The linear sampling method and energy conservation

In this paper we explain the linear sampling method and its performances under various scattering conditions by means of an analysis of the far-field equation based on the principle of energy conservation. Specifically, we consider the conservation of energy along the flow strips of the Poynting vector associated with the scattered field whose far-field pattern is one of the two terms in the far-field equation. The behavior of these flow lines is numerically investigated and theoretically described. Appropriate assumptions on the flow lines, based on the numerical results, allow characterizing a set of approximate solutions of the far-field equation which can be used to visualize the boundary of the scatterer in the framework of the linear sampling method. In particular, under the same assumptions, we can show that Tikhonov regularized solutions belong to this set of approximate solutions for appropriate choices of the regularization parameter.

[1]  D. Colton,et al.  The linear sampling method in inverse electromagnetic scattering theory , 2003 .

[2]  Fioralba Cakoni,et al.  Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects , 2006 .

[3]  J. Richmond.  Scattering by a dielectric cylinder of arbitrary cross section shape , 1965 .

[4]  T. Arens,et al.  Why linear sampling works , 2004 .

[5]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[6]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[7]  A. Tikhonov,et al.  Numerical Methods for the Solution of Ill-Posed Problems , 1995 .

[8]  C. Balanis.  Advanced Engineering Electromagnetics , 1989 .

[9]  A. A. Rizvi,et al.  POWER FLOW STRUCTURES IN TWO DIMENSIONAL ELECTROMAGNETIC FIELDS , 2000 .

[10]  Armin Lechleiter,et al.  The linear sampling method revisited , 2009 .

[11]  David Colton,et al.  An application of the reciprocity gap functional to inverse scattering theory , 2005 .

[12]  D. Colton,et al.  A simple method using Morozov's discrepancy principle for solving inverse scattering problems , 1997 .

[13]  Fioralba Cakoni,et al.  On the Mathematical Basis of the Linear Sampling Method , 2003 .

[14]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[15]  David Colton,et al.  Qualitative Methods in Inverse Scattering Theory , 1997 .

[16]  Houssem Haddar,et al.  Numerical and analytical studies of the linear sampling method in electromagnetic inverse scattering problems , 2003 .

[17]  Mario Bertero,et al.  Introduction to Inverse Problems in Imaging , 1998 .

[18]  Fioralba Cakoni,et al.  A Survey in Mathematics for Industry: Open problems in the qualitative approach to inverse electromagnetic scattering theory , 2005, European Journal of Applied Mathematics.

[19]  Karl F. Warnick,et al.  Behavior of the Regularized Sampling Inverse Scattering Method at Internal Resonance Frequencies , 2002 .

[20]  T. Isernia,et al.  On Simple Methods for Shape Reconstruction of Unknown Scatterers , 2007, IEEE Transactions on Antennas and Propagation.

[21]  Martin Hanke,et al.  Why linear sampling really seems to work , 2008 .

[22]  N I Grinberg,et al.  The Factorization Method for Inverse Problems , 2007 .