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Edge detection with mixed noise based on maximum a posteriori approach
A linear sampling method for inverse acoustic scattering by a locally rough interface
1. | School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China |
2. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China |
3. | LSEC, NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
4. | School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China |
This paper is concerned with the inverse problem of time-harmonic acoustic scattering by an unbounded, locally rough interface which is assumed to be a local perturbation of a plane. The purpose of this paper is to recover the local perturbation of the interface from the near-field measurement given on a straight line segment with a finite distance above the interface and generated by point sources. Precisely, we propose a novel version of the linear sampling method to recover the location and shape of the local perturbation of the interface numerically. Our method is based on a modified near-field operator equation associated with a special rough surface, constructed by reformulating the forward scattering problem into an equivalent integral equation formulation in a bounded domain, leading to a fast imaging algorithm. Numerical experiments are presented to illustrate the effectiveness of the imaging method.
References:
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G. Bao, J. Gao and P. Li,
Analysis of direct and inverse cavity scattering problems, Numer. Math. Theory Methods Appl., 4 (2011), 335-358.
doi: 10.4208/nmtma.2011.m1021. |
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G. Bao and P. Li,
Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.
doi: 10.1137/130916266. |
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G. Bao and P. Li,
Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899.
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[4] |
G. Bao and J. Lin,
Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.
doi: 10.1137/110824644. |
[5] |
G. Bao and J. Lin,
Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imaging, 7 (2013), 377-396.
doi: 10.3934/ipi.2013.7.377. |
[6] |
C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007, 23 pp.
doi: 10.1088/0266-5611/26/4/045007. |
[7] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006. |
[8] |
F. Cakoni, D. Gintides and H. Haddar,
The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237-255.
doi: 10.1137/090769338. |
[9] |
S. N. Chandler-Wilde and J. Elschner,
Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal., 42 (2010), 2554-2580.
doi: 10.1137/090776111. |
[10] |
S. N. Chandler-Wilde and R. Potthast,
The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 2967-3001.
doi: 10.1098/rspa.2002.0999. |
[11] |
S. N. Chandler-Wilde and B. Zhang,
Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers, SIAM J. Math. Anal., 30 (1999), 559-583.
doi: 10.1137/S0036141097328932. |
[12] |
L. Chorfi and P. Gaitan, Reconstruction of the interface between two-layered media using far-field measurements, Inverse Problems, 27 (2011), 075001, 19 pp.
doi: 10.1088/0266-5611/27/7/075001. |
[13] |
D. Colton and A. Kirsch,
A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.
doi: 10.1088/0266-5611/12/4/003. |
[14] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theorey, 4$^th$ edition, Springer, 2019.
doi: 10.1007/978-3-030-30351-8. |
[15] |
M. Ding, J. Li, K. Liu and J. Yang,
Imaging of locally rough surfaces by the linear sampling method with the near-field data, SIAM J. Imaging Sci., 10 (2017), 1579-1602.
doi: 10.1137/16M1097997. |
[16] |
G. Hu, X. Liu, B. Zhang and H. Zhang, A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces, Inverse Problems, 35 (2019), 025007, 20 pp.
doi: 10.1088/1361-6420/aaf3d6. |
[17] |
A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis. KIT, Germany, 2008. |
[18] |
A. Lechleiter and R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 39 (2017), B819–B839.
doi: 10.1137/16M1104111. |
[19] |
A. Lechleiter and R. Zhang, Reconstruction of local perturbations in periodic surfaces, Inverse Problems, 34 (2018), 035006, 17 pp.
doi: 10.1088/1361-6420/aaa7b1. |
[20] |
R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley, New York, 1986.
doi: 10.1007/978-3-663-10649-4. |
[21] |
P. Li,
Coupling of finite element and boundary integral method for electromagnetic scattering in a two-layered medium, J. Comput. Phys., 229 (2010), 481-497.
doi: 10.1016/j.jcp.2009.09.040. |
[22] |
J. Li, G. Sun and R. Zhang,
The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71 (2016), 1491-1502.
doi: 10.1016/j.camwa.2016.02.031. |
[23] |
J. Li and G. Sun,
A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.
doi: 10.1080/17415977.2014.922077. |
[24] |
J. Li, G. Sun and B. Zhang,
The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Anal., 96 (2017), 85-107.
doi: 10.1080/00036811.2016.1192141. |
[25] |
C. D. Lines and S. N. Chandler-Wilde,
A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.
doi: 10.1007/s00607-004-0109-8. |
[26] |
X. Liu, B. Zhang and H. Zhang,
A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imaging Sci., 11 (2018), 1629-1650.
doi: 10.1137/18M1166031. |
[27] |
X. Liu, B. Zhang and H. Zhang,
Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79 (2019), 153-176.
doi: 10.1137/18M1181407. |
[28] |
D. Natroshvili, T. Arens and S. N. Chandler-Wilde,
Uniqueness, existence, and integral equation formulations for interface scattering problems, Mem. Differential Equations Math. Phys., 30 (2003), 105-146.
|
[29] |
F. Qu, B. Zhang and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41 (2019), A3673–A3702.
doi: 10.1137/19M1240745. |
[30] |
D. G. Roy and S. Mudaliar,
Domain derivatives in dielectric rough surface scattering, IEEE Trans. Antennas Propagation, 63 (2015), 4486-4495.
doi: 10.1109/TAP.2015.2463682. |
[31] |
M. Thomas, Analysis of Rough Surface Scattering Problems, Ph.D Thesis, The University of Reading, UK, 2006. |
[32] |
X. Xu, B. Zhang and H. Zhang,
Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data, SIAM J. Imaging Sci., 12 (2019), 119-152.
doi: 10.1137/18M1210204. |
[33] |
J. Yang, J. Li and B. Zhang, Simultaneous recovery of a locally rough interface and its buried obstacles and homogeneous medium, arXiv: 2102.01855v1. |
[34] |
J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 035004, 17 pp.
doi: 10.1088/0266-5611/28/3/035004. |
[35] |
J. Yang, B. Zhang and R. Zhang,
Reconstruction of penetrable grating profiles, Inverse Problems Imaging, 7 (2013), 1393-1407.
doi: 10.3934/ipi.2013.7.1393. |
[36] |
H. Zhang,
Recovering unbounded rough surfaces with a direct imaging method, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 119-133.
doi: 10.1007/s10255-020-0916-5. |
[37] |
B. Zhang and S. N. Chandler-Wilde,
Integral equation methods for scattering by infinite rough surfaces, Math. Methods Appl. Sci., 26 (2003), 463-488.
doi: 10.1002/mma.361. |
[38] |
H. Zhang and B. Zhang,
A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.
doi: 10.1137/130908324. |
show all references
References:
[1] |
G. Bao, J. Gao and P. Li,
Analysis of direct and inverse cavity scattering problems, Numer. Math. Theory Methods Appl., 4 (2011), 335-358.
doi: 10.4208/nmtma.2011.m1021. |
[2] |
G. Bao and P. Li,
Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.
doi: 10.1137/130916266. |
[3] |
G. Bao and P. Li,
Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899.
doi: 10.1137/130944485. |
[4] |
G. Bao and J. Lin,
Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.
doi: 10.1137/110824644. |
[5] |
G. Bao and J. Lin,
Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imaging, 7 (2013), 377-396.
doi: 10.3934/ipi.2013.7.377. |
[6] |
C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems, 26 (2010), 045007, 23 pp.
doi: 10.1088/0266-5611/26/4/045007. |
[7] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006. |
[8] |
F. Cakoni, D. Gintides and H. Haddar,
The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237-255.
doi: 10.1137/090769338. |
[9] |
S. N. Chandler-Wilde and J. Elschner,
Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal., 42 (2010), 2554-2580.
doi: 10.1137/090776111. |
[10] |
S. N. Chandler-Wilde and R. Potthast,
The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 2967-3001.
doi: 10.1098/rspa.2002.0999. |
[11] |
S. N. Chandler-Wilde and B. Zhang,
Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers, SIAM J. Math. Anal., 30 (1999), 559-583.
doi: 10.1137/S0036141097328932. |
[12] |
L. Chorfi and P. Gaitan, Reconstruction of the interface between two-layered media using far-field measurements, Inverse Problems, 27 (2011), 075001, 19 pp.
doi: 10.1088/0266-5611/27/7/075001. |
[13] |
D. Colton and A. Kirsch,
A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.
doi: 10.1088/0266-5611/12/4/003. |
[14] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theorey, 4$^th$ edition, Springer, 2019.
doi: 10.1007/978-3-030-30351-8. |
[15] |
M. Ding, J. Li, K. Liu and J. Yang,
Imaging of locally rough surfaces by the linear sampling method with the near-field data, SIAM J. Imaging Sci., 10 (2017), 1579-1602.
doi: 10.1137/16M1097997. |
[16] |
G. Hu, X. Liu, B. Zhang and H. Zhang, A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces, Inverse Problems, 35 (2019), 025007, 20 pp.
doi: 10.1088/1361-6420/aaf3d6. |
[17] |
A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis. KIT, Germany, 2008. |
[18] |
A. Lechleiter and R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 39 (2017), B819–B839.
doi: 10.1137/16M1104111. |
[19] |
A. Lechleiter and R. Zhang, Reconstruction of local perturbations in periodic surfaces, Inverse Problems, 34 (2018), 035006, 17 pp.
doi: 10.1088/1361-6420/aaa7b1. |
[20] |
R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley, New York, 1986.
doi: 10.1007/978-3-663-10649-4. |
[21] |
P. Li,
Coupling of finite element and boundary integral method for electromagnetic scattering in a two-layered medium, J. Comput. Phys., 229 (2010), 481-497.
doi: 10.1016/j.jcp.2009.09.040. |
[22] |
J. Li, G. Sun and R. Zhang,
The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71 (2016), 1491-1502.
doi: 10.1016/j.camwa.2016.02.031. |
[23] |
J. Li and G. Sun,
A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.
doi: 10.1080/17415977.2014.922077. |
[24] |
J. Li, G. Sun and B. Zhang,
The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Anal., 96 (2017), 85-107.
doi: 10.1080/00036811.2016.1192141. |
[25] |
C. D. Lines and S. N. Chandler-Wilde,
A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.
doi: 10.1007/s00607-004-0109-8. |
[26] |
X. Liu, B. Zhang and H. Zhang,
A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imaging Sci., 11 (2018), 1629-1650.
doi: 10.1137/18M1166031. |
[27] |
X. Liu, B. Zhang and H. Zhang,
Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79 (2019), 153-176.
doi: 10.1137/18M1181407. |
[28] |
D. Natroshvili, T. Arens and S. N. Chandler-Wilde,
Uniqueness, existence, and integral equation formulations for interface scattering problems, Mem. Differential Equations Math. Phys., 30 (2003), 105-146.
|
[29] |
F. Qu, B. Zhang and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41 (2019), A3673–A3702.
doi: 10.1137/19M1240745. |
[30] |
D. G. Roy and S. Mudaliar,
Domain derivatives in dielectric rough surface scattering, IEEE Trans. Antennas Propagation, 63 (2015), 4486-4495.
doi: 10.1109/TAP.2015.2463682. |
[31] |
M. Thomas, Analysis of Rough Surface Scattering Problems, Ph.D Thesis, The University of Reading, UK, 2006. |
[32] |
X. Xu, B. Zhang and H. Zhang,
Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data, SIAM J. Imaging Sci., 12 (2019), 119-152.
doi: 10.1137/18M1210204. |
[33] |
J. Yang, J. Li and B. Zhang, Simultaneous recovery of a locally rough interface and its buried obstacles and homogeneous medium, arXiv: 2102.01855v1. |
[34] |
J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 035004, 17 pp.
doi: 10.1088/0266-5611/28/3/035004. |
[35] |
J. Yang, B. Zhang and R. Zhang,
Reconstruction of penetrable grating profiles, Inverse Problems Imaging, 7 (2013), 1393-1407.
doi: 10.3934/ipi.2013.7.1393. |
[36] |
H. Zhang,
Recovering unbounded rough surfaces with a direct imaging method, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 119-133.
doi: 10.1007/s10255-020-0916-5. |
[37] |
B. Zhang and S. N. Chandler-Wilde,
Integral equation methods for scattering by infinite rough surfaces, Math. Methods Appl. Sci., 26 (2003), 463-488.
doi: 10.1002/mma.361. |
[38] |
H. Zhang and B. Zhang,
A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.
doi: 10.1137/130908324. |







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