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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

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.

Citation: Houssem Haddar, Alexander Konschin. Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements. Inverse Problems and 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.

[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.

[3]

L. AudibertA. 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.

[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.

[5]

A.-S. Bonnet-Ben DhiaS. FlissC. 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.

[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.

[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.

[8]

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

[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.

[10]

F. CakoniH. 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.

[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.

[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.

[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.

[14]

G. HuY. 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.

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

[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.

[24]

J. YangB. 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.

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.

[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.

[3]

L. AudibertA. 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.

[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.

[5]

A.-S. Bonnet-Ben DhiaS. FlissC. 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.

[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.

[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.

[8]

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

[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.

[10]

F. CakoniH. 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.

[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.

[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.

[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.

[14]

G. HuY. 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.

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

[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.

[24]

J. YangB. 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.

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