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

Houssem Haddar; Alexander Konschin

doi:10.3934/ipi.2019067

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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 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. 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.
[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 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.
[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. 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.
[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. 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.
[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. 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.

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. 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.
[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 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.
[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. 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.
[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. 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.
[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. 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.

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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American Institute of Mathematical Sciences

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

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

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