Bifocusing method for localizing small objects with an incomplete multi-static response matrix from transverse electric polarized waves

Taeyoung Ha; Sangwoo Kang; Minyeob Lee; Won-Kwang Park; Seong-Ho Son; Taeyoung Ha; Sangwoo Kang; Minyeob Lee; Won-Kwang Park; Seong-Ho Son

doi:10.3934/math.20261024

AIMS Mathematics

2026, Volume 11, Issue 8: 25526-25552. doi: 10.3934/math.20261024

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Bifocusing method for localizing small objects with an incomplete multi-static response matrix from transverse electric polarized waves

1.
Division of Industrial Mathematics, National Institute for Mathematical Sciences, Daejeon 34047, Korea
2.
Graduate School of Data Science, Pusan National University, Pusan 46241, Korea
3.
Department of Mathematics, Kookmin University, Seoul 02707, Korea
4.
Department of ICT Convergence, Soonchunhyang University, Asan 31538, Korea
5.
Department of Mechanical Engineering, Soonchunhyang University, Asan 31538, Korea
^†These authors contributed equally to this work and share first authorship

Received: 23 June 2026 Revised: 28 July 2026 Accepted: 11 August 2026 Published: 18 August 2026
MSC : 78A46

This work investigated the application of the bifocusing method (BFM) in inverse scattering problems to detect small objects using transverse electric polarized waves. In practical experimental setups, restrictions on the scanning trajectory of receivers often lead to an incomplete multi-static response (MSR) matrix. Moreover, the direct application of the standard BFM imaging function-designed based on the structure of the measured scattered field-is mathematically problematic since its denominator can become zero at specific evaluation points. To overcome these limitations, we proposed a modified BFM-based imaging function and derived an asymptotic representation as an infinite series comprising Bessel functions of the first kind, the objects' material properties, and the receivers' angular aperture. This result elucidates the dependency of the imaging results on the sizes and permeabilities of the targets, and provides a logical basis for the frequency-dependent trade-off between image resolution and artifact generation. The validity of the theoretical result was corroborated through numerical simulations using noise-corrupted synthetic data for objects with diverse sizes and material properties, as well as a 2D Fresnel experimental dataset.
- bifocusing method,
- inverse scattering problems,
- transverse electric polarization,
- Bessel functions,
- multi-static response matrix,
- small-object detection,
- imaging functional
Citation: Taeyoung Ha, Sangwoo Kang, Minyeob Lee, Won-Kwang Park, Seong-Ho Son. Bifocusing method for localizing small objects with an incomplete multi-static response matrix from transverse electric polarized waves[J]. AIMS Mathematics, 2026, 11(8): 25526-25552. doi: 10.3934/math.20261024

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Abstract

This work investigated the application of the bifocusing method (BFM) in inverse scattering problems to detect small objects using transverse electric polarized waves. In practical experimental setups, restrictions on the scanning trajectory of receivers often lead to an incomplete multi-static response (MSR) matrix. Moreover, the direct application of the standard BFM imaging function-designed based on the structure of the measured scattered field-is mathematically problematic since its denominator can become zero at specific evaluation points. To overcome these limitations, we proposed a modified BFM-based imaging function and derived an asymptotic representation as an infinite series comprising Bessel functions of the first kind, the objects' material properties, and the receivers' angular aperture. This result elucidates the dependency of the imaging results on the sizes and permeabilities of the targets, and provides a logical basis for the frequency-dependent trade-off between image resolution and artifact generation. The validity of the theoretical result was corroborated through numerical simulations using noise-corrupted synthetic data for objects with diverse sizes and material properties, as well as a 2D Fresnel experimental dataset.

References

[1]	R. C. Aster, B. Borchers, C. H. Thurber, Parameter Estimation and Inverse Problems, 2 Eds, Amsterdam: Elsevier, 2013. https://doi.org/10.1016/C2009-0-61134-X
[2]	D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4 Eds, Berlin: Springer, 2019. https://doi.org/10.1007/978-3-030-30351-8
[3]	N. K. Nikolova, Introduction to Microwave Imaging, Cambridge: Cambridge University Press, 2017. https://doi.org/10.1017/9781316084267
[4]	E. Üregen, A. Yapar, Focusing-based Newton solution for electromagnetic inverse scattering problems, IEEE Trans. Antennas Propag., 72 (2024), 8611–8620. https://doi.org/10.1109/TAP.2024.3460193 doi: 10.1109/TAP.2024.3460193
[5]	Y. Qu, H. Dong, An inverse scattering problem for an open sound-hard crack from phaseless data, Inverse Probl. Imag., 23 (2026), 20–50. https://doi.org/10.3934/ipi.2025053 doi: 10.3934/ipi.2025053
[6]	H. Muramatsu, S. Kidera, TDOA prior-based contrast source inversion for microwave ablation zone quantitative imaging, IEEE Trans. Antennas Propag., 73 (2025), 3986–3999. https://doi.org/10.1109/TAP.2025.3538218 doi: 10.1109/TAP.2025.3538218
[7]	S. H. Kim, L. Pinar, A. H. Hielsher, Accelerated microwave tomographic imaging with a PDE-constrained optimization method, IEEE Trans. Comput. Imag., 11 (2025), 1614–1629. https://doi.org/10.1109/TCI.2025.3629414 doi: 10.1109/TCI.2025.3629414
[8]	H. Ammari, P. Garapon, F. Jouve, H. Kang, M. Lim, S. Yu, A new optimal control approach for the reconstruction of extended inclusions, SIAM J. Control. Optim., 51 (2013), 1372–1394. https://doi.org/10.1137/100808952 doi: 10.1137/100808952
[9]	Y. Chang, Y. Guo, D. Zhang, A forward-model-free iterative method for the inverse obstacle scattering problems in three dimensions, Inverse Probl. Imag., 25 (2026), 94–112. https://doi.org/10.3934/ipi.2026017 doi: 10.3934/ipi.2026017
[10]	J. K. Seo, E. J. Woo, Magnetic resonance electrical impedance tomography (MREIT), SIAM Rev., 53 (2011), 40–68. https://doi.org/10.1137/080742932 doi: 10.1137/080742932
[11]	V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. H. Nguyen, A. J. Sullivan, et al., An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data, Inverse Probl. Sci. Eng., 29 (2021), 712–735. https://doi.org/10.1080/17415977.2020.1802447 doi: 10.1080/17415977.2020.1802447
[12]	L. Ballerini, Particle swarm optimization in 3D medical image registration: a systematic review, Arch. Comput. Meth. Eng., 32 (2025), 311–318. https://doi.org/10.1007/s11831-024-10139-x doi: 10.1007/s11831-024-10139-x
[13]	Y. Wang, L. Lin, J. Lv, Numerical method for the inverse scattering problem of acoustic-elastic interaction by random periodic structure, J. Comput. Phys., 562 (2026), 115028. https://doi.org/10.1016/j.jcp.2026.115028 doi: 10.1016/j.jcp.2026.115028
[14]	O. Kwon, J. K. Seo, J. R. Yoon, A real-time algorithm for the location search of discontinuous conductivities with one measurement, Commun. Pur. Appl. Math., 55 (2002), 1–29. https://doi.org/10.1002/cpa.3009 doi: 10.1002/cpa.3009
[15]	W. K. Park, D. Lesselier, Reconstruction of thin electromagnetic inclusions by a level set method, Inverse Probl., 25 (2009), 085010. https://doi.org/10.1088/0266-5611/25/8/085010 doi: 10.1088/0266-5611/25/8/085010
[16]	K. Ito, B. Jin, J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Probl., 28 (2012), 025003. https://doi.org/10.1088/0266-5611/28/2/025003 doi: 10.1088/0266-5611/28/2/025003
[17]	S. Kang, M. Lambert, W. K. Park, Direct sampling method for imaging small dielectric inhomogeneities: analysis and improvement, Inverse Probl., 34 (2018), 095005. https://doi.org/10.1088/1361-6420/aacf1d doi: 10.1088/1361-6420/aacf1d
[18]	W. K. Park, Direct sampling method for retrieving small perfectly conducting cracks, J. Comput. Phys., 373 (2018), 648–661. https://doi.org/10.1016/j.jcp.2018.07.014 doi: 10.1016/j.jcp.2018.07.014
[19]	A. Kirsch, The factorization method for Maxwell's equations, Inverse Probl., 20 (2004), 117–134. https://doi.org/10.1088/0266-5611/20/6/S08 doi: 10.1088/0266-5611/20/6/S08
[20]	L. Borcea, S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Probl., 35 (2019), 124006. https://doi.org/10.1088/1361-6420/ab2c9b doi: 10.1088/1361-6420/ab2c9b
[21]	K. H. Leem, J. Liu, G. Pelekanos, An extended direct factorization method for inverse scattering with limited aperture data, Inverse Probl. Sci. Eng., 28 (2020), 754–776. https://doi.org/10.1080/17415977.2019.1647195 doi: 10.1080/17415977.2019.1647195
[22]	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
[23]	W. K. Park, Real-time detection of small anomaly from limited-aperture measurements in real-world microwave imaging, Mech. Syst. Signal Proc., 171 (2022), 108937. https://doi.org/10.1016/j.ymssp.2022.108937 doi: 10.1016/j.ymssp.2022.108937
[24]	W. K. Park, On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging, AIMS Math., 9 (2024), 21356–21382. https://doi.org/10.3934/math.20241037 doi: 10.3934/math.20241037
[25]	H. Ammari, E. Iakovleva, D. Lesselier, G. Perrusson, MUSIC type electromagnetic imaging of a collection of small three-dimensional inclusions, SIAM J. Sci. Comput., 29 (2007), 674–709. https://doi.org/10.1137/050640655 doi: 10.1137/050640655
[26]	W. K. Park, Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mech. Syst. Signal Proc., 153 (2021), 107501. https://doi.org/10.1016/j.ymssp.2020.107501 doi: 10.1016/j.ymssp.2020.107501
[27]	W. K. Park, Application and analysis of MUSIC algorithm for anomaly detection in microwave imaging without a switching device, Mech. Syst. Signal Proc., 243 (2026), 113705. https://doi.org/10.1016/j.ymssp.2025.113705 doi: 10.1016/j.ymssp.2025.113705
[28]	A. Kirsch, S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Probl., 16 (2000), 89–105. https://doi.org/10.1088/0266-5611/16/1/308 doi: 10.1088/0266-5611/16/1/308
[29]	H. F. Alqadah, A compressive multi-frequency linear sampling method for underwater acoustic imaging, IEEE Trans. Image Process., 25 (2016), 2444–2455. https://doi.org/10.1109/TIP.2016.2548243 doi: 10.1109/TIP.2016.2548243
[30]	P. Monk, M. Pena, V. Selgas, Multifrequency linear sampling method on experimental datasets, IEEE Trans. Antennas Propag., 71 (2023), 8788–8798. https://doi.org/10.1109/TAP.2023.3298974 doi: 10.1109/TAP.2023.3298974
[31]	R. Potthast, A study on orthogonality sampling, Inverse Probl., 26 (2010), 074015. https://doi.org/10.1088/0266-5611/26/7/074015 doi: 10.1088/0266-5611/26/7/074015
[32]	A. Cuccaro, A. Dell'Aversano, M. A. Maisto, G. Leone, R. Solimene, Microwave imaging of small scatterers: linking and comparing the beam-forming and the orthogonality sampling method, IEEE Trans. Antennas Propag., 73 (2025), 3177–3190. https://doi.org/10.1109/TAP.2025.3531661 doi: 10.1109/TAP.2025.3531661
[33]	W. K. Park, Inversion of two-dimensional Fresnel experimental dataset using orthogonality sampling method with single and multiple sources: the case of transverse magnetic polarized waves, Commun. Anal. Mech., 18 (2026), 142–171. https://doi.org/10.3934/cam.2026006 doi: 10.3934/cam.2026006
[34]	F. L. Louër, M. L. Rapún, Topological sensitivity for solving inverse multiple scattering problems in 3D electromagnetism. Part Ⅰ: one step method, SIAM J. Imag. Sci., 10 (2017), 1291–1321. https://doi.org/10.1137/17M1113850 doi: 10.1137/17M1113850
[35]	A. Carpio, M. Pena, M. L. Rapún, Processing the 2D and 3D Fresnel experimental databases via topological derivative methods, Inverse Probl., 37 (2021), 105012. https://doi.org/10.1088/1361-6420/ac21c8 doi: 10.1088/1361-6420/ac21c8
[36]	W. K. Park, Performance analysis of multi-frequency topological derivative for reconstructing perfectly conducting cracks, J. Comput. Phys., 335 (2017), 865–884. https://doi.org/10.1016/j.jcp.2017.02.007 doi: 10.1016/j.jcp.2017.02.007
[37]	Y. J. Kim, L. Jofre, F. D. Flaviis, M. Q. Feng, Microwave reflection tomographic array for damage detection of civil structures, IEEE Trans. Antennas Propag., 51 (2003), 3022–3032. https://doi.org/10.1109/TAP.2003.818786 doi: 10.1109/TAP.2003.818786
[38]	L. Jofre, A. Broquetas, J. Romeu, S. Blanch, A. P. Toda, X. Fabregas, et al., UWB tomographic radar imaging of penetrable and impenetrable objects, Proc. IEEE, 97 (2009), 451–464. https://doi.org/10.1109/JPROC.2008.2008854 doi: 10.1109/JPROC.2008.2008854
[39]	L. Jofre, A. P. Toda, J. M. J. Montana, P. C. Carrascosa, J. Romeu, S. Blanch, et al., UWB short-range bifocusing tomographic imaging, IEEE Trans. Instrum. Meas., 57 (2008), 2414–2420. https://doi.org/10.1109/TIM.2008.926382 doi: 10.1109/TIM.2008.926382
[40]	S. Kang, W. K. Park, S. H. Son, A qualitative analysis of 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
[41]	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
[42]	S. Kang, M. Lee, W. K. Park, S. H. Son, Application of the bifocusing method in microwave imaging by converting unknown measurement data into the constant, J. Korean Soc. Ind. Appl. Math., 28 (2024), 96–108. https://doi.org/10.12941/jksiam.2024.28.096 doi: 10.12941/jksiam.2024.28.096
[43]	J. Jeong, S. H. Son, Localization of small moving objects using only total fields in microwave imaging, Res. Phys., 61 (2024), 107777. https://doi.org/10.1016/j.rinp.2024.107777 doi: 10.1016/j.rinp.2024.107777
[44]	S. Kang, W. K. Park, A novel study on the bifocusing method in two-dimensional inverse scattering problem, AIMS Math., 8 (2023), 27080–27112. https://doi.org/10.3934/math.20231386 doi: 10.3934/math.20231386
[45]	S. Kang, M. Lee, W. K. Park, S. H. Son, Bifocusing-based imaging of small penetrable dielectric objects from Fresnel dataset, IEEE Trans. Antennas Propag., 73 (2025), 4795–4803. https://doi.org/10.1109/TAP.2025.3550328 doi: 10.1109/TAP.2025.3550328
[46]	W. K. Park, On the application of orthogonality sampling method for object detection in microwave imaging, IEEE Trans. Antennas Propag., 71 (2023), 934–946. https://doi.org/10.1109/TAP.2022.3220033 doi: 10.1109/TAP.2022.3220033
[47]	K. Belkebir, M. Saillard, Special section: Testing inversion algorithms against experimental data, Inverse Probl., 17 (2001), 1565–1571. https://doi.org/10.1088/0266-5611/21/6/S01 doi: 10.1088/0266-5611/21/6/S01
[48]	H. Ammari, H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Berlin: Springer-Verlag, 2004. https://doi.org/10.1007/b98245
[49]	T. Rao, X. Chen, Analysis of the time-reversal operator for a single cylinder under two-dimensional settings, J. Electromagn. Waves Appl., 20 (2006), 2153–2165. https://doi.org/10.1163/156939306779322503 doi: 10.1163/156939306779322503
[50]	S. Hou, K. Sølna, H. Zhao, A direct imaging algorithm for extended targets, Inverse Probl., 22 (2006), 1151–1178. https://doi.org/10.1088/0266-5611/22/4/003 doi: 10.1088/0266-5611/22/4/003
[51]	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
[52]	M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1996.

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