American Institute of Mathematical Sciences

Advanced
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), 035011, 20pp. 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), 2527-2539. 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), 757-798. 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), 085004, 33pp. doi: 10.1088/0266-5611/27/8/085004.  Google Scholar

[5]

H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[6]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 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), 845-867. doi: 10.1088/0266-5611/22/3/007.  Google Scholar

[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.  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, 56, Springer, New York, 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), 055003, 19pp. 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, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[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.  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), 1755-1762. doi: 10.1137/0522109.  Google Scholar

[15]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. 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), 035011, 20pp. 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), 2527-2539. 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), 757-798. 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), 085004, 33pp. doi: 10.1088/0266-5611/27/8/085004.  Google Scholar

[5]

H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[6]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 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), 845-867. doi: 10.1088/0266-5611/22/3/007.  Google Scholar

[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.  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, 56, Springer, New York, 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), 055003, 19pp. 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, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[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.  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), 1755-1762. doi: 10.1137/0522109.  Google Scholar

[15]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. doi: 10.1137/110836420.  Google Scholar

[1]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems & Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033
[2]

Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems & Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036
[3]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263
[4]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[5]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[6]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016
[7]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[8]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[9]

Lorenzo Audibert. The Generalized Linear Sampling and factorization methods only depends on the sign of contrast on the boundary. Inverse Problems & Imaging, 2017, 11 (6) : 1107-1119. doi: 10.3934/ipi.2017051
[10]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103
[11]

Deyue Zhang, Yukun Guo, Fenglin Sun, Hongyu Liu. Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Problems & Imaging, 2020, 14 (3) : 569-582. doi: 10.3934/ipi.2020026
[12]

Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749
[13]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[14]

Houssem Haddar, Alexander Konschin. Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements. Inverse Problems & Imaging, 2020, 14 (1) : 133-152. doi: 10.3934/ipi.2019067
[15]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[16]

Alexey Penenko. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems & Imaging, 2020, 14 (5) : 757-782. doi: 10.3934/ipi.2020035
[17]

Laurent Bourgeois, Arnaud Recoquillay. The Linear Sampling Method for Kirchhoff-Love infinite plates. Inverse Problems & Imaging, 2020, 14 (2) : 363-384. doi: 10.3934/ipi.2020016
[18]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163
[19]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
[20]

Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy