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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 and Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012
##### References:
 [1] T. Arens, D. 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. [2] A. B. Baggeroer, W. A. Kuperman and P. N. Mikhalevsky, An overview of matched field methods in ocean acoustics, IEEE Journal of Oceanic Engineering, 18 (1993), 401-424. [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. [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. [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. [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. [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. [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. [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. [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. [11] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, volume 88., SIAM, 2016. doi: 10.1137/1.9781611974461.ch1. [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. [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. [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. [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. [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. [17] A. Haack, J. 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. [18] A. Kirsch, The factorization method for maxwell's equations, Inverse Problems, 20 (2004), S117–S134. doi: 10.1088/0266-5611/20/6/S08. [19] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36., Oxford University Press, 2008. [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. [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. [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. [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. [24] R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015. doi: 10.1088/0266-5611/26/7/074015. [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. [26] J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, 1 (1997), 41-52. [27] J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652. [28] C. Tsogka, D. 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. [29] Y. Xu, C. 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.

show all references

##### References:
 [1] T. Arens, D. 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. [2] A. B. Baggeroer, W. A. Kuperman and P. N. Mikhalevsky, An overview of matched field methods in ocean acoustics, IEEE Journal of Oceanic Engineering, 18 (1993), 401-424. [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. [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. [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. [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. [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. [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. [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. [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. [11] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, volume 88., SIAM, 2016. doi: 10.1137/1.9781611974461.ch1. [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. [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. [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. [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. [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. [17] A. Haack, J. 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. [18] A. Kirsch, The factorization method for maxwell's equations, Inverse Problems, 20 (2004), S117–S134. doi: 10.1088/0266-5611/20/6/S08. [19] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36., Oxford University Press, 2008. [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. [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. [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. [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. [24] R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015. doi: 10.1088/0266-5611/26/7/074015. [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. [26] J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, 1 (1997), 41-52. [27] J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652. [28] C. Tsogka, D. 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. [29] Y. Xu, C. 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.
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
Three dimensional view of the generated mesh
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
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
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
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
Image of a large ball. Left: exact. Middle: three dimensional image when $k = 1$. Right: three dimensional image when $k = 3$
Iso-surface (with the same iso-value 0.4) image of the L-shape scatterer. Left: exact. Middle: $10\%$ noise. Right: $30\%$ noise
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