This paper considers the application of the orthogonality sampling method (OSM) with single and multiple sources for a fast identification of small objects in the limited-aperture inverse scattering problem. First, we apply the OSM with a single source and demonstrate that the indicator function of the OSM with a single source can be expressed by the Bessel function of order zero of the first kind, an infinite series of Bessel functions of nonzero integer order of the first kind, the range of the signal receiver, and the emitter location. We then explain that the objects can be identified using the OSM with a single source; however, the identification is strongly influenced by the location of the source and the applied frequency. To realize effective improvement, we consider the OSM with multiple sources. Based on the identified structure of the OSM with a single source, we propose an indicator function for the OSM with multiple sources and demonstrate that it can be expressed by the square of the Bessel function of order zero of the first kind and an infinite series of the square of the Bessel function of nonzero integer order of the first kind. This result shows that the locations of objects can be uniquely identified using the designed OSM. Simulation results with experimental data provided by the Institute Fresnel demonstrate the advantages and disadvantages of the OSM with a single source and how the proposed OSM with multiple sources behaves.
Citation: Won-Kwang Park. Inversion of two-dimensional Fresnel experimental dataset using orthogonality sampling method with single and multiple sources: the case of transverse magnetic polarized waves[J]. Communications in Analysis and Mechanics, 2026, 18(1): 142-171. doi: 10.3934/cam.2026006
This paper considers the application of the orthogonality sampling method (OSM) with single and multiple sources for a fast identification of small objects in the limited-aperture inverse scattering problem. First, we apply the OSM with a single source and demonstrate that the indicator function of the OSM with a single source can be expressed by the Bessel function of order zero of the first kind, an infinite series of Bessel functions of nonzero integer order of the first kind, the range of the signal receiver, and the emitter location. We then explain that the objects can be identified using the OSM with a single source; however, the identification is strongly influenced by the location of the source and the applied frequency. To realize effective improvement, we consider the OSM with multiple sources. Based on the identified structure of the OSM with a single source, we propose an indicator function for the OSM with multiple sources and demonstrate that it can be expressed by the square of the Bessel function of order zero of the first kind and an infinite series of the square of the Bessel function of nonzero integer order of the first kind. This result shows that the locations of objects can be uniquely identified using the designed OSM. Simulation results with experimental data provided by the Institute Fresnel demonstrate the advantages and disadvantages of the OSM with a single source and how the proposed OSM with multiple sources behaves.
| [1] | H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, In: Mathematics and Applications Series, Springer, Berlin, 2008. |
| [2] | H. Ammari, Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies, In: Lecture Notes in Mathematics, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-22990-9 |
| [3] |
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
|
| [4] |
N. Simonov, B.-R. Kim, K.-J. Lee, S.-I. Jeon, S.-H. Son, Advanced fast 3-D electromagnetic solver for microwave tomography imaging, IEEE Trans. Med. Imag., 36 (2017), 2160–2170. https://doi.org/10.1109/TMI.2017.2712800 doi: 10.1109/TMI.2017.2712800
|
| [5] |
E. J. Baranoski, Through-wall imaging: Historical perspective and future directions, J. Franklin Inst., 345 (2008), 556–569. https://doi.org/10.1016/j.jfranklin.2008.01.005 doi: 10.1016/j.jfranklin.2008.01.005
|
| [6] |
F. Soldovieri, R. Solimene, F. Ahmad, Sparse tomographic inverse scattering approach for through-the-wall radar imaging, IEEE Trans. Instrum. Meas., 61 (2012), 3340–3350. https://doi.org/10.1109/TIM.2012.2210454 doi: 10.1109/TIM.2012.2210454
|
| [7] |
M. Q. Feng, F. D. Flaviis, Y. J. Kim, Use of microwaves for damage detection of fiber reinforced polymer-wrapped concrete structures, J. Eng. Mech., 128 (2002), 172–183. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:2(172) doi: 10.1061/(ASCE)0733-9399(2002)128:2(172)
|
| [8] |
B. Wang, Q. Zhang, W. Zhao, Fast concrete crack detection method via L2 sparse representation, Electron. Lett., 54 (2018), 752–754. https://doi.org/10.1049/el.2018.0412 doi: 10.1049/el.2018.0412
|
| [9] |
F. Delbary, K. Erhard, R. Kress, R. Potthast, J. Schulz, Inverse electromagnetic scattering in a two-layered medium with an application to mine detection, Inverse Probl., 24 (2008), 015002. https://doi.org/10.1088/0266-5611/24/1/015002 doi: 10.1088/0266-5611/24/1/015002
|
| [10] |
P. Gao, L. Collins, P. M. Garber, N. Geng, L. Carin, Classification of landmine-like metal targets using wideband electromagnetic induction, IEEE Trans. Geosci. Remote Sens.Sens., 38 (2000), 1352–1361. https://doi.org/10.1109/36.843029 doi: 10.1109/36.843029
|
| [11] |
M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301–312. https://doi.org/10.1137/S0036144500368859 doi: 10.1137/S0036144500368859
|
| [12] | V. S. Chernyak, Fundamentals of Multisite Radar Systems: Multistatic Radars and Multiradar Systems, CRC Press, Routledge, 1998. |
| [13] |
J. Garnier, K. Sølna, Coherent interferometric imaging for synthetic aperture radar in the presence of noise, Inverse Probl., 24 (2008), 055001. https://doi.org/10.1088/0266-5611/24/5/055001 doi: 10.1088/0266-5611/24/5/055001
|
| [14] | Y. Á. López, M. García-Fernández, G. Álvarez-Narciandi, F. L. H. Andrés, Unmanned aerial vehicle-based ground-penetrating radar systems: A review, IEEE Geosci. Remote Sens. Mag., 10 (2022), 66–86. https://doi.org/10.1109/MGRS.2022.3160664 |
| [15] |
X. Liu, M. Serhir, M. Lambert, Detectability of underground electrical cables junction with a ground penetrating radar: electromagnetic simulation and experimental measurements, Constr. Build. Mater., 158 (2018), 1099–1110. https://doi.org/10.1016/j.conbuildmat.2017.10.038 doi: 10.1016/j.conbuildmat.2017.10.038
|
| [16] |
J. Jeong, W.-K. Park, S.-H. Son, Object localization in highly cluttered environments using neural network learning on microwave scattering data, Microw. Opt. Technol. Lett., 66 (2024), e70020. https://doi.org/10.1002/mop.70020 doi: 10.1002/mop.70020
|
| [17] |
J. Jin, F. Feng, W. Na, S. Yan, W. Liu, L. Zhu, et al., Recent advances in neural network-based inverse modeling techniques for microwave applications, Int. J. Numer. Model.: Electron. Netw. Devices Fields, 33 (2020), e2732. https://doi.org/10.1002/jnm.2732 doi: 10.1002/jnm.2732
|
| [18] |
P. Meng, Z. Xu, X. Wang, W. Yin, H. Liu, A novel method for solving the inverse spectral problem with incomplete data, J. Comput. Appl. Math., 463 (2025), 116525. https://doi.org/10.1016/j.cam.2025.116525 doi: 10.1016/j.cam.2025.116525
|
| [19] |
W. Yin, Z. Shen, P. Meng, H. Liu, An online interactive physics-informed adversarial network for solving mean field games, Eng. Anal. Bound. Elem., 169 (2024), 106002. https://doi.org/10.1016/j.enganabound.2024.106002 doi: 10.1016/j.enganabound.2024.106002
|
| [20] |
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
|
| [21] |
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
|
| [22] |
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
|
| [23] |
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
|
| [24] | 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. |
| [25] |
C. Y. Ahn, T. Ha, W.-K. Park, Direct sampling method for identifying magnetic inhomogeneities in limited-aperture inverse scattering problem, Comput. Math. Appl., 80 (2020), 2811–2829. https://doi.org/10.1016/j.camwa.2020.10.009 doi: 10.1016/j.camwa.2020.10.009
|
| [26] |
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
|
| [27] | S. Kang, M. Lambert, C. Y. Ahn, T. Ha, W.-K. Park, Single- and multi-frequency direct sampling methods in limited-aperture inverse scattering problem, IEEE Access, 8 (2020), 121637–121649. https://doi.org/10.1109/ACCESS.2020.3006341 |
| [28] |
J. Guo, G. Yan, J. Jin, J. Hu, The factorization method for cracks in inhomogeneous media, Appl. Math., 62 (2017), 509–533. https://doi.org/10.21136/AM.2017.0194-16 doi: 10.21136/AM.2017.0194-16
|
| [29] |
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
|
| [30] |
J. Yang, B. Zhang, H. Zhang, The factorization method for reconstructing a penetrable obstacle with unknown buried objects, SIAM J. Appl. Math., 73 (2012), 617–635. https://doi.org/10.1137/120883724 doi: 10.1137/120883724
|
| [31] |
W.-K. Park, Asymptotic properties of MUSIC-type imaging in two-dimensional inverse scattering from thin electromagnetic inclusions, SIAM J. Appl. Math., 75 (2015), 209–228. https://doi.org/10.1137/140975176 doi: 10.1137/140975176
|
| [32] |
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
|
| [33] |
Y. Zhong, X. Chen, MUSIC imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres, IEEE Trans. Antennas Propag., 55 (2007), 3542–3549. https://doi.org/10.1109/TAP.2007.910488 doi: 10.1109/TAP.2007.910488
|
| [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, 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
|
| [36] |
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
|
| [37] |
J. F. Funes, J. M. Perales, M.-L. Rapún, J. M. Vega, Defect detection from multi-frequency limited data via topological sensitivity, J. Math. Imaging Vis., 55 (2016), 19–35. https://doi.org/10.1007/s10851-015-0611-y doi: 10.1007/s10851-015-0611-y
|
| [38] |
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
|
| [39] |
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
|
| [40] |
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
|
| [41] |
R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Probl., 27 (2011), 085005. https://doi.org/10.1088/0266-5611/27/8/085005 doi: 10.1088/0266-5611/27/8/085005
|
| [42] |
C. Y. Ahn, S. Chae, W.-K. Park, Fast identification of short, sound-soft open arcs by the orthogonality sampling method in the limited-aperture inverse scattering problem, Appl. Math. Lett., 109 (2020), 106556. https://doi.org/10.1016/j.aml.2020.106556 doi: 10.1016/j.aml.2020.106556
|
| [43] |
I. Harris, D.-L. Nguyen, Orthogonality sampling method for the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 42 (2020), B722–B737. https://doi.org/10.1137/19M129783X doi: 10.1137/19M129783X
|
| [44] |
M. T. Bevacqua, T. Isernia, R. Palmeri, M. N. Akinci, L. Crocco, Physical insight unveils new imaging capabilities of orthogonality sampling method, IEEE Trans. Antennas Propag., 68 (2020), 4014–4021. https://doi.org/10.1109/TAP.2019.2963229 doi: 10.1109/TAP.2019.2963229
|
| [45] |
T. Le, D.-L. Nguyen, H. Schmidt, T. Truong, Imaging of 3D objects with experimental data using orthogonality sampling methods, Inverse Probl., 38 (2021), 025007. https://doi.org/10.1088/1361-6420/ac3d85 doi: 10.1088/1361-6420/ac3d85
|
| [46] |
K. Belkebir, M. Saillard, Special section: Testing inversion algorithms against experimental data, Inverse Probl., 17 (2001), 1565–1571. https://doi.org/10.1088/0266-5611/17/6/301 doi: 10.1088/0266-5611/17/6/301
|
| [47] |
J.-M. Geffrin, P. Sabouroux, C. Eyraud, Free space experimental scattering database continuation: experimental set-up and measurement precision, Inverse Probl., 21 (2005), S117–S130. https://doi.org/10.1088/0266-5611/21/6/S09 doi: 10.1088/0266-5611/21/6/S09
|
| [48] | D. M. Pozar, Microwave Engineering, 4th edition, John Wiley & Sons, Inc., 2011. |
| [49] | D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Problems, In: Mathematics and Applications Series, Springer, New York, 1998. https://doi.org/10.1007/978-3-662-03537-5 |
| [50] | N. Bleistein, J. W. Stockwell, J. K. Cohen, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, In: Interdisciplinary Applied Mathematics, Springer, New York, 2001. https://doi.org/10.1007/978-1-4613-0001-4 |
| [51] |
M. Slaney, A. C. Kak, L. E. Larsen, Limitations of imaging with first-order diffraction tomography, IEEE Trans. Microw. Theory Techn., 32 (1984), 860–874. https://doi.org/10.1109/TMTT.1984.1132783 doi: 10.1109/TMTT.1984.1132783
|
| [52] |
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
|
| [53] |
R. Solimene, A. Cuccaro, G. Ruvio, D. F. Tapia, M. O'Halloran, Beamforming and holography image formation methods: an analytic study, Opt. Express, 24 (2016), 9077–9093. https://doi.org/10.1364/OE.24.009077 doi: 10.1364/OE.24.009077
|
| [54] |
S. Kang, S. Chae, W.-K. Park, A study on the orthogonality sampling method corresponding to the observation directions configuration, Res. Phys., 33 (2022), 105108. https://doi.org/10.1016/j.rinp.2021.105108 doi: 10.1016/j.rinp.2021.105108
|
| [55] |
I. Harris, D.-L. Nguyen, T.-P. Nguyen, Direct sampling methods for isotropic and anisotropic scatterers with point source measurements, Inverse Probl. Imag., 16 (2022), 1137–1162. https://doi.org/10.3934/ipi.2022015 doi: 10.3934/ipi.2022015
|
| [56] | M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1996. |
| [57] |
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
|
| [58] |
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
|
| [59] |
R. Marklein, K. Balasubramanian, A. Qing, K. J. Langenberg, Linear and nonlinear iterative scalar inversion of multi-frequency multi-bistatic experimental electromagnetic scattering data, Inverse Probl., 17 (2001), 1597–1610. https://doi.org/10.1088/0266-5611/17/6/304 doi: 10.1088/0266-5611/17/6/304
|
| [60] | H. Ammari, J. Garnier, W. Jing, H. Kang, M. Lim, K. Sølna, et al., Mathematical and statistical methods for multistatic imaging, In: Lecture Notes in Mathematics, Springer, Cham, 2013. https://doi.org/10.1007/978-3-319-02585-8 |
| [61] |
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
|
| [62] |
C. Y. Ahn, S. Chae, S. Kang, K.-J. Lee, W.-K. Park, S.-H. Son, Orthogonality sampling method for identifying small anomalies in real-world microwave imaging, East Asian J. Appl. Math., 14 (2024), 293–313. https://doi.org/10.4208/eajam.2022-318.170723 doi: 10.4208/eajam.2022-318.170723
|
| [63] |
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
|
| [64] |
P. Jaccard, The distribution of the flora in the alpine zone, New Phytol., 11 (1912), 37–50. https://doi.org/10.1111/j.1469-8137.1912.tb05611.x doi: 10.1111/j.1469-8137.1912.tb05611.x
|
| [65] |
J.-M. Geffrin, P. Sabouroux, Continuing with the Fresnel database: experimental setup and improvements in 3D scattering measurements, Inverse Probl., 25 (2009), 024001. https://doi.org/10.1088/0266-5611/25/2/024001 doi: 10.1088/0266-5611/25/2/024001
|
| [66] |
C. Geng, M. Song, X. Wang, Y. Wang, Time-domain direct sampling method for inverse electromagnetic scattering with a single incident source, SIAM J. Imag. Sci., 18 (2025), 1208–1234. https://doi.org/10.1137/24M1701071 doi: 10.1137/24M1701071
|
| [67] |
Y. Guo, H. Li, X. Wang, A novel time-domain direct sampling approach for inverse scattering problems in acoustics, SIAM J. Appl. Math., 84 (2024), 2152–2174. https://doi.org/10.1137/23M1622854 doi: 10.1137/23M1622854
|
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Won-Kwang Park. Inversion of two-dimensional Fresnel experimental dataset using orthogonality sampling method with single and multiple sources: the case of transverse magnetic polarized waves[J]. Communications in Analysis and Mechanics, 2026, 18(1): 142-171. doi: 10.3934/cam.2026006
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