Abstract

This paper investigates the resolution and robustness of the multiple signal classification (MUSIC) method to locate small three-dimensional (3D) anisotropic scatterers near the medium interface in a multilayered background. An enhanced MUSIC algorithm developed for free-space background is extended to solve such a problem. Because its indicator is built in a stable signal subspace, which is continuous across the medium interface, better stability and higher resolution against noise are observed for the proposed method compared to the known standard MUSIC method. Numerical simulations with various medium interfaces and noise levels are conducted to verify the performance of the introduced MUSIC method.

© 2012 Optical Society of America

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  1. P. van den Berg and A. Abubakar, “Optical microscopy imaging using the contrast source inversion method,” J. Mod. Opt. 57, 756–764 (2010).
    [CrossRef]
  2. P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338–2347 (2005).
    [CrossRef]
  3. G. Bao and P. Li, “Numerical solution of inverse scattering for near-field optics,” Opt. Lett. 32, 1465–1467 (2007).
    [CrossRef]
  4. G. Oliveri, Y. Zhong, X. Chen, and A. Massa, “Multiresolution subspace-based optimization method for inverse scattering problems,” J. Opt. Soc. Am. A 28, 2057–2069 (2011).
    [CrossRef]
  5. D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034–1041 (2007).
    [CrossRef]
  6. E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619–3635 (2007).
    [CrossRef]
  7. X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25, 357–364 (2008).
    [CrossRef]
  8. P. C. Chaumet, K. Belkebir, and R. Lencrerot, “Three-dimensional optical imaging in layered media,” Opt. Express 14, 3415–3426 (2006).
    [CrossRef]
  9. Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
    [CrossRef]
  10. W. J. Walecki and F. Szondy, “Integrated quantum efficiency, reflectance, topography and stress metrology for solar cell manufacturing,” Proc. SPIE 7064, 70640A (2008).
    [CrossRef]
  11. H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,” Multiscale Model. Simul. 3, 597–628 (2005).
    [CrossRef]
  12. W. Zhang and A. Hoorfar, “Through-the-wall target localization with time reversal MUSIC method,” Prog. Electromagn. Res. 106, 75–89 (2010).
    [CrossRef]
  13. E. Iakovleva, S. Gdoura, and D. Lesselier, “Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging,” IEEE Trans. Antennas Propag. 55, 2598–2609 (2007).
    [CrossRef]
  14. X. Chen and Y. Zhong, “MUSIC electromagnetic imaging with enchanced resolution for small inclusions,” Inverse Probl. 25, 015008 (2009).
    [CrossRef]
  15. X. Chen, “Multiple signal classification method for detecting point-like scatterers embedded in an inhomogeneous background medium,” J. Acoust. Soc. Am. 127, 2392–2397 (2010).
    [CrossRef]
  16. X. Chen, “Signal-subspace method approach to the intensity-only electromagnetic inverse scattering problem,” J. Opt. Soc. Am. A 25, 2018–2024 (2008).
    [CrossRef]
  17. Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542–3549 (2007).
    [CrossRef]
  18. C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
    [CrossRef]
  19. A. Dubois, K. Belkebir, and M. Saillard, “Localization and characterization of twodimensional targets buried in a cluttered environment,” Inverse Probl. 20, S63–S79 (2004).
    [CrossRef]
  20. A. Devaney, “Time reversal imaging of obscured targets from multistatic data,” IEEE Trans. Antennas Propag. 53, 1600–1610 (2005).
    [CrossRef]
  21. L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).
  22. X. Chen, Y. Zhong, and K. Agarwal, “Subspace methods for solving electromagnetic inverse scattering problems,” Methods Appl. Anal. 17, 407–432 (2010).

2011

2010

P. van den Berg and A. Abubakar, “Optical microscopy imaging using the contrast source inversion method,” J. Mod. Opt. 57, 756–764 (2010).
[CrossRef]

W. Zhang and A. Hoorfar, “Through-the-wall target localization with time reversal MUSIC method,” Prog. Electromagn. Res. 106, 75–89 (2010).
[CrossRef]

X. Chen, “Multiple signal classification method for detecting point-like scatterers embedded in an inhomogeneous background medium,” J. Acoust. Soc. Am. 127, 2392–2397 (2010).
[CrossRef]

X. Chen, Y. Zhong, and K. Agarwal, “Subspace methods for solving electromagnetic inverse scattering problems,” Methods Appl. Anal. 17, 407–432 (2010).

2009

X. Chen and Y. Zhong, “MUSIC electromagnetic imaging with enchanced resolution for small inclusions,” Inverse Probl. 25, 015008 (2009).
[CrossRef]

2008

2007

D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Inverse scattering for frequency-scanned full-field optical coherence tomography,” J. Opt. Soc. Am. A 24, 1034–1041 (2007).
[CrossRef]

E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619–3635 (2007).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542–3549 (2007).
[CrossRef]

G. Bao and P. Li, “Numerical solution of inverse scattering for near-field optics,” Opt. Lett. 32, 1465–1467 (2007).
[CrossRef]

E. Iakovleva, S. Gdoura, and D. Lesselier, “Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging,” IEEE Trans. Antennas Propag. 55, 2598–2609 (2007).
[CrossRef]

2006

2005

H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,” Multiscale Model. Simul. 3, 597–628 (2005).
[CrossRef]

P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338–2347 (2005).
[CrossRef]

A. Devaney, “Time reversal imaging of obscured targets from multistatic data,” IEEE Trans. Antennas Propag. 53, 1600–1610 (2005).
[CrossRef]

2004

A. Dubois, K. Belkebir, and M. Saillard, “Localization and characterization of twodimensional targets buried in a cluttered environment,” Inverse Probl. 20, S63–S79 (2004).
[CrossRef]

2002

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

1996

C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
[CrossRef]

1994

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).

Abubakar, A.

P. van den Berg and A. Abubakar, “Optical microscopy imaging using the contrast source inversion method,” J. Mod. Opt. 57, 756–764 (2010).
[CrossRef]

Agarwal, K.

X. Chen, Y. Zhong, and K. Agarwal, “Subspace methods for solving electromagnetic inverse scattering problems,” Methods Appl. Anal. 17, 407–432 (2010).

Ammari, H.

H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,” Multiscale Model. Simul. 3, 597–628 (2005).
[CrossRef]

Bao, G.

Belkebir, K.

P. C. Chaumet, K. Belkebir, and R. Lencrerot, “Three-dimensional optical imaging in layered media,” Opt. Express 14, 3415–3426 (2006).
[CrossRef]

A. Dubois, K. Belkebir, and M. Saillard, “Localization and characterization of twodimensional targets buried in a cluttered environment,” Inverse Probl. 20, S63–S79 (2004).
[CrossRef]

Boppart, S. A.

Bryan, J.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Carney, P. S.

Chaumet, P. C.

Chen, X.

G. Oliveri, Y. Zhong, X. Chen, and A. Massa, “Multiresolution subspace-based optimization method for inverse scattering problems,” J. Opt. Soc. Am. A 28, 2057–2069 (2011).
[CrossRef]

X. Chen, “Multiple signal classification method for detecting point-like scatterers embedded in an inhomogeneous background medium,” J. Acoust. Soc. Am. 127, 2392–2397 (2010).
[CrossRef]

X. Chen, Y. Zhong, and K. Agarwal, “Subspace methods for solving electromagnetic inverse scattering problems,” Methods Appl. Anal. 17, 407–432 (2010).

X. Chen and Y. Zhong, “MUSIC electromagnetic imaging with enchanced resolution for small inclusions,” Inverse Probl. 25, 015008 (2009).
[CrossRef]

X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25, 357–364 (2008).
[CrossRef]

X. Chen, “Signal-subspace method approach to the intensity-only electromagnetic inverse scattering problem,” J. Opt. Soc. Am. A 25, 2018–2024 (2008).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542–3549 (2007).
[CrossRef]

Devaney, A.

A. Devaney, “Time reversal imaging of obscured targets from multistatic data,” IEEE Trans. Antennas Propag. 53, 1600–1610 (2005).
[CrossRef]

Devaney, A. J.

Dubois, A.

A. Dubois, K. Belkebir, and M. Saillard, “Localization and characterization of twodimensional targets buried in a cluttered environment,” Inverse Probl. 20, S63–S79 (2004).
[CrossRef]

Fink, M.

C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
[CrossRef]

Gdoura, S.

E. Iakovleva, S. Gdoura, and D. Lesselier, “Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging,” IEEE Trans. Antennas Propag. 55, 2598–2609 (2007).
[CrossRef]

Guo, P.

Hernandez, R. D.

Hoorfar, A.

W. Zhang and A. Hoorfar, “Through-the-wall target localization with time reversal MUSIC method,” Prog. Electromagn. Res. 106, 75–89 (2010).
[CrossRef]

Iakovleva, E.

E. Iakovleva, S. Gdoura, and D. Lesselier, “Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging,” IEEE Trans. Antennas Propag. 55, 2598–2609 (2007).
[CrossRef]

H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,” Multiscale Model. Simul. 3, 597–628 (2005).
[CrossRef]

Joines, W.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Kooi, P. S.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).

Lencrerot, R.

Leong, M. S.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).

Lesselier, D.

E. Iakovleva, S. Gdoura, and D. Lesselier, “Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging,” IEEE Trans. Antennas Propag. 55, 2598–2609 (2007).
[CrossRef]

H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,” Multiscale Model. Simul. 3, 597–628 (2005).
[CrossRef]

Lev-Ari, H.

Li, L. W.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).

Li, P.

Liu, Q. H.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Manneville, S.

C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
[CrossRef]

Marengo, E. A.

Marks, D. L.

Massa, A.

Nolte, L.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Oliveri, G.

Prada, C.

C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
[CrossRef]

Ralston, T. S.

Saillard, M.

A. Dubois, K. Belkebir, and M. Saillard, “Localization and characterization of twodimensional targets buried in a cluttered environment,” Inverse Probl. 20, S63–S79 (2004).
[CrossRef]

Spoliansky, D.

C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
[CrossRef]

Szondy, F.

W. J. Walecki and F. Szondy, “Integrated quantum efficiency, reflectance, topography and stress metrology for solar cell manufacturing,” Proc. SPIE 7064, 70640A (2008).
[CrossRef]

van den Berg, P.

P. van den Berg and A. Abubakar, “Optical microscopy imaging using the contrast source inversion method,” J. Mod. Opt. 57, 756–764 (2010).
[CrossRef]

Walecki, W. J.

W. J. Walecki and F. Szondy, “Integrated quantum efficiency, reflectance, topography and stress metrology for solar cell manufacturing,” Proc. SPIE 7064, 70640A (2008).
[CrossRef]

Wang, T.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Ybarra, G.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Yeo, T. S.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).

Zhang, W.

W. Zhang and A. Hoorfar, “Through-the-wall target localization with time reversal MUSIC method,” Prog. Electromagn. Res. 106, 75–89 (2010).
[CrossRef]

Zhang, Z. Q.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Zhong, Y.

G. Oliveri, Y. Zhong, X. Chen, and A. Massa, “Multiresolution subspace-based optimization method for inverse scattering problems,” J. Opt. Soc. Am. A 28, 2057–2069 (2011).
[CrossRef]

X. Chen, Y. Zhong, and K. Agarwal, “Subspace methods for solving electromagnetic inverse scattering problems,” Methods Appl. Anal. 17, 407–432 (2010).

X. Chen and Y. Zhong, “MUSIC electromagnetic imaging with enchanced resolution for small inclusions,” Inverse Probl. 25, 015008 (2009).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542–3549 (2007).
[CrossRef]

IEEE Trans. Antennas Propag.

E. Iakovleva, S. Gdoura, and D. Lesselier, “Multistatic response matrix of a 3-D inclusion in half space and MUSIC imaging,” IEEE Trans. Antennas Propag. 55, 2598–2609 (2007).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542–3549 (2007).
[CrossRef]

A. Devaney, “Time reversal imaging of obscured targets from multistatic data,” IEEE Trans. Antennas Propag. 53, 1600–1610 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, G. Ybarra, L. Nolte, and W. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech. 50, 123–133 (2002).
[CrossRef]

Inverse Probl.

X. Chen and Y. Zhong, “MUSIC electromagnetic imaging with enchanced resolution for small inclusions,” Inverse Probl. 25, 015008 (2009).
[CrossRef]

A. Dubois, K. Belkebir, and M. Saillard, “Localization and characterization of twodimensional targets buried in a cluttered environment,” Inverse Probl. 20, S63–S79 (2004).
[CrossRef]

J. Acoust. Soc. Am.

X. Chen, “Multiple signal classification method for detecting point-like scatterers embedded in an inhomogeneous background medium,” J. Acoust. Soc. Am. 127, 2392–2397 (2010).
[CrossRef]

C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: detection and selective focusing on two scatterers, ”J. Acoust. Soc. Am. 99, 2067–2076(1996).
[CrossRef]

J. Electromagn. Waves Appl.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “On the eigenfunction expansion of dyadic Green’s function in planarly stratified media,” J. Electromagn. Waves Appl. 8, 663–678 (1994).

J. Mod. Opt.

P. van den Berg and A. Abubakar, “Optical microscopy imaging using the contrast source inversion method,” J. Mod. Opt. 57, 756–764 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Methods Appl. Anal.

X. Chen, Y. Zhong, and K. Agarwal, “Subspace methods for solving electromagnetic inverse scattering problems,” Methods Appl. Anal. 17, 407–432 (2010).

Multiscale Model. Simul.

H. Ammari, E. Iakovleva, and D. Lesselier, “A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,” Multiscale Model. Simul. 3, 597–628 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

W. J. Walecki and F. Szondy, “Integrated quantum efficiency, reflectance, topography and stress metrology for solar cell manufacturing,” Proc. SPIE 7064, 70640A (2008).
[CrossRef]

Prog. Electromagn. Res.

W. Zhang and A. Hoorfar, “Through-the-wall target localization with time reversal MUSIC method,” Prog. Electromagn. Res. 106, 75–89 (2010).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Problem sketch of inverse scattering: (a) the 3D view of the problem where the green plane indicates the medium interface; (b) the 2D view of scatterer locations on y=z plane, where the square domain of interest is denoted by blue points and the red points are the locations of scatterers.

Fig. 2.
Fig. 2.

Singular values of MSR matrices with 30 dB white Gaussian noise under different medium interfaces: (a) [z0,z1]=[0,0.2]λ0, (b) [z0,z1]=[0.2,0.2]λ0, (c) [z0,z1]=[0,0.05]λ0, and (d) [z0,z1]=[0,0.12]λ0.

Fig. 3.
Fig. 3.

10-base-logarithm pseudospectrums of MUSIC algorithms with different medium interfaces. The white dotted line represents the interface location. Each column corresponds to results with one given interface configuration: the first column [z0,z1]=[0,0.2]λ0, the second column [z0,z1]=[0.2,0.2]λ0, the third column [z0,z1]=[0,0.05]λ0, and the fourth column [z0,z1]=[0,0.12]λ0. Each row corresponds to the results of a method: the first row enhanced MUSIC, the second row standard MUSIC with a¯1=(1,1,1)T, the third row standard MUSIC with a¯2=(1,0,0)T, and the fourth row standard MUSIC with a¯3=(0,0,1)T.

Fig. 4.
Fig. 4.

Singular values of MSR matrices with different noise levels where medium interfaces are at [z0,z1]=[0,0.05]λ0: (a) 25 dB, (b) 20 dB, and (c) 15 dB.

Fig. 5.
Fig. 5.

10-base-logarithm pseudospectrums of MUSIC algorithms with different noise levels where medium interfaces are at [z0,z1]=[0,0.05]λ0. The white dotted line represents the interface location. Each column corresponds to results with one given noise level: the first column 25 dB, the second column 20 dB, and the third column 15 dB. Each row corresponds to results of a method: the first row enhanced MUSIC, the second row standard MUSIC with a¯1=(1,1,1)T, the third row standard MUSIC with a¯2=(1,0,0)T, and the fourth row standard MUSIC with a¯3=(0,0,1)T.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

ϵ¯¯m=[ϵmxxϵmxyϵmxzϵmyxϵmyyϵmyzϵmzxϵmzyϵmzz]=Ξ¯¯E,m1[ϵm(1)000ϵm(2)000ϵm(3)]Ξ¯¯E,m,
E¯tin(s¯j)=E¯0in(s¯j)+m=1,mjM{iωμ0G¯¯(s¯j,s¯m)·ξ¯¯m(ϵb(s¯m))·E¯tin(s¯m)}j=1,2,,M,
E¯sca(r¯l)=m=1M{iωμ0G¯¯(r¯l,s¯m)·ξ¯¯m(ϵb(s¯m))·E¯tin(s¯m)},l=1,2,,N,
ξ¯¯m(ϵb(s¯m))=i4πkb(s¯m)am3ϵb(s¯m)μ0Ξ¯¯E,m1·diag[ϵm(1)ϵb(s¯m)ϵm(1)+2ϵb(s¯m),ϵm(2)ϵb(s¯m)ϵm(2)+2ϵb(s¯m),ϵm(3)ϵb(s¯m)ϵm(3)+2ϵb(s¯m)]·Ξ¯¯E,m
A¯¯=R¯¯·Λ¯¯·(I¯¯Φ¯¯·Λ¯¯)1·R¯¯T,
US=span{u¯p,σp>0}=span{u¯1,u¯2,,u¯K},UN=span{u¯p,σp=0}=span{u¯K+1,u¯K+2,,u¯3N},
Q¯¯(s¯)·a¯USif and only ifs¯{s¯1,s¯2,,s¯M}
W1(a¯,s¯)=1i=K+13N|u¯iH·Q¯¯(s¯)·a¯|2,
cos(θ(a¯,s¯))=|q¯(a¯,s¯)||v¯(a¯,s¯)|,
W1(a¯,s¯)=1(1cos2(θ))|v¯(a¯,s¯)|.
W2(a¯opt(s¯),s¯)=11cos2(θmin(s¯)).
a¯opt(s¯)=argmaxa¯i=1L|u¯iH·Q¯¯(s¯)·a¯|2|Q¯¯(s¯)·a¯|2,
US=span{u¯1,u¯2,,u¯K},UN=span{u¯K+1,u¯K+2,,u¯3N},
G¯x(r¯+,s¯)=G¯x(r¯,s¯),G¯y(r¯+,s¯)=G¯y(r¯,s¯),ϵr(r¯+)G¯z(r¯+,s¯)=ϵr(r¯)G¯z(r¯,s¯),
Q¯¯(s¯)=[Q¯x,Q¯y,Q¯z]=[G¯¯T(r¯1,s¯),G¯¯T(r¯2,s¯),,G¯¯T(r¯N,s¯)]T,
=[G¯¯(s¯,r¯1),G¯¯(s¯,r¯2),,G¯¯(s¯,r¯N)]T.
Q¯x(s¯+)=Q¯x(s¯),Q¯y(s¯+)=Q¯y(s¯),ϵr(s¯+)Q¯z(s¯+)=ϵr(s¯)Q¯z(s¯).

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