Identifying defects in an unknown background using differential measurements

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August 2015, 9(3): 625-643. doi: 10.3934/ipi.2015.9.625

Identifying defects in an unknown background using differential measurements

Lorenzo Audibert ^1, , Alexandre Girard ^1, and Houssem Haddar ^2,

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

Received July 2014 Revised February 2015 Published July 2015

Abstract
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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.

Keywords: factorization method, linear sampling method, Inverse scattering problems, differential measurements., qualitative methods.

Mathematics Subject Classification: 35R60, 35R30, 65M3.

Citation: Lorenzo Audibert, Alexandre Girard, Houssem Haddar. Identifying defects in an unknown background using differential measurements. Inverse Problems and Imaging, 2015, 9 (3) : 625-643. doi: 10.3934/ipi.2015.9.625

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,, , ().
[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.

show all references

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,, , ().
[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.

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