Abstract

An innovative method for the localization of multiple sparse metallic targets is proposed. Starting from the local shape function (LSF) formulation of the inverse scattering problem and exploiting the multitask Bayesian compressive sensing (MT-BCS) paradigm, a two-step approach is described where, after a first estimation of the LSF scattering amplitudes, the reconstruction of the metallic objects is yielded through a thresholding and voting step. Selected numerical examples are presented to analyze the accuracy, the robustness, and the computational efficiency of the LSF–MT-BCS technique.

© 2013 Optical Society of America

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2013 (2)

C.-C. Chiu, C.-H. Sun, C.-L. Li, and C.-H. Huang, “Comparative study of some population-based optimization algorithms on inverse scattering of a two-dimensional perfectly conducting cylinder in dielectric slab medium,” IEEE Trans. Geosci. Remote Sens. 51, 2302–2315 (2013).
[CrossRef]

L. Poli, G. Oliveri, P. Rocca, and A. Massa, “Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under TE illuminations,” IEEE Trans. Geosci. Remote Sens. 51, 2920–2936 (2013).

2012 (4)

L. Poli, G. Oliveri, and A. Massa, “Microwave imaging within the first-order Born approximation by means of the contrast-field Bayesian compressive sensing,” IEEE Trans. Antennas Propag. 60, 2865–2879 (2012).
[CrossRef]

G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag. 60, 2309–2326 (2012).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “Reliable diagnosis of large linear arrays—Bayesian compressive sensing approach,” IEEE Trans. Antennas Propag. 60, 4627–4636 (2012).
[CrossRef]

G. Oliveri, L. Poli, P. Rocca, and A. Massa, “Bayesian compressive optical imaging within the Rytov approximation,” Opt. Lett. 37, 1760–1762 (2012).
[CrossRef]

2011 (6)

J. Shen, X. Chen, Y. Zhong, and L. Ran, “Inverse scattering problem in presence of a conducting cylinder,” Opt. Express 19, 10698–10706 (2011).
[CrossRef]

A. Brancaccio, G. Leone, and R. Solimene, “Fault detection in metallic grid scattering,” J. Opt. Soc. Am. A 28, 2588–2599 (2011).
[CrossRef]

F. Viani, G. Oliveri, and A. Massa, “Compressive sensing pattern matching techniques for synthesizing planar sparse arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

G. Oliveri and A. Massa, “Bayesian compressive sampling for pattern synthesis with maximally sparse non-uniform linear arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “A Bayesian compressive sampling-based inversion for imaging sparse scatterers,” IEEE Trans. Geosci. Remote Sens. 49, 3993–4006 (2011).
[CrossRef]

P. Rocca, G. Oliveri, and A. Massa, “Differential evolution as applied to electromagnetics,” IEEE Antennas Propag. Mag. 53, 38–49 (2011).
[CrossRef]

2010 (3)

Y. Alvarez-Lopez, A. Dominguez-Casas, C. Garcia-Gonzalez, and F. Las-Heras, “Geometry reconstruction of metallic bodies using the sources reconstruction method,” IEEE Antennas Wireless Propag. Lett. 9, 1197–1200 (2010).
[CrossRef]

J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
[CrossRef]

K. Agarwal, X. Chen, and Y. Zhong, “A multipole-expansion based linear sampling method for solving inverse scattering problems,” Opt. Express 18, 6366–6381 (2010).
[CrossRef]

2009 (2)

S. Ji, D. Dunson, and L. Carin, “Multitask compressive sensing,” IEEE Trans. Signal Process. 57(4), 92–106 (2009).
[CrossRef]

P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems 25, 1–41 (2009).

2008 (4)

I. T. Rekanos, “Shape reconstruction of a perfectly conducting scatterer using differential evolution and particle swarm optimization,” IEEE Trans. Geosci. Remote Sens. 46, 1967–1974 (2008).
[CrossRef]

R. Solimene, A. Brancaccio, J. Romano, and R. Pierri, “Localizing thin metallic cylinders by a 2.5-D linear distributional approach: experimental results,” IEEE Trans. Antennas Propag. 56, 2630–2637 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[CrossRef]

2007 (1)

R. G. Baraniuk, “Compressive sampling,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

2006 (1)

R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).
[CrossRef]

2005 (3)

Y.-C. Chen, Y.-F. Chen, C.-C. Chiu, and C.-Y. Chang, “Image reconstruction of a buried perfectly conducting cylinder illuminated by transverse electric waves,” Int. J. Imaging Syst. Technol. 15, 261–265 (2005).
[CrossRef]

W. Chien and C.-C. Chiu, “Using NU-SSGA to reduce the searching time in inverse problem of a buried metallic object,” IEEE Trans. Antennas Propag. 53, 3128–3134 (2005).
[CrossRef]

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029 (2005).
[CrossRef]

2004 (2)

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042–3047 (2004).
[CrossRef]

T. Miwa and I. Arai, “Super-resolution imaging for point reflectors near transmitting and receiving array,” IEEE Trans. Antennas Propag. 52, 220–229 (2004).
[CrossRef]

2003 (5)

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).
[CrossRef]

E. Cekli and H. A. Cirpan, “Unconditional maximum likelihood approach for localization of near-field sources: algorithm and performance analysis,” J. Inst. Electron. Commun. Eng. Jpn. 57, 9–15 (2003).

Y. Zhou, J. Li, and H. Ling, “Shape inversion of metallic cavities using hybrid genetic algorithm combined with tabu list,” Electron. Lett. 39, 280–281 (2003).
[CrossRef]

A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy,” IEEE Trans. Antennas Propag. 51, 1251–1262 (2003).
[CrossRef]

A. Liseno, R. Pierri, and F. Soldovieri, “Shape identification by physical optics: the two-dimensional TE case,” J. Opt. Soc. Am. A 20, 1827–1830 (2003).
[CrossRef]

2002 (3)

2001 (2)

A. Qing, C. K. Lee, and L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

C. Ramananjaona, M. Lambert, and D. Lesselier, “Shape inversion from TM and TE real data by controlled evolution of level sets,” Inverse Probl. 17, 1585–1595 (2001).
[CrossRef]

2000 (2)

A. Qing and C. K. Lee, “Microwave imaging of parallel perfectly conducting cylinders using real-coded genetic algorithm coupled with Newton–Kantorovich method” Prog. Electromagn. Res. 28, 275–294 (2000).
[CrossRef]

K. A. Michalski, “Electromagnetic imaging of circular–cylindrical conductors and tunnels using a differential evolution algorithm,” Microw. Opt. Technol. Lett. 27, 330–334 (2000).
[CrossRef]

1999 (1)

D. Asteli and B. Ottersten, “The effect of local scattering on direction of arrival estimation with MUSIC,” IEEE Trans. Signal Process. 47, 3220–3234 (1999).
[CrossRef]

1997 (1)

T. Takenaka, Z. Q. Meng, T. Tanaka, and W. C. Chew, “Local shape function combined with genetic algorithm applied to inverse scattering for strips,” Microw. Opt. Technol. Lett. 16, 337–341 (1997).
[CrossRef]

1994 (2)

G. P. Otto and W. C. Chew, “Microwave inverse scattering—local shape function imaging for improved resolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
[CrossRef]

R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

1993 (1)

1992 (2)

W. C. Chew and G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shape functions,” IEEE Microw. Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
[CrossRef]

1989 (1)

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

1986 (1)

D. Colton and P. Monk, “A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region II,” SIAM J. Appl. Math. 46, 506–523 (1986).
[CrossRef]

Agarwal, K.

Alvarez-Lopez, Y.

Y. Alvarez-Lopez, A. Dominguez-Casas, C. Garcia-Gonzalez, and F. Las-Heras, “Geometry reconstruction of metallic bodies using the sources reconstruction method,” IEEE Antennas Wireless Propag. Lett. 9, 1197–1200 (2010).
[CrossRef]

Arai, I.

T. Miwa and I. Arai, “Super-resolution imaging for point reflectors near transmitting and receiving array,” IEEE Trans. Antennas Propag. 52, 220–229 (2004).
[CrossRef]

Asteli, D.

D. Asteli and B. Ottersten, “The effect of local scattering on direction of arrival estimation with MUSIC,” IEEE Trans. Signal Process. 47, 3220–3234 (1999).
[CrossRef]

Baraniuk, R.

Baraniuk, R. G.

R. G. Baraniuk, “Compressive sampling,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

Benedetti, M.

P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems 25, 1–41 (2009).

Borderies, P.

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).
[CrossRef]

Brancaccio, A.

A. Brancaccio, G. Leone, and R. Solimene, “Fault detection in metallic grid scattering,” J. Opt. Soc. Am. A 28, 2588–2599 (2011).
[CrossRef]

R. Solimene, A. Brancaccio, J. Romano, and R. Pierri, “Localizing thin metallic cylinders by a 2.5-D linear distributional approach: experimental results,” IEEE Trans. Antennas Propag. 56, 2630–2637 (2008).
[CrossRef]

Bruno, A. C.

E. P. Ribeiro, A. C. Bruno, P. C. Ribeiro, J. Szczupak, and O. G. Symko, “Image of a two-dimensional magnetic moment distribution: application to detect small metallic objects in the human body,” in 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 1992), Vol. 5, 2178–2179.

Bucci, O. M.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Candes, E. J.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

Carin, L.

S. Ji, D. Dunson, and L. Carin, “Multitask compressive sensing,” IEEE Trans. Signal Process. 57(4), 92–106 (2009).
[CrossRef]

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[CrossRef]

Carlin, M.

G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag. 60, 2309–2326 (2012).
[CrossRef]

Cekli, E.

E. Cekli and H. A. Cirpan, “Unconditional maximum likelihood approach for localization of near-field sources: algorithm and performance analysis,” J. Inst. Electron. Commun. Eng. Jpn. 57, 9–15 (2003).

Chang, C.-Y.

Y.-C. Chen, Y.-F. Chen, C.-C. Chiu, and C.-Y. Chang, “Image reconstruction of a buried perfectly conducting cylinder illuminated by transverse electric waves,” Int. J. Imaging Syst. Technol. 15, 261–265 (2005).
[CrossRef]

Chen, X.

Chen, Y.-C.

Y.-C. Chen, Y.-F. Chen, C.-C. Chiu, and C.-Y. Chang, “Image reconstruction of a buried perfectly conducting cylinder illuminated by transverse electric waves,” Int. J. Imaging Syst. Technol. 15, 261–265 (2005).
[CrossRef]

Chen, Y.-F.

Y.-C. Chen, Y.-F. Chen, C.-C. Chiu, and C.-Y. Chang, “Image reconstruction of a buried perfectly conducting cylinder illuminated by transverse electric waves,” Int. J. Imaging Syst. Technol. 15, 261–265 (2005).
[CrossRef]

Chew, W. C.

T. Takenaka, Z. Q. Meng, T. Tanaka, and W. C. Chew, “Local shape function combined with genetic algorithm applied to inverse scattering for strips,” Microw. Opt. Technol. Lett. 16, 337–341 (1997).
[CrossRef]

G. P. Otto and W. C. Chew, “Microwave inverse scattering—local shape function imaging for improved resolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
[CrossRef]

W. C. Chew and G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shape functions,” IEEE Microw. Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
[CrossRef]

Chien, W.

W. Chien and C.-C. Chiu, “Using NU-SSGA to reduce the searching time in inverse problem of a buried metallic object,” IEEE Trans. Antennas Propag. 53, 3128–3134 (2005).
[CrossRef]

Chiu, C.-C.

C.-C. Chiu, C.-H. Sun, C.-L. Li, and C.-H. Huang, “Comparative study of some population-based optimization algorithms on inverse scattering of a two-dimensional perfectly conducting cylinder in dielectric slab medium,” IEEE Trans. Geosci. Remote Sens. 51, 2302–2315 (2013).
[CrossRef]

W. Chien and C.-C. Chiu, “Using NU-SSGA to reduce the searching time in inverse problem of a buried metallic object,” IEEE Trans. Antennas Propag. 53, 3128–3134 (2005).
[CrossRef]

Y.-C. Chen, Y.-F. Chen, C.-C. Chiu, and C.-Y. Chang, “Image reconstruction of a buried perfectly conducting cylinder illuminated by transverse electric waves,” Int. J. Imaging Syst. Technol. 15, 261–265 (2005).
[CrossRef]

Cirpan, H. A.

E. Cekli and H. A. Cirpan, “Unconditional maximum likelihood approach for localization of near-field sources: algorithm and performance analysis,” J. Inst. Electron. Commun. Eng. Jpn. 57, 9–15 (2003).

Colton, D.

D. Colton and P. Monk, “A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region II,” SIAM J. Appl. Math. 46, 506–523 (1986).
[CrossRef]

Devaney, A.

Devaney, A. J.

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042–3047 (2004).
[CrossRef]

Dominguez-Casas, A.

Y. Alvarez-Lopez, A. Dominguez-Casas, C. Garcia-Gonzalez, and F. Las-Heras, “Geometry reconstruction of metallic bodies using the sources reconstruction method,” IEEE Antennas Wireless Propag. Lett. 9, 1197–1200 (2010).
[CrossRef]

Donelli, M.

P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems 25, 1–41 (2009).

Dorney, T.

Dunson, D.

S. Ji, D. Dunson, and L. Carin, “Multitask compressive sensing,” IEEE Trans. Signal Process. 57(4), 92–106 (2009).
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O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
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P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems 25, 1–41 (2009).

Garcia-Gonzalez, C.

Y. Alvarez-Lopez, A. Dominguez-Casas, C. Garcia-Gonzalez, and F. Las-Heras, “Geometry reconstruction of metallic bodies using the sources reconstruction method,” IEEE Antennas Wireless Propag. Lett. 9, 1197–1200 (2010).
[CrossRef]

Gruber, F. K.

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042–3047 (2004).
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W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
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Huang, C.-H.

C.-C. Chiu, C.-H. Sun, C.-L. Li, and C.-H. Huang, “Comparative study of some population-based optimization algorithms on inverse scattering of a two-dimensional perfectly conducting cylinder in dielectric slab medium,” IEEE Trans. Geosci. Remote Sens. 51, 2302–2315 (2013).
[CrossRef]

Jen, L.

A. Qing, C. K. Lee, and L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
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S. Ji, D. Dunson, and L. Carin, “Multitask compressive sensing,” IEEE Trans. Signal Process. 57(4), 92–106 (2009).
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S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
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R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
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K. J. Langenberg, K. Mayer, A. Zimmer, and C. Kohl, “Nondestructive evaluation of embedded structures in concrete: modeling and tomographic imaging,” in Proceedings of the URSI International Symposium on Electromagnetic Theory (URSI EMTS, 2004), pp. 1203–1205.

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C. Ramananjaona, M. Lambert, and D. Lesselier, “Shape inversion from TM and TE real data by controlled evolution of level sets,” Inverse Probl. 17, 1585–1595 (2001).
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K. J. Langenberg, K. Mayer, A. Zimmer, and C. Kohl, “Nondestructive evaluation of embedded structures in concrete: modeling and tomographic imaging,” in Proceedings of the URSI International Symposium on Electromagnetic Theory (URSI EMTS, 2004), pp. 1203–1205.

Las-Heras, F.

Y. Alvarez-Lopez, A. Dominguez-Casas, C. Garcia-Gonzalez, and F. Las-Heras, “Geometry reconstruction of metallic bodies using the sources reconstruction method,” IEEE Antennas Wireless Propag. Lett. 9, 1197–1200 (2010).
[CrossRef]

Lee, C. K.

A. Qing, C. K. Lee, and L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

A. Qing and C. K. Lee, “Microwave imaging of parallel perfectly conducting cylinders using real-coded genetic algorithm coupled with Newton–Kantorovich method” Prog. Electromagn. Res. 28, 275–294 (2000).
[CrossRef]

Leone, G.

Lesselier, D.

C. Ramananjaona, M. Lambert, and D. Lesselier, “Shape inversion from TM and TE real data by controlled evolution of level sets,” Inverse Probl. 17, 1585–1595 (2001).
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Li, C.-L.

C.-C. Chiu, C.-H. Sun, C.-L. Li, and C.-H. Huang, “Comparative study of some population-based optimization algorithms on inverse scattering of a two-dimensional perfectly conducting cylinder in dielectric slab medium,” IEEE Trans. Geosci. Remote Sens. 51, 2302–2315 (2013).
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Li, J.

Y. Zhou, J. Li, and H. Ling, “Shape inversion of metallic cavities using hybrid genetic algorithm combined with tabu list,” Electron. Lett. 39, 280–281 (2003).
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Y. Zhou, J. Li, and H. Ling, “Shape inversion of metallic cavities using hybrid genetic algorithm combined with tabu list,” Electron. Lett. 39, 280–281 (2003).
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R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).
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R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029 (2005).
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A. Liseno, R. Pierri, and F. Soldovieri, “Shape identification by physical optics: the two-dimensional TE case,” J. Opt. Soc. Am. A 20, 1827–1830 (2003).
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A. Liseno and R. Pierri, “Imaging perfectly conducting objects as support of induced currents: Kirchhoff approximation and frequency diversity,” J. Opt. Soc. Am. A 19, 1308–1318 (2002).
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Liu, Q. H.

W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
[CrossRef]

Marengo, E. A.

F. K. Gruber, E. A. Marengo, and A. J. Devaney, “Time-reversal imaging with multiple signal classification considering multiple scattering between the targets,” J. Acoust. Soc. Am. 115, 3042–3047 (2004).
[CrossRef]

Massa, A.

L. Poli, G. Oliveri, P. Rocca, and A. Massa, “Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under TE illuminations,” IEEE Trans. Geosci. Remote Sens. 51, 2920–2936 (2013).

G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag. 60, 2309–2326 (2012).
[CrossRef]

L. Poli, G. Oliveri, and A. Massa, “Microwave imaging within the first-order Born approximation by means of the contrast-field Bayesian compressive sensing,” IEEE Trans. Antennas Propag. 60, 2865–2879 (2012).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “Reliable diagnosis of large linear arrays—Bayesian compressive sensing approach,” IEEE Trans. Antennas Propag. 60, 4627–4636 (2012).
[CrossRef]

G. Oliveri, L. Poli, P. Rocca, and A. Massa, “Bayesian compressive optical imaging within the Rytov approximation,” Opt. Lett. 37, 1760–1762 (2012).
[CrossRef]

F. Viani, G. Oliveri, and A. Massa, “Compressive sensing pattern matching techniques for synthesizing planar sparse arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “A Bayesian compressive sampling-based inversion for imaging sparse scatterers,” IEEE Trans. Geosci. Remote Sens. 49, 3993–4006 (2011).
[CrossRef]

G. Oliveri and A. Massa, “Bayesian compressive sampling for pattern synthesis with maximally sparse non-uniform linear arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

P. Rocca, G. Oliveri, and A. Massa, “Differential evolution as applied to electromagnetics,” IEEE Antennas Propag. Mag. 53, 38–49 (2011).
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P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems 25, 1–41 (2009).

Mayer, K.

K. J. Langenberg, K. Mayer, A. Zimmer, and C. Kohl, “Nondestructive evaluation of embedded structures in concrete: modeling and tomographic imaging,” in Proceedings of the URSI International Symposium on Electromagnetic Theory (URSI EMTS, 2004), pp. 1203–1205.

Meng, Z. Q.

T. Takenaka, Z. Q. Meng, T. Tanaka, and W. C. Chew, “Local shape function combined with genetic algorithm applied to inverse scattering for strips,” Microw. Opt. Technol. Lett. 16, 337–341 (1997).
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L. Poli, G. Oliveri, P. Rocca, and A. Massa, “Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under TE illuminations,” IEEE Trans. Geosci. Remote Sens. 51, 2920–2936 (2013).

G. Oliveri, P. Rocca, and A. Massa, “Reliable diagnosis of large linear arrays—Bayesian compressive sensing approach,” IEEE Trans. Antennas Propag. 60, 4627–4636 (2012).
[CrossRef]

G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag. 60, 2309–2326 (2012).
[CrossRef]

L. Poli, G. Oliveri, and A. Massa, “Microwave imaging within the first-order Born approximation by means of the contrast-field Bayesian compressive sensing,” IEEE Trans. Antennas Propag. 60, 2865–2879 (2012).
[CrossRef]

G. Oliveri, L. Poli, P. Rocca, and A. Massa, “Bayesian compressive optical imaging within the Rytov approximation,” Opt. Lett. 37, 1760–1762 (2012).
[CrossRef]

F. Viani, G. Oliveri, and A. Massa, “Compressive sensing pattern matching techniques for synthesizing planar sparse arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “A Bayesian compressive sampling-based inversion for imaging sparse scatterers,” IEEE Trans. Geosci. Remote Sens. 49, 3993–4006 (2011).
[CrossRef]

G. Oliveri and A. Massa, “Bayesian compressive sampling for pattern synthesis with maximally sparse non-uniform linear arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

P. Rocca, G. Oliveri, and A. Massa, “Differential evolution as applied to electromagnetics,” IEEE Antennas Propag. Mag. 53, 38–49 (2011).
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W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
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G. P. Otto and W. C. Chew, “Microwave inverse scattering—local shape function imaging for improved resolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
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W. C. Chew and G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shape functions,” IEEE Microw. Guided Wave Lett. 2, 284–286 (1992).
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Pierri, R.

R. Solimene, A. Brancaccio, J. Romano, and R. Pierri, “Localizing thin metallic cylinders by a 2.5-D linear distributional approach: experimental results,” IEEE Trans. Antennas Propag. 56, 2630–2637 (2008).
[CrossRef]

R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).
[CrossRef]

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029 (2005).
[CrossRef]

A. Liseno, R. Pierri, and F. Soldovieri, “Shape identification by physical optics: the two-dimensional TE case,” J. Opt. Soc. Am. A 20, 1827–1830 (2003).
[CrossRef]

A. Liseno and R. Pierri, “Imaging perfectly conducting objects as support of induced currents: Kirchhoff approximation and frequency diversity,” J. Opt. Soc. Am. A 19, 1308–1318 (2002).
[CrossRef]

Poli, L.

L. Poli, G. Oliveri, P. Rocca, and A. Massa, “Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under TE illuminations,” IEEE Trans. Geosci. Remote Sens. 51, 2920–2936 (2013).

L. Poli, G. Oliveri, and A. Massa, “Microwave imaging within the first-order Born approximation by means of the contrast-field Bayesian compressive sensing,” IEEE Trans. Antennas Propag. 60, 2865–2879 (2012).
[CrossRef]

G. Oliveri, L. Poli, P. Rocca, and A. Massa, “Bayesian compressive optical imaging within the Rytov approximation,” Opt. Lett. 37, 1760–1762 (2012).
[CrossRef]

Qing, A.

A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy,” IEEE Trans. Antennas Propag. 51, 1251–1262 (2003).
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A. Qing, C. K. Lee, and L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

A. Qing and C. K. Lee, “Microwave imaging of parallel perfectly conducting cylinders using real-coded genetic algorithm coupled with Newton–Kantorovich method” Prog. Electromagn. Res. 28, 275–294 (2000).
[CrossRef]

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C. Ramananjaona, M. Lambert, and D. Lesselier, “Shape inversion from TM and TE real data by controlled evolution of level sets,” Inverse Probl. 17, 1585–1595 (2001).
[CrossRef]

Ran, L.

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I. T. Rekanos, “Shape reconstruction of a perfectly conducting scatterer using differential evolution and particle swarm optimization,” IEEE Trans. Geosci. Remote Sens. 46, 1967–1974 (2008).
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I. T. Rekanos and T. D. Tsiboukis, “An inverse scattering technique for microwave imaging of binary objects,” IEEE Trans. Microw. Theory Tech. 50, 1439–1441 (2002).
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Ribeiro, P. C.

E. P. Ribeiro, A. C. Bruno, P. C. Ribeiro, J. Szczupak, and O. G. Symko, “Image of a two-dimensional magnetic moment distribution: application to detect small metallic objects in the human body,” in 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 1992), Vol. 5, 2178–2179.

Rocca, P.

L. Poli, G. Oliveri, P. Rocca, and A. Massa, “Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under TE illuminations,” IEEE Trans. Geosci. Remote Sens. 51, 2920–2936 (2013).

G. Oliveri, P. Rocca, and A. Massa, “Reliable diagnosis of large linear arrays—Bayesian compressive sensing approach,” IEEE Trans. Antennas Propag. 60, 4627–4636 (2012).
[CrossRef]

G. Oliveri, L. Poli, P. Rocca, and A. Massa, “Bayesian compressive optical imaging within the Rytov approximation,” Opt. Lett. 37, 1760–1762 (2012).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “A Bayesian compressive sampling-based inversion for imaging sparse scatterers,” IEEE Trans. Geosci. Remote Sens. 49, 3993–4006 (2011).
[CrossRef]

P. Rocca, G. Oliveri, and A. Massa, “Differential evolution as applied to electromagnetics,” IEEE Antennas Propag. Mag. 53, 38–49 (2011).
[CrossRef]

P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems 25, 1–41 (2009).

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R. Solimene, A. Brancaccio, J. Romano, and R. Pierri, “Localizing thin metallic cylinders by a 2.5-D linear distributional approach: experimental results,” IEEE Trans. Antennas Propag. 56, 2630–2637 (2008).
[CrossRef]

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029 (2005).
[CrossRef]

Saillard, M.

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).
[CrossRef]

Schatzberg, A.

Shen, J.

Soldovieri, F.

R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).
[CrossRef]

A. Liseno, R. Pierri, and F. Soldovieri, “Shape identification by physical optics: the two-dimensional TE case,” J. Opt. Soc. Am. A 20, 1827–1830 (2003).
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A. Brancaccio, G. Leone, and R. Solimene, “Fault detection in metallic grid scattering,” J. Opt. Soc. Am. A 28, 2588–2599 (2011).
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R. Solimene, A. Brancaccio, J. Romano, and R. Pierri, “Localizing thin metallic cylinders by a 2.5-D linear distributional approach: experimental results,” IEEE Trans. Antennas Propag. 56, 2630–2637 (2008).
[CrossRef]

R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).
[CrossRef]

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029 (2005).
[CrossRef]

Sun, C.-H.

C.-C. Chiu, C.-H. Sun, C.-L. Li, and C.-H. Huang, “Comparative study of some population-based optimization algorithms on inverse scattering of a two-dimensional perfectly conducting cylinder in dielectric slab medium,” IEEE Trans. Geosci. Remote Sens. 51, 2302–2315 (2013).
[CrossRef]

Symes, W.

Symko, O. G.

E. P. Ribeiro, A. C. Bruno, P. C. Ribeiro, J. Szczupak, and O. G. Symko, “Image of a two-dimensional magnetic moment distribution: application to detect small metallic objects in the human body,” in 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 1992), Vol. 5, 2178–2179.

Szczupak, J.

E. P. Ribeiro, A. C. Bruno, P. C. Ribeiro, J. Szczupak, and O. G. Symko, “Image of a two-dimensional magnetic moment distribution: application to detect small metallic objects in the human body,” in 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 1992), Vol. 5, 2178–2179.

Takenaka, T.

T. Takenaka, Z. Q. Meng, T. Tanaka, and W. C. Chew, “Local shape function combined with genetic algorithm applied to inverse scattering for strips,” Microw. Opt. Technol. Lett. 16, 337–341 (1997).
[CrossRef]

Tanaka, T.

T. Takenaka, Z. Q. Meng, T. Tanaka, and W. C. Chew, “Local shape function combined with genetic algorithm applied to inverse scattering for strips,” Microw. Opt. Technol. Lett. 16, 337–341 (1997).
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J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
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I. T. Rekanos and T. D. Tsiboukis, “An inverse scattering technique for microwave imaging of binary objects,” IEEE Trans. Microw. Theory Tech. 50, 1439–1441 (2002).
[CrossRef]

van den Berg, P. M.

R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

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F. Viani, G. Oliveri, and A. Massa, “Compressive sensing pattern matching techniques for synthesizing planar sparse arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
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W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
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E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
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Wang, Y.-M.

W. C. Chew, L. Gurel, Y.-M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, “A generalized recursive algorithm for wave-scattering solutions in two dimensions,” IEEE Trans. Microwave Theory Tech. 40, 716–723 (1992).
[CrossRef]

Wright, S. J.

J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE 98, 948–958 (2010).
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S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[CrossRef]

Zhong, Y.

Zhou, Y.

Y. Zhou, J. Li, and H. Ling, “Shape inversion of metallic cavities using hybrid genetic algorithm combined with tabu list,” Electron. Lett. 39, 280–281 (2003).
[CrossRef]

Zimmer, A.

K. J. Langenberg, K. Mayer, A. Zimmer, and C. Kohl, “Nondestructive evaluation of embedded structures in concrete: modeling and tomographic imaging,” in Proceedings of the URSI International Symposium on Electromagnetic Theory (URSI EMTS, 2004), pp. 1203–1205.

Electron. Lett. (1)

Y. Zhou, J. Li, and H. Ling, “Shape inversion of metallic cavities using hybrid genetic algorithm combined with tabu list,” Electron. Lett. 39, 280–281 (2003).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

P. Rocca, G. Oliveri, and A. Massa, “Differential evolution as applied to electromagnetics,” IEEE Antennas Propag. Mag. 53, 38–49 (2011).
[CrossRef]

IEEE Antennas Wireless Propag. Lett. (1)

Y. Alvarez-Lopez, A. Dominguez-Casas, C. Garcia-Gonzalez, and F. Las-Heras, “Geometry reconstruction of metallic bodies using the sources reconstruction method,” IEEE Antennas Wireless Propag. Lett. 9, 1197–1200 (2010).
[CrossRef]

IEEE Microw. Guided Wave Lett. (1)

W. C. Chew and G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shape functions,” IEEE Microw. Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

IEEE Signal Process. Mag. (2)

R. G. Baraniuk, “Compressive sampling,” IEEE Signal Process. Mag. 24, 118–121 (2007).
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E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
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IEEE Trans. Antennas Propag. (12)

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
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W. Chien and C.-C. Chiu, “Using NU-SSGA to reduce the searching time in inverse problem of a buried metallic object,” IEEE Trans. Antennas Propag. 53, 3128–3134 (2005).
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R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029 (2005).
[CrossRef]

L. Poli, G. Oliveri, and A. Massa, “Microwave imaging within the first-order Born approximation by means of the contrast-field Bayesian compressive sensing,” IEEE Trans. Antennas Propag. 60, 2865–2879 (2012).
[CrossRef]

G. Oliveri and A. Massa, “Bayesian compressive sampling for pattern synthesis with maximally sparse non-uniform linear arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag. 60, 2309–2326 (2012).
[CrossRef]

F. Viani, G. Oliveri, and A. Massa, “Compressive sensing pattern matching techniques for synthesizing planar sparse arrays,” IEEE Trans. Antennas Propag. 59, 467–481 (2011).
[CrossRef]

G. Oliveri, P. Rocca, and A. Massa, “Reliable diagnosis of large linear arrays—Bayesian compressive sensing approach,” IEEE Trans. Antennas Propag. 60, 4627–4636 (2012).
[CrossRef]

T. Miwa and I. Arai, “Super-resolution imaging for point reflectors near transmitting and receiving array,” IEEE Trans. Antennas Propag. 52, 220–229 (2004).
[CrossRef]

A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy,” IEEE Trans. Antennas Propag. 51, 1251–1262 (2003).
[CrossRef]

R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).
[CrossRef]

R. Solimene, A. Brancaccio, J. Romano, and R. Pierri, “Localizing thin metallic cylinders by a 2.5-D linear distributional approach: experimental results,” IEEE Trans. Antennas Propag. 56, 2630–2637 (2008).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (6)

I. T. Rekanos, “Shape reconstruction of a perfectly conducting scatterer using differential evolution and particle swarm optimization,” IEEE Trans. Geosci. Remote Sens. 46, 1967–1974 (2008).
[CrossRef]

G. Micolau, M. Saillard, and P. Borderies, “DORT method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sens. 41, 1813–1820 (2003).
[CrossRef]

C.-C. Chiu, C.-H. Sun, C.-L. Li, and C.-H. Huang, “Comparative study of some population-based optimization algorithms on inverse scattering of a two-dimensional perfectly conducting cylinder in dielectric slab medium,” IEEE Trans. Geosci. Remote Sens. 51, 2302–2315 (2013).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

LSF–MT-BCS calibration (multiple scatterers, =λ/6, O=S=10). Behavior of the average integral error, ξ¯, versus (a) β1 and β2 when η=ηopt and (b) η when β1=β1opt and β2=β2opt.

Fig. 2.
Fig. 2.

Illustrative example (multiple scatterers, =λ/6, O=S=10, SNR=20dB). (a), (c), (e) Binary function γ and (b), (d) amplitude coefficients av (v=1) of (a), (b) the actual PEC profile and the LSF–MT-BCS (c), (d) single-view (v=1) and (e) multiview (v=1,,V) reconstructions.

Fig. 3.
Fig. 3.

Numerical assessment (multiple L-shaped scatterers, O=3, S=9). (a) Actual and (b)–(e) retrieved binary function γ with (b), (c) the LSF–MT-BCS and (d), (e) the LSF–ST-BCS when (b), (d) SNR=30dB and (c), (e) SNR=5dB.

Fig. 4.
Fig. 4.

Numerical assessment (multiple L-shaped scatterers, O=3, S=9). Behavior of the integral error ξ versus SNR.

Fig. 5.
Fig. 5.

Numerical assessment (multiple L-shaped scatterers, O=4, S=12, SNR=5dB). (a) Actual and retrieved binary function γ with (b) the LSFMTBCS and (c) the LSFSTBCS.

Fig. 6.
Fig. 6.

Numerical assessment (multiple L-shaped scatterers, SNR{5,10,20}dB). Behavior of the integral error ξ versus S.

Fig. 7.
Fig. 7.

Numerical assessment (statistical analysis, SNR=10dB). Behavior of the average (points) and maximum and minimum (error bars) values of the integral error ξ versus S in the presence of (a) =λ/6 and (b) =λ/3 scatterers.

Fig. 8.
Fig. 8.

Numerical assessment (multiple squares profile, S=8, =λ/6, SNR=10dB). (a) Behavior of the integral error ξ versus V when M=27. (b)–(e) Reconstructed distributions when (b), (c) V=5 and (d), (e) V=15 by applying (b), (d) the LSF–ST-BCS and (c), (e) the LSF–MT-BCS.

Fig. 9.
Fig. 9.

Numerical assessment (multiple squares profile, S=8, =λ/6, SNR=10dB). (a) Behavior of the integral error ξ versus the total number of measurement points when V=M. Reconstructed distributions when M=V=15 by applying (b) the LSF–ST-BCS and (c) the LSF–MT-BCS.

Fig. 10.
Fig. 10.

Numerical assessment (multiple squares profiles, =λ/6, SN/50, SNR=10dB). (a) Behavior of the integral error ξ and the CPU time versus L[λ,5λ]. (b), (c) Reconstructed distributions when L=5λ by applying (b) the LSF–ST-BCS and (c) the LSF–MT-BCS.

Equations (22)

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γn={1,rnΩPEC0,rnΩPECn=1,,N.
ev(r)=n=1N[H0(1)(2πλ|rrn|)anv]rΩobs
av=[diag(γ)·Ψ+I]1·diag(γ)·Φ·fv,
ψnp={J0(2πδλ)H0(1)(2πλ|rnrp|)H0(1)(2πδλ)np0n=p,
ϕnq=J0(2πδλ)H0(1)(2πδλ)Jq(2πλ|rn|)exp[iqarctan(ynxn)],
fv(r)q=QQ{Jq(2πλ|r|)exp[iqarctan(yx)]fqv}
Hv{Hvmn=H0(1)(2πλ|rmvrn|),m=1,,M,n=1,,N}
A^varg{maxAv[P(Av|Ev)]},v=1,,V,
P(Av|Ev)=P(Ev|Av)P(Av)P(Ev)
P(Av)P(Av|α,σ)P(α)P(σ)dαdσ,
P(Av|Ev)=P(Av|Ev,α)P(α|Ev)dα,
A^v=[diag(α^)+(Kv)Kv]1(Kv)Evv=1,,V,
Jv{Jmn=J0(2πλ|rmvrn|),m=1,,M,n=1,,N}
Yv{Ymn=Y0(2πλ|rmvrn|),m=1,,M,n=1,,N},
I(α)=12v=1V{log(|C|)+(2N+2β1)log[(Ev)CEv+2β2]},
a^v{a^nv=A^vn+iA^vn+N,n=1,,N}v=1,,V.
γ^nvH(|a^nv|maxn{|a^nv|}η)n=1,,N,
γ^nH(v=1Vγ^nvV0.5)n=1,,N.
rn=[LN(n1)modN,LN(n1)N]n=1,,N
θv=2π(v1)V,v=1,,V.
rmv=[ρcos(2π(m1)M),ρsin(2π(m1)M)]m=1,,M,
ξ1Nn=1N|γ^nγn|2n=1N|γn|2=n=1N|γ^nγn|2N×S

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