The no-response approach and its relation to non-iterative methods for the inverse scattering

Honda, Naofumi; Nakamura, Gen; Potthast, Roland; Sini, Mourad

doi:10.1007/s10231-006-0030-1

Cited by 20 publications

(16 citation statements)

References 17 publications

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“…If a ∈ D, then the limit in (3.18) is conjectured to be ∞, see [11]. However up to now, we do not have the full answer.…”

Section: Presentation Of the Resultsmentioning

confidence: 82%

“…The functionẼ 0, 1)). By Theorem 7.1 of [11], there is a C 2 strict deformation family {D l(s) a } of a and ∂Ω. That is, each ∂D l(s) a is C 2 diffeomorphic to the unit circle and {D l(s) a } satisfies the following properties:…”

Section: Presentation Of the Resultsmentioning

confidence: 99%

“…However, we can also state the natural version of the probe method directly from the far field data, without reducing the far field to the near field, see [11]. For a review of these methods, the readers are referred to [22,23] and for some relations between them to [11,20].…”

Section: Introduction and Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Reconstruction of the Shape and Surface Impedance from Acoustic Scattering Data for an Arbitrary Cylinder

Liu¹,

Nakamura²,

Sini³

2007

SIAM J. Appl. Math.

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Abstract. The inverse scattering for an obstacle D ⊂ R 2 with mixed boundary condition can be considered as a prototype for radar detection of complex obstacles with coated and non-coated parts of the boundary. We construct some indicator functions for this inverse problem using the far-field pattern directly, without the necessity to transform the far-field to the near field. Based on the careful singularity analysis, these indicator functions enable us to reconstruct the shape of the obstacle and distinguish the coated from the non-coated part of the boundary. Moreover, an explicit representation formula for the surface impedance in the coated part of the boundary is also given. Our reconstruction scheme reveals that the coated part of the obstacle is less visible than the non-coated one, which corresponds to the physical fact that the coated boundary absorbs some part of the scattered wave. Numerics are presented for the reconstruction formulas, which show that both the boundary shape and the surface impedance in the coated part of boundary can be reconstructed accurately. The theoretical reconstruction techniques proposed in this work can be applied in the practical 3-dimensional electromagnetic inverse scattering problems with hopeful numerical performances, which are of great importance in the design of non-detectable obstacles.

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“…If a ∈ D, then the limit in (3.18) is conjectured to be ∞, see [11]. However up to now, we do not have the full answer.…”

Section: Presentation Of the Resultsmentioning

confidence: 82%

Section: Presentation Of the Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction of the Shape and Surface Impedance from Acoustic Scattering Data for an Arbitrary Cylinder

Liu¹,

Nakamura²,

Sini³

2007

SIAM J. Appl. Math.

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“…In a recent work [17,19], we started such studies by taking as pilot the probing method (as the probe and the singular sources methods, see [11,20]). Precisely, using multipoles of order two as point sources, we derived in [17] the asymptotic expansion of the indicator functions with respect to the source point.…”

Section: Introductionmentioning

confidence: 99%

On the Accuracy of the Numerical Detection of Complex Obstacles from Far Field Data Using the Probe Method

Liu¹,

Sini²

2009

SIAM J. Sci. Comput.

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We deal with the acoustic inverse scattering problem for detecting an obstacle with mixed boundary conditions from the far field map. We show how the geometrical properties and the material parameter distributed on the surface are involved in the obstacle reconstruction numerically. The main advance of this research is the numerical analysis and implementation of our recent theoretical work [J.J. Liu and M. Sini, How to make the reconstruction of obstacles more (or less) accurate from exterior measurements, RICAM preprint 2008-04, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria, 2008] regarding the higher order asymptotic expansion of the probe indicator function and the introduction of complex-valued surface impedance. An efficient numerical implementation scheme is proposed based on the properties of minimum norm solutions. Precisely, using the relation between the surface impedance and the obstacle curvature contained in the higher order expansion of the indicators, we reveal how the obstacle curvature and the surface impedance on the coated part influence the blowing-up behavior. Using this theoretical result, we can specify the complex surface impedance in terms of the obstacle curvature to make the reconstruction more (or less) accurate. By establishing properties of the minimum norm solution for approximating the singular sources, efficient realizations for approximating the multipoles by the Herglotz wave function and therefore the implementations of the probe methods are developed with an error estimate. We finally show extensive numerical tests explaining how and to what extent the coupling relation between the curvature and the surface impedance changes the accuracy of the reconstruction of the obstacles.

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“…An introduction into the mathematical theory of inverse problems for acoustic and electromagnetic waves can be found in (Colton and Kress, 1998). A survey about several more recent methods is given by (Potthast, 2006) and a comparative study of some of these methods can be found in (Honda et al, 2007).…”

Section: Introductionmentioning

confidence: 99%

The No Response Test for the reconstruction of polyhedral objects in electromagnetics

Potthast

Sini

2010

Journal of Computational and Applied Mathematics

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We develope a No Response Test for the reconstruction of some polyhedral obstacle from one or few time-harmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a response when the scatterer is inside of this test domain. This is the first formulation of the No Response Test for electromagnetics. We will prove convergence of the method for testing a non-vibrating domain B whether the far field pattern of some scattered time-harmonic field is analytically extendable into the interior of B. We will describe algorithmical realizations of the No Response Test. Finally, we will show the feasibility of the method by reconstruction of polygonal objects in three dimensions.

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The no-response approach and its relation to non-iterative methods for the inverse scattering

Cited by 20 publications

References 17 publications

Reconstruction of the Shape and Surface Impedance from Acoustic Scattering Data for an Arbitrary Cylinder

Reconstruction of the Shape and Surface Impedance from Acoustic Scattering Data for an Arbitrary Cylinder

On the Accuracy of the Numerical Detection of Complex Obstacles from Far Field Data Using the Probe Method

The No Response Test for the reconstruction of polyhedral objects in electromagnetics

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