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27
A Regularized Sampling Method for Solving Three Dimensional Inverse Scattering Problems
 SIAM J. Sci. Comput
, 2000
"... The inverse scattering problem under consideration is to determine the shape of an obstacle in R³ from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region. A method for solving this nonlinear and impr ..."
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Cited by 17 (2 self)
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The inverse scattering problem under consideration is to determine the shape of an obstacle in R&sup3; from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region. A method for solving this nonlinear and improperly posed problem is presented which is based on solving a linear integral equation of the first kind and avoids the use of nonlinear optimization methods. Numerical examples are given showing the practicality of this new approach.
The Herglotz wave function, the Vekua transform and the enclosure method
 Hiroshima Math. J
"... Abstract. This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal soundhard obstacles and piecewise linear cracks from the far fi ..."
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Cited by 11 (8 self)
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Abstract. This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal soundhard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua transform. By virtue of the construction, one can avoid any restriction on the wave number in the extraction formulae. An attempt for the case when the far field pattern is given on limited angles is also given.
Multifrequency Inverse Obstacle Scattering . . .
, 2003
"... This work is a study of strategies for obstacle reconstruction from multifrequency far field scattering data. We outline two strategies for obstacle reconstruction from multifrequency far field scattering data: the point source method proposed by Potthast for solving inverse scattering problems with ..."
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Cited by 6 (5 self)
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This work is a study of strategies for obstacle reconstruction from multifrequency far field scattering data. We outline two strategies for obstacle reconstruction from multifrequency far field scattering data: the point source method proposed by Potthast for solving inverse scattering problems with single frequency data in the resonance region, and filtered backprojection techniques based on the physical optics approximation for high frequency scattering. Our implementation of the point source method can be viewed as a generalized filtered backprojection algorithm, the key to which is the construction of the filter used in the backprojection operator. Numerical examples indicate that the critical factor for reconstructions in multifrequency settings is the frequency dependence of the filter.
A Time Domain Point Source Method for Inverse Scattering by Rough Surfaces
"... In this paper we propose a new method to determine the location and shape of an unbounded rough surface from measurements of scattered electromagnetic waves. The proposed method is based on the point source method of Potthast (IMA J.Appl.Math., 61:119140, 1998) for inverse scattering by bounded obs ..."
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In this paper we propose a new method to determine the location and shape of an unbounded rough surface from measurements of scattered electromagnetic waves. The proposed method is based on the point source method of Potthast (IMA J.Appl.Math., 61:119140, 1998) for inverse scattering by bounded obstacles. We propose a version for inverse rough surface scattering which can reconstruct the total field when the incident field is not necessarily time harmonic. We present numerical results for the case of a perfectly conducting surface in TE polarization, in which case a homogeneous Dirichlet condition applies on the boundary. The results show great accuracy of reconstruction of the total field and of the prediction of the surface location. 1
Inverse crack problem and probe method
 Cubo
, 2006
"... A problem of extracting information about the location and shape of unknown cracks in a background medium from the DirichlettoNeumann map is considered. An application of a new formulation of the probe method introduced by the author to the problem is given. The method is based on: the blowup prop ..."
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Cited by 5 (2 self)
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A problem of extracting information about the location and shape of unknown cracks in a background medium from the DirichlettoNeumann map is considered. An application of a new formulation of the probe method introduced by the author to the problem is given. The method is based on: the blowup property of sequences of special solutions of the governing equation for the background medium which are related to a singular solution of the equation; an explicit lower bound of an L2norm of the gradient of the socalled reflected solution. AMS: 35R30
Image synthesis for inverse obstacle scattering using the eigenfunction expansion theorem
 Computing
, 2005
"... In recent years several new inverse scattering techniques have been developed that determine the boundary of an unknown obstacle by reconstructing the surrounding scattered field. In the case of sound soft obstacles, the boundary is usually found as the minimum contour of the total field. In this no ..."
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Cited by 4 (2 self)
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In recent years several new inverse scattering techniques have been developed that determine the boundary of an unknown obstacle by reconstructing the surrounding scattered field. In the case of sound soft obstacles, the boundary is usually found as the minimum contour of the total field. In this note we derive a different approach for imaging the boundary from the reconstructed fields based on a generalization of the eigenfunction expansion theorem. The aim of this alternative approach is the construction of higher contrast images than is currently obtained with the minimum contour approach. AMS Subject Classification: 35R30, 35P25, 68U10, 94A08 Key words: inverse problems, scattering theory, image processing, eigenfunction expansion
Two sides of probe method and obstacle with impedance boundary condition
 Hokkaido Math. J
"... Abstract An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated DirichlettoNeumann ma ..."
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Cited by 4 (4 self)
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Abstract An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated DirichlettoNeumann map. The main result is a characterization of the unknown obstacle via the sequences that are constructed by the DirichlettoNeumann map, under smallness conditions on the wave number and the upper bound of the impedance. Moreover two alternative simple proofs of a previous result of ChengLiuNakamura which are based on only some energy estimates, an analysis of the blowup of the energy of socalled reflected solutions and an application of the enclosure method to the problem are also given.
The Linear Sampling Method for ThreeDimensional Inverse Scattering Problems
"... The inverse scattering problem under consideration is to determine the shape of an obstacle in R 3 from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region. A method for solving this nonlinear and improperly ..."
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Cited by 1 (0 self)
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The inverse scattering problem under consideration is to determine the shape of an obstacle in R 3 from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region. A method for solving this nonlinear and improperly posed problem is presented which is based on solving a linear integral equation of the first kind and avoids the use of nonlinear optimization methods. Numerical examples are given showing the practicality of this new approach. 1 Introduction Inverse scattering theory is concerned with determining the shape of a scattering obstacle from a knowledge of the scattered acoustic or electromagnetic wave corresponding to a given incident field. Although of considerable importance in various areas of science and technology, the mathematical and numerical analysis of such problems is of relatively recent origin. There have been a number of successful reconstruction algorithms proposed for the three...
A PointSource Method in Inverse Acoustic Scattering.
"... We present a pointsource method to solve inverse acoustic obstacle scattering problems. The pointsource method gives a new and explicit procedure how to compute from the far field pattern u 1 of a scattered field an approximation for the total acoustic field u near the unknown obstacle. The poin ..."
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Cited by 1 (1 self)
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We present a pointsource method to solve inverse acoustic obstacle scattering problems. The pointsource method gives a new and explicit procedure how to compute from the far field pattern u 1 of a scattered field an approximation for the total acoustic field u near the unknown obstacle. The pointsource method then searches for the boundary of the obstacle as a zerocurve of u or @u @ , respectively, where is normal to the unknown domain D. A threedimensional example for the numerical reconstruction of a scatterer consisting of two separate components is provided. 1 Introduction. We consider the scattering of acoustic waves from a soundsoft or soundhard scatterer D. We assume D to be bounded in IR 3 and the boundary @D sufficiently smooth. An incident timeharmonic acoustic wave u i satisfies the Helmholtz equation 4u + 2 u = 0 in IR 3 , where denotes the wave number. The direct scattering problem is to find a scattered field u s which solves the Helmholtz equati...
On Source Analysis by Wave Splitting with Applications in Inverse Scattering of Multiple Obstacles
, 2006
"... We study wave splitting procedures for acoustic or electromagnetic scattering problems. The idea of these procedures is to split some scattered field into a sum of fields coming from different spatial regions such that this information can be used either for inversion algorithms or for active noise ..."
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We study wave splitting procedures for acoustic or electromagnetic scattering problems. The idea of these procedures is to split some scattered field into a sum of fields coming from different spatial regions such that this information can be used either for inversion algorithms or for active noise control. Splitting algorithms can be based on general boundary layer potential representation or Green’s representation formula. We will prove the unique decomposition of scattered wave outside the specified reference domain G and the unique decomposition of farfield pattern with respect to different reference domain G. Further, we employ the splitting technique for field reconstruction for a scatterer with two or more separate components, by combining it with the point source method for wave recovery. Using the decomposition of scattered wave as well as its farfield pattern, the wave splitting procedure proposed in this paper gives an efficient way to the computation of scattered wave near the obstacle, from which the multiple obstacles which cause the farfield pattern can be reconstructed separately. This considerably extends the range of the decomposition methods in the area of inverse scattering. Finally, we will provide numerical examples to prove the feasibility of the splitting method.