American Institute of Mathematical Sciences

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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

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 & 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).  doi: 10.1088/0266-5611/30/3/035011.  Google Scholar

[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.  doi: 10.1088/0266-5611/23/6/015.  Google Scholar

[3]

L. Borcea, F. González del Cueto, G. Papanicolaou and C. Tsogka, Filtering deterministic layer effects in imaging,, SIAM Rev., 54 (2012), 757.  doi: 10.1137/120880975.  Google Scholar

[4]

L. Borcea, J. Garnier, G. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/8/085004.  Google Scholar

[5]

H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations,, Universitext, (2011).   Google Scholar

[6]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction,, Interaction of Mechanics and Mathematics, (2006).   Google Scholar

[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.  doi: 10.1088/0266-5611/22/3/007.  Google Scholar

[8]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, 3rd edition, (2013).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[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, (2007).  doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[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).  doi: 10.1088/0266-5611/28/5/055003.  Google Scholar

[11]

H. Haddar, Sampling 2d, 2013,, , ().   Google Scholar

[12]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford Lecture Series in Mathematics and its Applications, (2008).   Google Scholar

[13]

A. I. Nachman, L. Päivärinta and A. Teirilä, On imaging obstacles inside inhomogeneous media,, J. Funct. Anal., 252 (2007), 490.  doi: 10.1016/j.jfa.2007.06.020.  Google Scholar

[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.  doi: 10.1137/0522109.  Google Scholar

[15]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341.  doi: 10.1137/110836420.  Google Scholar

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).  doi: 10.1088/0266-5611/30/3/035011.  Google Scholar

[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.  doi: 10.1088/0266-5611/23/6/015.  Google Scholar

[3]

L. Borcea, F. González del Cueto, G. Papanicolaou and C. Tsogka, Filtering deterministic layer effects in imaging,, SIAM Rev., 54 (2012), 757.  doi: 10.1137/120880975.  Google Scholar

[4]

L. Borcea, J. Garnier, G. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/8/085004.  Google Scholar

[5]

H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations,, Universitext, (2011).   Google Scholar

[6]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction,, Interaction of Mechanics and Mathematics, (2006).   Google Scholar

[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.  doi: 10.1088/0266-5611/22/3/007.  Google Scholar

[8]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, 3rd edition, (2013).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[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, (2007).  doi: 10.1007/978-0-387-49808-9_4.  Google Scholar

[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).  doi: 10.1088/0266-5611/28/5/055003.  Google Scholar

[11]

H. Haddar, Sampling 2d, 2013,, , ().   Google Scholar

[12]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford Lecture Series in Mathematics and its Applications, (2008).   Google Scholar

[13]

A. I. Nachman, L. Päivärinta and A. Teirilä, On imaging obstacles inside inhomogeneous media,, J. Funct. Anal., 252 (2007), 490.  doi: 10.1016/j.jfa.2007.06.020.  Google Scholar

[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.  doi: 10.1137/0522109.  Google Scholar

[15]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341.  doi: 10.1137/110836420.  Google Scholar

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