• Previous Article
    Synthetic-Aperture Radar image based positioning in GPS-denied environments using Deep Cosine Similarity Neural Networks
  • IPI Home
  • This Issue
  • Next Article
    Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification
August  2021, 15(4): 745-762. doi: 10.3934/ipi.2021012

A sampling type method in an electromagnetic waveguide

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Received  July 2020 Revised  November 2020 Published  August 2021 Early access  January 2021

We propose a sampling type method to image scatterer in an electromagnetic waveguide. The waveguide terminates at one end and the measurements are on the other end and in the far field. The imaging function is based on integrating the measurements and a known function over the measurement surface directly. The design and analysis of such imaging function are based on a factorization of a data operator given by the measurements. We show by analysis that the imaging function peaks inside the scatterer, where the coercivity of the factorized operator and the design of the known function play a central role. Finally, numerical examples are provided to demonstrate the performance of the imaging method.

Citation: Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems & Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012
References:
[1]

T. ArensD. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM Journal on Applied Mathematics, 71 (2011), 753-772.  doi: 10.1137/100806333.  Google Scholar

[2]

A. B. BaggeroerW. A. Kuperman and P. N. Mikhalevsky, An overview of matched field methods in ocean acoustics, IEEE Journal of Oceanic Engineering, 18 (1993), 401-424.   Google Scholar

[3]

L. Borcea and S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Problems, 35 (2019), 124006, 33pp. doi: 10.1088/1361-6420/ab2c9b.  Google Scholar

[4]

L. Borcea and D. L. Nguyen, Imaging with electromagnetic waves in terminating waveguides, Inverse Problems and Imaging, 10 (2016), 915-941.  doi: 10.3934/ipi.2016027.  Google Scholar

[5]

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

[6]

L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[7]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018, 20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[8]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19pp. doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[10]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, 188. Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[11]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, volume 88., SIAM, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[12]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, volume 80., SIAM, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[13]

J. Chen and G. Huang, A direct imaging method for inverse electromagnetic scattering problem in rectangular waveguide, Communications in Computational Physics, 23 (2018), 1415-1433.  doi: 10.4208/cicp.oa-2017-0048.  Google Scholar

[14]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[15]

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

[16]

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005, 23pp. doi: 10.1088/0266-5611/27/8/085005.  Google Scholar

[17]

A. HaackJ. Schreyer and G. Jackel, State-of-the-art of non-destructive testing methods for determining the state of a tunnel lining, Tunnelling and Underground Space Technology Incorporating Trenchless Technology Research, 4 (1995), 413-431.   Google Scholar

[18]

A. Kirsch, The factorization method for maxwell's equations, Inverse Problems, 20 (2004), S117–S134. doi: 10.1088/0266-5611/20/6/S08.  Google Scholar

[19] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36., Oxford University Press, 2008.   Google Scholar
[20]

X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20pp. doi: 10.1088/1361-6420/aa777d.  Google Scholar

[21]

P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747.  doi: 10.3934/ipi.2012.6.709.  Google Scholar

[22]

P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain, Inverse Problems, 32 (2016), 055001, 26pp. doi: 10.1088/0266-5611/32/5/055001.  Google Scholar

[23]

P. Monk, V. Selgas and F. Yang, Near-field linear sampling method for an inverse problem in an electromagnetic waveguide, Inverse Problems, 35 (2019), 065001, 27pp. doi: 10.1088/1361-6420/ab0cdc.  Google Scholar

[24]

R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015. doi: 10.1088/0266-5611/26/7/074015.  Google Scholar

[25]

P. Rizzo, A. Marzani, J. Bruck, et al., Ultrasonic guided waves for nondestructive evaluation/structural health monitoring of trusses, Measurement Science and Technology, 21 (2010), 045701. Google Scholar

[26]

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, 1 (1997), 41-52.   Google Scholar

[27]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.  Google Scholar

[28]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM Journal on Imaging Sciences, 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.  Google Scholar

[29]

Y. XuC. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

show all references

References:
[1]

T. ArensD. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM Journal on Applied Mathematics, 71 (2011), 753-772.  doi: 10.1137/100806333.  Google Scholar

[2]

A. B. BaggeroerW. A. Kuperman and P. N. Mikhalevsky, An overview of matched field methods in ocean acoustics, IEEE Journal of Oceanic Engineering, 18 (1993), 401-424.   Google Scholar

[3]

L. Borcea and S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Problems, 35 (2019), 124006, 33pp. doi: 10.1088/1361-6420/ab2c9b.  Google Scholar

[4]

L. Borcea and D. L. Nguyen, Imaging with electromagnetic waves in terminating waveguides, Inverse Problems and Imaging, 10 (2016), 915-941.  doi: 10.3934/ipi.2016027.  Google Scholar

[5]

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

[6]

L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[7]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018, 20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[8]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19pp. doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[10]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, 188. Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[11]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, volume 88., SIAM, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[12]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, volume 80., SIAM, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[13]

J. Chen and G. Huang, A direct imaging method for inverse electromagnetic scattering problem in rectangular waveguide, Communications in Computational Physics, 23 (2018), 1415-1433.  doi: 10.4208/cicp.oa-2017-0048.  Google Scholar

[14]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[15]

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

[16]

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005, 23pp. doi: 10.1088/0266-5611/27/8/085005.  Google Scholar

[17]

A. HaackJ. Schreyer and G. Jackel, State-of-the-art of non-destructive testing methods for determining the state of a tunnel lining, Tunnelling and Underground Space Technology Incorporating Trenchless Technology Research, 4 (1995), 413-431.   Google Scholar

[18]

A. Kirsch, The factorization method for maxwell's equations, Inverse Problems, 20 (2004), S117–S134. doi: 10.1088/0266-5611/20/6/S08.  Google Scholar

[19] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36., Oxford University Press, 2008.   Google Scholar
[20]

X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20pp. doi: 10.1088/1361-6420/aa777d.  Google Scholar

[21]

P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747.  doi: 10.3934/ipi.2012.6.709.  Google Scholar

[22]

P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain, Inverse Problems, 32 (2016), 055001, 26pp. doi: 10.1088/0266-5611/32/5/055001.  Google Scholar

[23]

P. Monk, V. Selgas and F. Yang, Near-field linear sampling method for an inverse problem in an electromagnetic waveguide, Inverse Problems, 35 (2019), 065001, 27pp. doi: 10.1088/1361-6420/ab0cdc.  Google Scholar

[24]

R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015. doi: 10.1088/0266-5611/26/7/074015.  Google Scholar

[25]

P. Rizzo, A. Marzani, J. Bruck, et al., Ultrasonic guided waves for nondestructive evaluation/structural health monitoring of trusses, Measurement Science and Technology, 21 (2010), 045701. Google Scholar

[26]

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, 1 (1997), 41-52.   Google Scholar

[27]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.  Google Scholar

[28]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM Journal on Imaging Sciences, 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.  Google Scholar

[29]

Y. XuC. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

Figure 1.  Plot of $ |\mathcal{H} \Psi ({\boldsymbol{x}}_*;{\boldsymbol{z}}) {\boldsymbol{e}}_j|^2 $ as a function of the sampling point $ {\boldsymbol{z}} $ for a rectangular waveguide with cross section $ (0,10)\times(0,10) $ where $ {\boldsymbol{x}}_* = (5,5,-5) $ and $ k = 3 $. Top left: Iso-surface plot with iso-value $ 0.6 $. Top right: $ x_1x_2 $-cross section image. Bottom left: $ x_2x_3 $-cross section image. Bottom right: $ x_1x_3 $-cross section image
Figure 2.  Three dimensional view of the generated mesh
Figure 3.  Image of a cuboid. Top left: exact. Top middle: three dimensional image using iso-surface plot with iso-value $ 0.6 $. Top right: three dimensional image using iso-surface plot with iso-value $ 0.4 $. At the bottom, we plot the cross section images where the exact geometry is indicated by the dashed line. Bottom left: $ x_1x_2 $-cross section image. Bottom middle: $ x_2x_3 $-cross section image. Bottom right: $ x_1x_3 $-cross section image
Figure 4.  Image of a L-shape scatterer. Top left: exact. Top middle: three dimensional image using iso-surface plot with iso-value $ 0.5 $. Top right: three dimensional image using iso-surface plot with iso-value $ 0.4 $. At the bottom, we plot the cross section images where the exact geometry is indicated by the dashed line. Bottom left: $ x_1x_2 $-cross section image. Bottom middle: $ x_2x_3 $-cross section image. Bottom right: $ x_1x_3 $-cross section image
Figure 5.  Image of a cylindrical scatterer. Top left: exact. Top middle: $ k = 3 $; three dimensional image using iso-surface plot with iso-value $ 0.4 $. Top right: $ k = 5 $; three dimensional image using iso-surface plot with iso-value $ 0.3 $. In the middle ($ k = 3 $) and bottom ($ k = 5 $) row, we plot the cross section images where the exact geometry is indicated by the dashed line. Middle/bottom row left: $ x_1x_2 $-cross section image. Middle/bottom row middle: $ x_2x_3 $-cross section image. Middle/bottom row right: $ x_1x_3 $-cross section image
Figure 6.  Top: image of two balls. Bottom: image of three balls. Left: exact. Middle: three dimensional image using iso-surface plot (with iso-value 0.2). Right: $ x_1x_2 $-cross section image
Figure 7.  Image of a large ball. Left: exact. Middle: three dimensional image when $ k = 1 $. Right: three dimensional image when $ k = 3 $
Figure 8.  Iso-surface (with the same iso-value 0.4) image of the L-shape scatterer. Left: exact. Middle: $ 10\% $ noise. Right: $ 30\% $ noise
[1]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems & Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033

[2]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709

[3]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[4]

Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems & Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036

[5]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[6]

Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547

[7]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[8]

Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051

[9]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951

[10]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

[11]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

[12]

Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435

[13]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[14]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

[15]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[16]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263

[17]

Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems & Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023

[18]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431

[19]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257

[20]

Keji Liu. Scattering by impenetrable scatterer in a stratified ocean waveguide. Inverse Problems & Imaging, 2019, 13 (6) : 1349-1365. doi: 10.3934/ipi.2019059

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (134)
  • HTML views (255)
  • Cited by (0)

Other articles
by authors

[Back to Top]