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). 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

[1]

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
[2]

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
[3]

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
[4]

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
[5]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709
[14]

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
[15]

Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123
[16]

Xiaodong Liu. The factorization method for scatterers with different physical properties. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 563-577. doi: 10.3934/dcdss.2015.8.563
[17]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[18]

Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231
[19]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[20]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences