}_\theta $, respectively, of the electric far-field pattern measured on the unit sphere and generated by infinitely many sets of superpositions of two electromagnetic plane waves with different directions and polarizations. Our proof is mainly based on Rellich’s lemma and the Stratton–Chu formula for radiating solutions to the Maxwell equations.

scholarly journals Near-field phase profile and far-field contrast modulation for micro-lens arrays at the refraction limit

2020 ◽  
Vol 2 (12) ◽  
Author(s):  
M. Yousefi ◽  
D. Nečesal ◽  
T. Scharf ◽  
M. Rossi
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Gaussian Beam ◽  
Near Field ◽  
Field Pattern ◽  
Far Field ◽  
High Contrast ◽  
Field Phase ◽  
Far Field Pattern ◽  
Micro Lens

Abstract We investigate the far-field pattern generation for a micro-lens array (MLA) illuminated under different conditions. Plane wave and Gaussian beam illumination are considered for an MLA with a small diameter of 27 microns and 30 microns period. At these dimensions, the optical effects are governed by diffraction and refraction and sometimes the regime is called the refraction limit. For Gaussian beam illumination, a high contrast dot pattern can be obtained in the far field according to the self-imaging theory for point source illumination and it is investigated in the simulation part. Also, we designed an interference microscopy setup to record both the phase and intensity in near field behind the MLA and also in the far field. The new instrument allows us to change illumination conditions from plane wave to point source. We then experimentally compare the near-field phase modulation and resulting far-field intensity for different conditions. For plane wave illumination, a high contrast pattern is observed in the far field. For the Gaussian beam illumination, the contrast of the far-field pattern depends on the distance of the source and MLA resulting in high contrast and a larger field of view only for particular distances depending on the interference of the Gaussian beam curved phase front and the MLA.

The electromagnetic inverse-scattering problem for partly coated Lipschitz domains

2004 ◽  
Vol 134 (4) ◽  
pp. 661-682 ◽  
Author(s):  
Fioralba Cakoni ◽  
David Colton ◽  
Peter Monk
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattered Wave ◽  
Field Pattern ◽  
Electromagnetic Plane Wave ◽  
Linear Sampling Method ◽  
Time Harmonic ◽  
Far Field Pattern ◽  
Electromagnetic Plane

We consider the inverse-scattering problem of determining the shape of a partly coated obstacle in R3 from a knowledge of the incident time-harmonic electromagnetic plane wave and the electric far-field pattern of the scattered wave. A justification is given of the linear sampling method in this case and numerical examples are provided showing the practicality of our method.

scholarly journals An inverse scattering problem for acoustic waves in a spherically stratified medium

1977 ◽  
Vol 20 (3) ◽  
pp. 257-263 ◽  
Author(s):  
David Colton
Keyword(s):  
Acoustic Waves ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattered Wave ◽  
Stratified Medium ◽  
Field Pattern ◽  
Soft Sphere ◽  
Far Field Pattern ◽  
Scattering Of Acoustic Waves ◽  
Local Speed

The inverse problem we will consider in this paper has its origins in the following problem connected with the scattering of acoustic waves in a nonhomogeneous medium. Let an incoming plane acoustic wave of frequency ω moving in the direction of the z axis be scattered off a “soft” sphere Ω of radius one which is surrounded by a pocket of rarefied or condensed air in which the local speed of sound is given by c(r) where r = |x| for x ∈ R3. Let us(x)eiωt be the velocity potential of the scattered wave and let r, θ, φ be spherical coordinates in R3. Then from a knowledge of the far field pattern f(θ, φ, λ) for λ = ω/c0 contained in some finite interval 0 < λ0 ≤ λ ≤ λ1,, we would like to determine the unknown function c(r).

scholarly journals Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers

2019 ◽  
Vol 9 (20) ◽  
pp. 4446 ◽  
Author(s):  
Chi Young Ahn ◽  
Taeyoung Ha ◽  
Won-Kwang Park
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Pattern ◽  
Theoretical Reason ◽  
Order Zero ◽  
Expansion Formula ◽  
Kirchhoff Migration ◽  
Simulation Results ◽  
Far Field Pattern ◽  
Better Than

In this paper, we take into account a two-dimensional inverse scattering problem for localizing small electromagnetic anomalies when they are surrounded by small, randomly distributed electromagnetic scatterers. Generally, subspace migration is considered to be an improved version of Kirchhoff migration; however, for the problem considered here, simulation results have confirmed that Kirchhoff migration is better than subspace migration, though the reasons for this have not been investigated theoretically. In order to explain theoretical reason, we explored that the imaging function of Kirchhoff migration can be expressed by the size, permittivity, permeability of anomalies and random scatterers, and the Bessel function of the first kind of order zero and one. Considered approach is based on the fact that the far-field pattern can be represented using an asymptotic expansion formula in the presence of such anomalies and random scatterers. We also present results of numerical simulations to validate the discovered imaging function structures.

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