On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging

Won-Kwang Park; Won-Kwang Park

doi:10.3934/math.20241037

AIMS Mathematics

2024, Volume 9, Issue 8: 21356-21382. doi: 10.3934/math.20241037

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On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging

Won-Kwang Park ^,

Department of Information Security, Cryptology, and Mathematics, Kookmin University, Seoul, 02707, Korea

Received: 01 April 2024 Revised: 18 June 2024 Accepted: 21 June 2024 Published: 02 July 2024
MSC : 78A46

We apply subspace migration (SM) for fast identification of a small object in microwave imaging. Most research in this area is performed under the assumption that the diagonal elements of the scattering matrix can be easily measured if the transmitter and the receiver are in the same location. Unfortunately, it is very difficult to measure such elements in most real-world microwave imaging. To address this issue, several studies have been conducted with the unknown diagonal elements set to zero. In this paper, we generalize the imaging problem by using SM to set the diagonal elements of the scattering matrix to a constant. To demonstrate the applicability of SM and its dependence on the constant, we show that the imaging function of SM can be represented by an infinite series of Bessel functions of integer order, antenna number and arrangement, and the applied constant. This result allows us to discover additional properties, such as the unique determination of the object. We also demonstrated simulation results using synthetic data to back up the theoretical result.
- microwave imaging,
- subspace migration,
- scattering matrix,
- Bessel functions,
- simulation results
Citation: Won-Kwang Park. On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging[J]. AIMS Mathematics, 2024, 9(8): 21356-21382. doi: 10.3934/math.20241037

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Abstract

We apply subspace migration (SM) for fast identification of a small object in microwave imaging. Most research in this area is performed under the assumption that the diagonal elements of the scattering matrix can be easily measured if the transmitter and the receiver are in the same location. Unfortunately, it is very difficult to measure such elements in most real-world microwave imaging. To address this issue, several studies have been conducted with the unknown diagonal elements set to zero. In this paper, we generalize the imaging problem by using SM to set the diagonal elements of the scattering matrix to a constant. To demonstrate the applicability of SM and its dependence on the constant, we show that the imaging function of SM can be represented by an infinite series of Bessel functions of integer order, antenna number and arrangement, and the applied constant. This result allows us to discover additional properties, such as the unique determination of the object. We also demonstrated simulation results using synthetic data to back up the theoretical result.

References

[1]	H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, vol. 62 of Mathematics and Applications Series, Springer, Berlin, 2008. https://doi.org/10.1007/978-3-540-79553-7
[2]	R. Chandra, A. J. Johansson, M. Gustafsson, F. Tufvesson, A microwave imaging-based technique to localize an in-body RF source for biomedical applications, IEEE T. Bio-Med. Eng., 62 (2015), 1231–1241. https://doi.org/10.1109/TBME.2014.2367117 doi: 10.1109/TBME.2014.2367117
[3]	M. Haynes, J. Stang, M. Moghaddam, Real-time microwave imaging of differential temperature for thermal therapy monitoring, IEEE T. Bio-Med. Eng., 61 (2014), 1787–1797. https://doi.org/10.1109/TBME.2014.2307072 doi: 10.1109/TBME.2014.2307072
[4]	J. Y. Kim, K. J. Lee, B. R. Kim, S. I. Jeon, S. H. Son, Numerical and experimental assessments of focused microwave thermotherapy system at 925MHz, ETRI J., 41 (2019), 850–862. https://doi.org/10.4218/etrij.2018-0088 doi: 10.4218/etrij.2018-0088
[5]	L. Collins, P. Gao, D. Schofield, J. Moulton, L. Majakowsky, L. Reidy, et al., A statistical approach to landmine detection using broadband electromagnetic data, IEEE T. Geosci. Remote, 40 (2002), 950–962. https://doi.org/10.1109/TGRS.2002.1006387 doi: 10.1109/TGRS.2002.1006387
[6]	P. Gao, L. Collins, P. M. Garber, N. Geng, L. Carin, Classification of landmine-like metal targets using wideband electromagnetic induction, IEEE T. Geosci. Remote Sens., 38 (2000), 1352–1361. https://doi.org/10.1109/ICASSP.1999.758404 doi: 10.1109/ICASSP.1999.758404
[7]	Y. J. Kim, L. Jofre, F. D. Flaviis, M. Q. Feng, Microwave reflection tomographic array for damage detection of civil structures, IEEE T. Antenn. Propag., 51 (2003), 3022–3032. https://doi.org/10.1109/TAP.2003.818786 doi: 10.1109/TAP.2003.818786
[8]	C. B. Smith, E. M. Hernandez, Non-negative constrained inverse eigenvalue problems–application to damage identification, Mech. Syst. Signal Proc., 129 (2019), 629–644. https://doi.org/10.1016/j.ymssp.2019.04.052 doi: 10.1016/j.ymssp.2019.04.052
[9]	V. S. Chernyak, Fundamentals of Multisite Radar Systems: Multistatic Radars and Multiradar Systems, CRC Press, Routledge, 1998. https://doi.org/10.1201/9780203755228
[10]	I. Stojanovic, W. C. Karl, Imaging of moving targets with multi-static SAR using an overcomplete dictionary, IEEE J.-STSP., 4 (2010), 164–176. https://doi.org/10.1109/JSTSP.2009.2038982 doi: 10.1109/JSTSP.2009.2038982
[11]	T. Rubæk, P. M. Meaney, P. Meincke, K. D. Paulsen, Nonlinear microwave imaging for breast-cancer screening using Gauss–Newton's method and the CGLS inversion algorithm, IEEE T. Antenn. Propag., 55 (2007), 2320–2331. https://doi.org/10.1109/TAP.2007.901993 doi: 10.1109/TAP.2007.901993
[12]	D. Ireland, K. Bialkowski, A. Abbosh, Microwave imaging for brain stroke detection using Born iterative method, IET Microw. Antenna. P., 7 (2013), 909–915. https://doi.org/10.1049/iet-map.2013.0054 doi: 10.1049/iet-map.2013.0054
[13]	G. Oliveri, N. Anselmi, A. Massa, Compressive sensing imaging of non-sparse 2D scatterers by a total-variation approach within the Born approximation, IEEE T. Antenn. Propag., 62 (2014), 5157–5170. https://doi.org/10.1109/TAP.2014.2344673 doi: 10.1109/TAP.2014.2344673
[14]	A. Franchois, C. Pichot, Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method, IEEE T. Antenn. Propag., 45 (1997), 203–215. https://doi.org/10.1109/8.560338 doi: 10.1109/8.560338
[15]	O. Dorn, D. Lesselier, Level set methods for inverse scattering, Inverse Probl., 22 (2006), R67–R131. https://doi.org/10.1088/0266-5611/22/4/R01 doi: 10.1088/0266-5611/22/4/R01
[16]	O. Kwon, J. K. Seo, J. R. Yoon, A real-time algorithm for the location search of discontinuous conductivities with one measurement, Comm. Pur. Appl. Math., 55 (2002), 1–29. https://doi.org/10.1002/cpa.3009 doi: 10.1002/cpa.3009
[17]	W. K. Park, D. Lesselier, Reconstruction of thin electromagnetic inclusions by a level set method, Inverse Probl., 25 (2009), Article No. 085010. https://doi.org/10.1088/0266-5611/25/8/085010 doi: 10.1088/0266-5611/25/8/085010
[18]	S. H. Son, W. K. Park, Application of the bifocusing method in microwave imaging without background information, J. Korean Soc. Ind. Appl. Math., 27 (2023), 109–122. https://doi.org/10.12941/jksiam.2023.27.109 doi: 10.12941/jksiam.2023.27.109
[19]	H. Ammari, H. Kang, E. Kim, K. Louati, M. Vogelius, A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements, Numer. Math., 108 (2008), 501–528. https://doi.org/10.1007/s00211-007-0130-x doi: 10.1007/s00211-007-0130-x
[20]	W. K. Park, Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mech. Syst. Signal Proc., 153 (2021), Article No. 107501. https://doi.org/10.1016/j.ymssp.2020.107501 doi: 10.1016/j.ymssp.2020.107501
[21]	Y. T. Chow, K. Ito, K. Liu, J. Zou, Direct sampling method for diffusive optical tomography, SIAM J. Sci. Comput., 37 (2015), A1658–A1684. https://doi.org/10.1137/14097519X doi: 10.1137/14097519X
[22]	Y. T. Chow, K. Ito, J. Zou, A direct sampling method for electrical impedance tomography, Inverse Probl., 30 (2014), Article No. 095003. https://doi.org/10.1088/0266-5611/30/9/095003 doi: 10.1088/0266-5611/30/9/095003
[23]	S. Amstutz, N. Dominguez, Topological sensitivity analysis in the context of ultrasonic non-destructive testing, Eng. Anal. Bound. Elem., 32 (2008), 936–947. https://doi.org/10.1016/j.enganabound.2007.09.008 doi: 10.1016/j.enganabound.2007.09.008
[24]	F. L. Louër, M. L. Rapún, Detection of multiple impedance obstacles by non-iterative topological gradient based methods, J. Comput. Phys., 388 (2019), 534–560. https://doi.org/10.1016/j.jcp.2019.03.023 doi: 10.1016/j.jcp.2019.03.023
[25]	W. K. Park, A novel study on the orthogonality sampling method in microwave imaging without background information, Appl. Math. Lett., 145 (2023), Article No. 108766. https://doi.org/10.1016/j.aml.2023.108766 doi: 10.1016/j.aml.2023.108766
[26]	T. Le, D. L. Nguyen, H. Schmidt, T. Truong, Imaging of 3D objects with experimental data using orthogonality sampling methods, Inverse Probl., 38 (2021), Article No. 025007. https://doi.org/10.1088/1361-6420/ac3d85 doi: 10.1088/1361-6420/ac3d85
[27]	S. Coşğun, E. Bilgin, M. Çayören, Microwave imaging of breast cancer with factorization method: SPIONs as contrast agent, Med. Phys., 47 (2020), 3113–3122. https://doi.org/10.1002/mp.14156 doi: 10.1002/mp.14156
[28]	B. Harrach, J. K. Seo, E. J. Woo, Factorization method and its physical justification in frequency-difference electrical impedance tomography, IEEE T. Biomed. Eng., 29 (2010), 1918–1926. https://doi.org/10.1109/tmi.2010.2053553 doi: 10.1109/tmi.2010.2053553
[29]	H. F. Alqadah, A compressive multi-frequency linear sampling method for underwater acoustic imaging, IEEE T. Image Process., 25 (2016), 2444–2455. https://doi.org/10.1109/TIP.2016.2548243 doi: 10.1109/TIP.2016.2548243
[30]	M. G. Aram, M. Haghparast, M. S. Abrishamian, A. Mirtaheri, Comparison of imaging quality between linear sampling method and time reversal in microwave imaging problems, Inverse Probl. Sci. Eng., 24 (2016), 1347–1363. https://doi.org/10.1080/17415977.2015.1104308 doi: 10.1080/17415977.2015.1104308
[31]	H. Ammari, J. Garnier, H. Kang, M. Lim, K. Sølna, Multistatic imaging of extended targets, SIAM J. Imag. Sci., 5 (2012), 564–600. https://doi.org/10.1137/10080631X doi: 10.1137/10080631X
[32]	L. Borcea, G. Papanicolaou, F. G. Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imag. Sci., 1 (2008), 75–114. https://doi.org/10.1137/07069290X doi: 10.1137/07069290X
[33]	W. K. Park, On the identification of small anomaly in microwave imaging without homogeneous background information, AIMS Math., 8 (2023), 27210–27226. https://doi.org/10.3934/math.20231392 doi: 10.3934/math.20231392
[34]	H. Ammari, J. Garnier, H. Kang, W. K. Park, K. Sølna, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math., 71 (2011), 68–91. https://doi.org/10.1137/100800130 doi: 10.1137/100800130
[35]	W. K. Park, Multi-frequency subspace migration for imaging of perfectly conducting, arc-like cracks in full- and limited-view inverse scattering problems, J. Comput. Phys., 283 (2015), 52–80. https://doi.org/10.1016/j.jcp.2014.11.036 doi: 10.1016/j.jcp.2014.11.036
[36]	W. K. Park, Shape identification of open sound-hard arcs without priori information in limited-view inverse scattering problem, Comput. Math. Appl., 128 (2022), 55–68. https://doi.org/10.1016/j.camwa.2022.10.010 doi: 10.1016/j.camwa.2022.10.010
[37]	W. K. Park, Fast location search of small anomaly by using microwave, Int. J. Appl. Electromagn. Mech., 59 (2019), 1505–1510. https://doi.org/10.3233/JAE-171107 doi: 10.3233/JAE-171107
[38]	W. K. Park, Real-time microwave imaging of unknown anomalies via scattering matrix, Mech. Syst. Signal Proc., 118 (2019), 658–674. https://doi.org/10.1016/j.ymssp.2018.09.012 doi: 10.1016/j.ymssp.2018.09.012
[39]	W. K. Park, Real-time detection of small anomaly from limited-aperture measurements in real-world microwave imaging, Mech. Syst. Signal Proc., 171 (2022), Article No. 108937. https://doi.org/10.1016/j.ymssp.2022.108937 doi: 10.1016/j.ymssp.2022.108937
[40]	S. H. Son, H. J. Kim, K. J. Lee, J. Y. Kim, J. M. Lee, S. I. Jeon, et al., Experimental measurement system for 3–6GHz microwave breast tomography, J. Electromagn. Eng. Sci., 15 (2015), 250–257. https://doi.org/10.5515/JKIEES.2015.15.4.250 doi: 10.5515/JKIEES.2015.15.4.250
[41]	S. H. Son, K. J. Lee, W. K. Park, Real-time tracking of moving objects from scattering matrix in real-world microwave imaging, AIMS Math., 9 (2024), 13570–13588. https://doi.org/10.3934/math.2024662 doi: 10.3934/math.2024662
[42]	S. H. Son, N. Simonov, H. J. Kim, J. M. Lee, S. I. Jeon, Preclinical prototype development of a microwave tomography system for breast cancer detection, ETRI J., 32 (2010), 901–910. https://doi.org/10.4218/etrij.10.0109.0626 doi: 10.4218/etrij.10.0109.0626
[43]	S. Kang, W. K. Park, S. H. Son, A qualitative analysis of the bifocusing method for a real-time anomaly detection in microwave imaging, Comput. Math. Appl., 137 (2023), 93–101. https://doi.org/10.1016/j.camwa.2023.02.017 doi: 10.1016/j.camwa.2023.02.017
[44]	K. J. Lee, S. H. Son, W. K. Park, A real-time microwave imaging of unknown anomaly with and without diagonal elements of scattering matrix, Results Phys., 17 (2020), Article No. 103104. https://doi.org/10.1016/j.rinp.2020.103104 doi: 10.1016/j.rinp.2020.103104
[45]	W. K. Park, On the application of orthogonality sampling method for object detection in microwave imaging, IEEE T. Antenn. Propag., 71 (2023), 934–946. https://doi.org/10.1109/TAP.2022.3220033 doi: 10.1109/TAP.2022.3220033
[46]	S. H. Son, K. J. Lee, W. K. Park, Application and analysis of direct sampling method in real-world microwave imaging, Appl. Math. Lett., 96 (2019), 47–53. https://doi.org/10.1016/j.aml.2019.04.016 doi: 10.1016/j.aml.2019.04.016
[47]	M. Slaney, A. C. Kak, L. E. Larsen, Limitations of imaging with first-order diffraction tomography, IEEE T. Microw. Theory, 32 (1984), 860–874. https://doi.org/10.1109/TMTT.1984.1132783 doi: 10.1109/TMTT.1984.1132783
[48]	D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Problems, vol. 93 of Mathematics and Applications Series, Springer, New York, 1998. https://doi.org/10.1007/978-3-030-30351-8
[49]	L. J. Landau, Bessel functions: monotonicity and bounds, J. London Math. Soc., 61 (2000), 197–215. https://doi.org/10.1112/S0024610799008352 doi: 10.1112/S0024610799008352
[50]	M. H. Ding, H. Liu, G. H. Zheng, On inverse problems for several coupled PDF systems arising in mathematical biology, J. Math. Biology, 87 (2023), Article No. 86. https://doi.org/10.1007/s00285-023-02021-4 doi: 10.1007/s00285-023-02021-4
[51]	H. Liu, C. W. K. Lo, Determining a parabolic system by boundary observation of its non-negative solutions with biological applications, Inverse Probl., 40 (2024), Article No. 025009. https://doi.org/10.1088/1361-6420/ad149f doi: 10.1088/1361-6420/ad149f
[52]	Y. Yin, W. Yin, P. Meng, H. Liu, The interior inverse scattering problem for a two-layered cavity using the Bayesian method, Inverse Probl. Imag., 16 (2022), 673–690. https://doi.org/10.3934/ipi.2021069 doi: 10.3934/ipi.2021069

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