Figure 1.
Example for the refractive index $ n = n_p + q $
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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 |
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 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 6.
Results for Example $ 4 $ with $ k^2 = 0.09 $ , $ \nu = 10^{-3} $ and with $ k^2 = 3 $ , $ \nu = 1 $
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