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Low-frequency electromagnetic imaging using sensitivity functions and beamforming. (English) Zbl 1447.78010

The authors propose a new scheme for performing imaging using low-frequency electromagnetic waves in inhomogeneous media. The main idea is based on a novel computational filtering scheme and the introduction of algorithms for simulating electromagnetic signals in complex media. The method is flexible and possesses higher imaging capabilities than high-frequency methods in media involving conductive materials. A detailed numerical analysis is included in the last section, involving different situations, in view to describe the performance and capabilities of the imaging scheme.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35H99 Close-to-elliptic equations
35Q60 PDEs in connection with optics and electromagnetic theory
35R30 Inverse problems for PDEs
65R32 Numerical methods for inverse problems for integral equations
65T50 Numerical methods for discrete and fast Fourier transforms
78A45 Diffraction, scattering
78A55 Technical applications of optics and electromagnetic theory
92C55 Biomedical imaging and signal processing
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[1] H. Ammari, J. Chen, Z. Chen, J. Garnier, and D. Volkov, Target detection and characterization from electromagnetic induction data, J. Math. Pures Appl. (9), 101 (2014), pp. 54-75. · Zbl 1280.35145
[2] S. Andrieux, T. Baranger, and A. B. Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems, 22 (2006), 115. · Zbl 1089.35084
[3] J.-F. Aubry, M. Tanter, J. Gerber, J.-L. Thomas, and M. Fink, Optimal focusing by spatio-temporal inverse filter. II. Experiments. Application to focusing through absorbing and reverberating media, J. Acoust. Soc. Am., 110 (2001), pp. 48-58.
[4] A. H. Baker, E. R. Jessup, and T. Manteuffel, A technique for accelerating the convergence of restarted GMRES, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 962-984. · Zbl 1086.65025
[5] J. G. Berryman and R. V. Kohn, Variational constraints for electrical-impedance tomography, Phys. Rev. Lett., 65 (1990), pp. 325-328.
[6] L. Borcea, Electrical impedance tomography, Inverse problems, 18 (2002), R99. · Zbl 1031.35147
[7] L. Borcea, T. Callaghan, J. Garnier, and G. Papanicolaou, A universal filter for enhanced imaging with small arrays, Inverse Problems, 26 (2009), 015006. · Zbl 1190.94007
[8] L. Borcea, G. A. Gray, and Y. Zhang, Variationally constrained numerical solution of electrical impedance tomography, Inverse Problems, 19 (2003), pp. 1159-1184. · Zbl 1054.35122
[9] A. Borges, J. De Oliveira, J. Velez, C. Tavares, F. Linhares, and A. Peyton, Development of electromagnetic tomography (EMT) for industrial applications. Part 2: Image reconstruction and software framework, in Proceedings of the 1st World Congress on Industrial Process Tomography, Buxton Press, Buxton, England, 1999, pp. 219-225.
[10] A. Borsic, B. M. Graham, A. Adler, and W. R. Lionheart, In vivo impedance imaging with total variation regularization, IEEE Trans. Med. Imaging, 29 (2010), pp. 44-54.
[11] S. Brooks, A. Gelman, G. Jones, and X.-L. Meng, Handbook of Markov Chain Monte Carlo, CRC, Boca Raton, FL, 2011. · Zbl 1218.65001
[12] V. Chitturi and F. Nagi, Spatial resolution in electrical impedance tomography: A topical review, J. Electr. Bioimped., 8 (2017), pp. 66-78.
[13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Appl. Math. Sci. 93, Springer, New York, 2012. · Zbl 1425.35001
[14] Z. Cui, Q. Wang, Q. Xue, W. Fan, L. Zhang, Z. Cao, B. Sun, H. Wang, and W. Yang, A review on image reconstruction algorithms for electrical capacitance/resistance tomography, Sensor Rev., 36 (2016), pp. 429-445.
[15] O. Dorn, H. Bertete-Aguirre, J. Berryman, and G. Papanicolaou, A nonlinear inversion method for \(3\) D electromagnetic imaging using adjoint fields, Inverse Problems, 15 (1999), pp. 1523-1558. · Zbl 0943.35101
[16] O. Dorn, H. Bertete-Aguirre, J. Berryman, and G. Papanicolaou, Sensitivity analysis of a nonlinear inversion method for 3D electromagnetic imaging in anisotropic media, Inverse Problems, 18 (2002), pp. 285-318. · Zbl 0997.78009
[17] M. M. Dunlop, M. A. Iglesias, and A. M. Stuart, Hierarchical Bayesian level set inversion, Stat. Comput., 27 (2017), pp. 1555-1584. · Zbl 1384.62084
[18] M. M. Dunlop and A. M. Stuart, The Bayesian formulation of EIT: Analysis and algorithms, Inverse Probl. Imaging, 10 (2016), pp. 1007-1036, https://doi.org/10.3934/ipi.2016030. · Zbl 1348.62104
[19] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, Philadelphia, 1999. · Zbl 0939.49002
[20] H. Garde and K. Knudsen, 3D reconstruction for partial data electrical impedance tomography using a sparsity prior, in Proceedings of the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (2014), American Institute of Mathematical Sciences, Springfield, MO, 2015, pp. 495-504, https://doi.org/10.3934/proc.2015.0495. · Zbl 1335.65088
[21] H. Garde and K. Knudsen, Sparsity prior for electrical impedance tomography with partial data, Inverse Probl. Sci. Eng., 24 (2016), pp. 524-541.
[22] H. Garde and S. Staboulis, The regularized monotonicity method: Detecting irregular indefinite inclusions, Inverse Probl. Imaging, 13 (2019), pp. 93-116, https://doi.org/10.3934/ipi.2019006. · Zbl 1407.35226
[23] Matthias Gehre, Bangti Jin, and Xiliang Lu, An analysis of finite element approximation in electrical impedance tomography, Inverse Problems, 30 (2014), 045013. · Zbl 1288.78033
[24] P. F. Grant and M. M. Lowery, Effect of dispersive conductivity and permittivity in volume conductor models of deep brain stimulation, IEEE Trans. Biomed. Eng., 57 (2010), pp. 2386-2393.
[25] H. Griffiths, Magnetic induction tomography, Meas. Sci. Tech., 12 (2001), pp. 1126-1131.
[26] O. S. Haddadin and E. S. Ebbini, Ultrasonic focusing through inhomogeneous media by application of the inverse scattering problem, J. Acoust. Soc. Amer., 104 (1998), pp. 313-325.
[27] B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), pp. 3382-3403. · Zbl 1282.35413
[28] B. Harrach and M. Ullrich, Resolution guarantees in electrical impedance tomography, IEEE Trans. Med. Imaging, 34 (2015), pp. 1513-1521.
[29] B. Jin, T. Khan, and P. Maass, A reconstruction algorithm for electrical impedance tomography based on sparsity regularization, Internat. J. Numer. Methods Engrg., 89 (2012), pp. 337-353. · Zbl 1242.78016
[30] M. A. Kemp, M. Franzi, A. Haase, E. Jongewaard, M. T. Whittaker, M. Kirkpatrick, and R. Sparr, A high Q piezoelectric resonator as a portable VLF transmitter, Nat. Commun., 10 (2019), 1715.
[31] R. V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems, 6 (1990), pp. 389-414. · Zbl 0718.65089
[32] A. Korzhenevskii and V. Cherepenin, Magnetic induction tomography, J. Commun. Technol. Electron., 42 (1997), pp. 469-474.
[33] X. Li, S. K. Davis, S. C. Hagness, D. W. Van der Weide, and B. D. Van Veen, Microwave imaging via space-time beamforming: Experimental investigation of tumor detection in multilayer breast phantoms, IEEE Trans. Microwave Theory Techniq., 52 (2004), pp. 1856-1865.
[34] R. Merwa, K. Hollaus, O. Biro, and H. Scharfetter, Detection of brain oedema using magnetic induction tomography: A feasibility study of the likely sensitivity and detectability, Physiol. Meas., 25 (2004), pp. 347-354.
[35] R. Merwa, K. Hollaus, B. Brandstätter, and H. Scharfetter, Numerical solution of the general \(3\) D eddy current problem for magnetic induction tomography (spectroscopy), Physiol. Meas., 24 (2003), pp. 545-554.
[36] P. D. Mountcastle, N. A. Goodman, and C. J. Morgan, Generalized adaptive radar signal processing, in Proceedings of the 25th Army Science Conference, Defense Technical Information Center, Fort Belvoir, Fairfax County, VA, 2008.
[37] S. L. Penny, Inspection Techniques for Determining Graphite Core Deterioration for Nuclear Applications, PhD Thesis, The University of Manchester, Manchester, England, United Kingdom, 2016.
[38] A. Peyton, M. Beck, A. Borges, J. De Oliveira, G. Lyon, Z. Yu, M. Brown, and J. Ferrerra, Development of electromagnetic tomography (EMT) for industrial applications. Part 1: Sensor design and instrumentation, in 1st World Congress on Industrial Process Tomography, Buxton Press, Buxton, England, 1999, pp. 306-312.
[39] V. S. Ryaben’kii and S. V. Tsynkov, A Theoretical Introduction to Numerical Analysis, Chapman and Hall/CRC, Boca Raton, FL, 2006.
[40] D. S. Shumakov, A. S. Beaverstone, and N. K. Nikolova, Optimal illumination schemes for near-field microwave imaging, Progr. Electromagn. Res., 157 (2016), pp. 93-110.
[41] E. Somersalo, M. Cheney, and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), pp. 1023-1040. · Zbl 0759.35055
[42] J. A. Stratton, Electromagnetic Theory, Wiley, Hoboken, NJ, 2007. · JFM 67.1119.01
[43] A. Tamburrino, Monotonicity based imaging methods for elliptic and parabolic inverse problems, J. Inverse Ill-Posed Probl., 14 (2006), pp. 633-642. · Zbl 1111.35127
[44] A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, Inverse Problems, 18 (2002), pp. 1809-1830. · Zbl 1034.35154
[45] A. Tamburrino, G. Rubinacci, M. Soleimani, and W. Lionheart, A noniterative inversion method for electrical resistance, capacitance and inductance tomography for two phase materials, in Proceedings of the 3rd World Congress on Industrial Process Tomography, Banff, Canada, Virtual Centre for Industrial Process Tomotraphy, Leeds, England, 2003, pp. 233-238.
[46] M. Tanter, J.-F. Aubry, J. Gerber, J.-L. Thomas, and M. Fink, Optimal focusing by spatio-temporal inverse filter. I. Basic principles, J. Acoust. Soc. Amer., 110 (2001), pp. 37-47.
[47] A. N. Tikhonov, A. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-posed Problems, Math. Appl. 328, Springer, Dordrecht, The Netherlands, 2011. · Zbl 0831.65059
[48] M. Tong Harris, M. H. Langston, P.-D. Letourneau, G. Papanicolaou, J. Ezick, and R. Lethin, Fast large-scale algorithm for electromagnetic wave propagation in \(3\) D media, in 2019 IEEE High Performance Extreme Computing Conference (HPEC), IEEE, Piscataway, NJ, 2019.
[49] G. Uhlmann, Electrical impedance tomography and Calderón’s problem, Inverse Problems, 25 (2009), 123011. · Zbl 1181.35339
[50] F. Vico, L. Greengard, and M. Ferrando, Fast convolution with free-space Green’s functions, J. Comput. Phys., 323 (2016), pp. 191-203. · Zbl 1415.65269
[51] F. Vignon, J. de Rosny, J.-F. Aubry, and M. Fink, Optimal adaptive focusing through heterogeneous media with the minimally invasive inverse filter, J. Acoust. Soc. Amer., 122 (2007), pp. 2715-2724.
[52] W. Yang and L. Peng, Image reconstruction algorithms for electrical capacitance tomography, Meas. Sci. Tech., 14 (2002), R1.
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