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February  2020, 14(1): 133-152. doi: 10.3934/ipi.2019067

Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements

Houssem Haddar 1, and  Alexander Konschin 2,,

1. 

CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
2. 

Center for Industrial Mathematics, University of Bremen, Bibliothekstraẞe 5, 28359 Bremen, Germany

* Corresponding author: Alexander Konschin

Received  April 2019 Revised  August 2019 Published  November 2019

Fund Project: A. Konschin was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 281474342/GRK2224/1

We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting.

Keywords: Factorization method, inhomogeneous periodic media, far field measurements, reconstruction of defects, inverse problem.
Mathematics Subject Classification: Primary: 78A46, 35R30; Secondary: 35Q60.
Citation: 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
References:
[1]

T. Arens and N. I. Grinberg, A complete factorization method for scattering by periodic structures, Computing, 75 (2005), 111-132.  doi: 10.1007/s00607-004-0092-0.  Google Scholar

[2]

T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems, 19 (2003), 1195-1211.  doi: 10.1088/0266-5611/19/5/311.  Google Scholar

[3]

L. Audibert, A. Girard and H. Haddar, Identifying defects in an unknown background using differential measurements, Inverse Probl. Imaging, 9 (2015), 625-643.  doi: 10.3934/ipi.2015.9.625.  Google Scholar

[4]

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), 20pp.  doi: 10.1088/0266-5611/30/3/035011.  Google Scholar

[5]

A.-S. Bonnet-Ben Dhia, S. Fliss, C. Hazard and A. Tonnoir, A Rellich type theorem for the Helmholtz equation in a conical domain, C. R. Math. Acad. Sci. Paris, 354 (2016), 27-32.  doi: 10.1016/j.crma.2015.10.015.  Google Scholar

[6]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.  Google Scholar

[7]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 31pp.  doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[8]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[10]

F. Cakoni, H. Haddar and T.-P. Nguyen, New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 35 (2019), 31pp.  doi: 10.1088/1361-6420/aaecfd.  Google Scholar

[11]

D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[12]

J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions, Methods Appl. Anal., 18 (2011), 215-244.  doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar

[13]

H. Haddar and T.-P. Nguyen, Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers, Comput. Math. Appl., 74 (2017), 2831-2855.  doi: 10.1016/j.camwa.2017.07.015.  Google Scholar

[14]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 25pp.  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[15]

A. Kirsch and N. I. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, Oxford, 2008. doi: 10.1093/acprof:oso/9780199213535.001.0001.  Google Scholar

[16]

A. Konschin and A. Lechleiter, Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements, Inverse Problems, 35 (2019), 114006.  doi: 10.1088/1361-6420/ab1c66.  Google Scholar

[17]

A. Lechleiter, Imaging of periodic dielectrics, BIT, 50 (2010), 59-83.  doi: 10.1007/s10543-010-0255-7.  Google Scholar

[18]

A. Lechleiter, The Floquet-Bloch transform and scattering from locally perturbed periodic surfaces, J. Math. Anal. Appl., 446 (2017), 605-627.  doi: 10.1016/j.jmaa.2016.08.055.  Google Scholar

[19]

A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci., 6 (2013), 1111-1139.  doi: 10.1137/120903968.  Google Scholar

[20]

D.-L. Nguyen, Spectral methods for direct and inverse scattering from periodic structures, Ph.D thesis, Ecole Polytechnique, Palaiseau, France, 2012. Google Scholar

[21]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, Ph.D thesis, Karlsruher Institut für Technologie, 2010. doi: 10.5445/KSP/1000019400.  Google Scholar

[22]

J. Sarannen and G. Vainikko, Periodic Integral and Pseudodifferential Equations, Springer, Berlin, 2002. Google Scholar

[23]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, in Recent Advances in Scientific Computing and Applications, Contemp. Math., 586, Amer. Math. Soc., Providence, RI, 2013, 341–351. doi: 10.1090/conm/586/11652.  Google Scholar

[24]

J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 17pp.  doi: 10.1088/0266-5611/28/3/035004.  Google Scholar

show all references

References:
[1]

T. Arens and N. I. Grinberg, A complete factorization method for scattering by periodic structures, Computing, 75 (2005), 111-132.  doi: 10.1007/s00607-004-0092-0.  Google Scholar

[2]

T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems, 19 (2003), 1195-1211.  doi: 10.1088/0266-5611/19/5/311.  Google Scholar

[3]

L. Audibert, A. Girard and H. Haddar, Identifying defects in an unknown background using differential measurements, Inverse Probl. Imaging, 9 (2015), 625-643.  doi: 10.3934/ipi.2015.9.625.  Google Scholar

[4]

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), 20pp.  doi: 10.1088/0266-5611/30/3/035011.  Google Scholar

[5]

A.-S. Bonnet-Ben Dhia, S. Fliss, C. Hazard and A. Tonnoir, A Rellich type theorem for the Helmholtz equation in a conical domain, C. R. Math. Acad. Sci. Paris, 354 (2016), 27-32.  doi: 10.1016/j.crma.2015.10.015.  Google Scholar

[6]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.  Google Scholar

[7]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 31pp.  doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[8]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An Introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[10]

F. Cakoni, H. Haddar and T.-P. Nguyen, New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 35 (2019), 31pp.  doi: 10.1088/1361-6420/aaecfd.  Google Scholar

[11]

D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[12]

J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions, Methods Appl. Anal., 18 (2011), 215-244.  doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar

[13]

H. Haddar and T.-P. Nguyen, Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers, Comput. Math. Appl., 74 (2017), 2831-2855.  doi: 10.1016/j.camwa.2017.07.015.  Google Scholar

[14]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 25pp.  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[15]

A. Kirsch and N. I. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, Oxford, 2008. doi: 10.1093/acprof:oso/9780199213535.001.0001.  Google Scholar

[16]

A. Konschin and A. Lechleiter, Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements, Inverse Problems, 35 (2019), 114006.  doi: 10.1088/1361-6420/ab1c66.  Google Scholar

[17]

A. Lechleiter, Imaging of periodic dielectrics, BIT, 50 (2010), 59-83.  doi: 10.1007/s10543-010-0255-7.  Google Scholar

[18]

A. Lechleiter, The Floquet-Bloch transform and scattering from locally perturbed periodic surfaces, J. Math. Anal. Appl., 446 (2017), 605-627.  doi: 10.1016/j.jmaa.2016.08.055.  Google Scholar

[19]

A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci., 6 (2013), 1111-1139.  doi: 10.1137/120903968.  Google Scholar

[20]

D.-L. Nguyen, Spectral methods for direct and inverse scattering from periodic structures, Ph.D thesis, Ecole Polytechnique, Palaiseau, France, 2012. Google Scholar

[21]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, Ph.D thesis, Karlsruher Institut für Technologie, 2010. doi: 10.5445/KSP/1000019400.  Google Scholar

[22]

J. Sarannen and G. Vainikko, Periodic Integral and Pseudodifferential Equations, Springer, Berlin, 2002. Google Scholar

[23]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, in Recent Advances in Scientific Computing and Applications, Contemp. Math., 586, Amer. Math. Soc., Providence, RI, 2013, 341–351. doi: 10.1090/conm/586/11652.  Google Scholar

[24]

J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 17pp.  doi: 10.1088/0266-5611/28/3/035004.  Google Scholar

Figure 1.  Example for the refractive index $ n = n_p + q $
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Figure 2.  Three periods of the parameters and results for Example $ 1 $
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Figure 3.  Three periods of main parameters for the numerical experiments
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Figure 4.  Results for Example $ 2 $ with $ k^2 = 0.4 $, $ q = q_1 $, $ \nu = 10^{-3} $ and $ \nu = 10^{-6} $
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Figure 5.  Results for Example $ 3 $ with $ k^2 = 1.8 $, $ \nu = 1 $ and $ \nu = 10^{-3} $
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Figure 6.  Results for Example $ 4 $ with $ k^2 = 0.09 $, $ \nu = 10^{-3} $ and with $ k^2 = 3 $, $ \nu = 1 $
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Figure 7.  Results for FM and LSM with $ k^2 = 3 $ and $ 1\% $ relative noise
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