Identifying defects in an unknown background using differential measurements

Lorenzo  Audibert; Alexandre  Girard; Houssem Haddar

doi:10.3934/ipi.2015.9.625

Article Contents

2015, Volume 9, Issue 3: 625-643. Doi: 10.3934/ipi.2015.9.625

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Identifying defects in an unknown background using differential measurements

1.
EDF R&D, Departement STEP, 6 quai Watier, 78401, Chatou CEDEX
2.
INRIA Saclay Ile de France/Ecole Polytechnique, CMAP, Route de Saclay, 91128 Palaiseau Cedex

Received: June 30, 2014

Revised: January 31, 2015

Published: July 2015

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Abstract

We present a new qualitative imaging method capable of selecting defects in complex and unknown background from differential measurements of farfield operators: i.e. far measurements of scattered waves in the cases with and without defects. Indeed, the main difficulty is that the background physical properties are unknown. Our approach is based on a new exact characterization of a scatterer domain in terms of the far field operator range and the link with solutions to so-called interior transmission problems. We present the theoretical foundations of the method and some validating numerical experiments in a two dimensional setting.

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Mathematics Subject Classification: 35R60, 35R30, 65M32.

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References

[1]	L. Audibert and H. Haddar, A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements, Inverse Problems, 30 (2014), 035011, 20pp.doi: 10.1088/0266-5611/30/3/035011.
[2]	G. Bal, L. Carin, D. Liu and K. Ren, Experimental validation of a transport-based imaging method in highly scattering environments, Inverse Problems, 23 (2007), 2527-2539.doi: 10.1088/0266-5611/23/6/015.
[3]	L. Borcea, F. González del Cueto, G. Papanicolaou and C. Tsogka, Filtering deterministic layer effects in imaging, SIAM Rev., 54 (2012), 757-798.doi: 10.1137/120880975.
[4]	L. Borcea, J. Garnier, G. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004, 33pp.doi: 10.1088/0266-5611/27/8/085004.
[5]	H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[6]	F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.
[7]	F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects, Inverse Problems, 22 (2006), 845-867.doi: 10.1088/0266-5611/22/3/007.
[8]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edition, Applied Mathematical Sciences, 93, Springer, New York, 2013.doi: 10.1007/978-1-4614-4942-3.
[9]	J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Stochastic Modelling and Applied Probability, 56, Springer, New York, 2007.doi: 10.1007/978-0-387-49808-9_4.
[10]	Y. Grisel, V. Mouysset, P.-A. Mazet and J.-P. Raymond, Determining the shape of defects in non-absorbing inhomogeneous media from far-field measurements, Inverse Problems, 28 (2012), 055003, 19pp.doi: 10.1088/0266-5611/28/5/055003.
[11]	H. Haddar, Sampling 2d, 2013, http://sourceforge.net/projects/samplings-2d/.
[12]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.
[13]	A. I. Nachman, L. Päivärinta and A. Teirilä, On imaging obstacles inside inhomogeneous media, J. Funct. Anal., 252 (2007), 490-516.doi: 10.1016/j.jfa.2007.06.020.
[14]	B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., 22 (1991), 1755-1762.doi: 10.1137/0522109.
[15]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.doi: 10.1137/110836420.