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

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

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500007185 ◽

1991 ◽

Vol 34 (2) ◽

pp. 295-311 ◽

Cited By ~ 8

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.

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

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxaa024 ◽

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.

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

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2013-0061 ◽

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.

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

Mathematics ◽

10.3390/math9172087 ◽

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.

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A SIMPLE REGULARIZATION METHOD FOR SOLVING ACOUSTICAL INVERSE SCATTERING PROBLEMS

Journal of Computational Acoustics ◽

10.1142/s0218396x01000644 ◽

2001 ◽

Vol 09 (02) ◽

pp. 565-573 ◽

Cited By ~ 1

Author(s):

MICHELE PIANA

Keyword(s):

Regularization Parameter ◽

Integral Kernel ◽

Field Pattern ◽

Far Field ◽

Scattering Problems ◽

Ill Posed ◽

Ill Conditioning ◽

Far Field Pattern ◽

The Waves ◽

Morozov’S Discrepancy Principle

The problem of determining the shape of an object from far-field data is considered. We present a method, originally formulated in Ref. 1 and furtherly modified in Ref. 3, for the solution of this ill-posed nonlinear inverse problem whose main features are: • the method is exact, that is no low- or high-frequency approximation is considered; • it is not necessary to know the number of scatterers and whether or not the scatterers are penetrable by the waves; • if the medium is not penetrable, it is not necessary to know whether the obstacle is sound-hard or sound-soft; • in the case of an inhomogeneous scatterer, the method provides the shape of the inhomogeneity. The method is particularly simple since it requires only the solution of a linear Fredholm integral equation of the first kind whose integral kernel is the far-field pattern. The numerical instability due to ill-conditioning can be reduced by using regularization algorithms such as Tikhonov method where the regularization parameter is chosen by using Morozov's discrepancy principle generalized to the case where the noise affects the kernel of the integral operator.

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The inverse scattering problem for a cylinder

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500017005 ◽

1979 ◽

Vol 84 (1-2) ◽

pp. 135-143 ◽

Cited By ~ 10

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.

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Effect of transverse mode structure on the far field pattern of metal-metal terahertz quantum cascade lasers

Journal of Applied Physics ◽

10.1063/1.3043795 ◽

2008 ◽

Vol 104 (12) ◽

pp. 124513 ◽

Cited By ~ 14

Author(s):

P. Gellie ◽

W. Maineult ◽

A. Andronico ◽

G. Leo ◽

C. Sirtori ◽

...

Keyword(s):

Quantum Cascade Lasers ◽

Transverse Mode ◽

Field Pattern ◽

Far Field ◽

Mode Structure ◽

Quantum Cascade ◽

Metal Metal ◽

Far Field Pattern ◽

Cascade Lasers

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Far-field patterns of line sources mounted between four-dielectric substrates

Canadian Journal of Physics ◽

10.1139/p92-023 ◽

1992 ◽

Vol 70 (2-3) ◽

pp. 173-178 ◽

Cited By ~ 1

Author(s):

Ioanna Diamandi ◽

Costas Mertzianidis ◽

John N. Sahalos

Keyword(s):

Remote Sensing ◽

Line Source ◽

Dielectric Substrate ◽

Microstrip Antennas ◽

Field Pattern ◽

Substrate Thickness ◽

Far Field ◽

Dielectric Substrates ◽

Field Patterns ◽

Far Field Pattern

The far-field pattern characteristics of line sources lying between the slabs of a four-dielectric substrate configuration are presented. The patterns are calculated for several cases of the substrate thickness as well as for several line-source locations. The considerations that are made give useful applications in remote sensing and microstrip antennas.

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