Abstract

The problem of determining the shape of perfectly conducting objects from knowledge of the scattered electric field is considered. The formulation of the problem accommodates the nature of the distribution of the induced surface current density. Thus, as the unknown representing the object’s contour, a single layer distribution is chosen so that the contour of the scatterer is described by its support. The nonlinear unknown-data mapping is then linearized by means of the Kirchhoff approximation, and the problem is recast as the inversion of a linear operator acting on a distribution space. An extension of the singular value decomposition approach to solve the linearized problem is provided and numerical results are presented.

© 2002 Optical Society of America

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References

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  1. B. Borden, “Some issues in inverse synthetic aperture radar image reconstruction,” Inverse Probl. 13, 571–584 (1997).
    [CrossRef]
  2. A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
    [CrossRef]
  3. K. J. Langenberg, “Elastic wave inverse scattering as applied to nondestructive evaluation,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike eds. (Adam Hilger, Bristol, UK, 1992).
  4. R. Kress, “Numerical methods in inverse obstacle scattering,” Aust. New Zeal. Ind. Appl. Math. J. 42, C44–C67 (2000).
  5. A. Kirsch, “The domain derivative and two applications in inverse scattering theory,” Inverse Probl. 9, 81–96 (1993).
    [CrossRef]
  6. A. Qing, C. K. Lee, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
    [CrossRef]
  7. R. D. Marger, N. Bleistein, “An examination of the limited aperture problem of physical optics inverse scattering,” IEEE Trans. Antennas Propag. AP-26, 695–699 (1979).
  8. K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).
  9. R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
    [CrossRef]
  10. Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
    [CrossRef]
  11. D. S. Jones, Methods in Electromagnetic Wave Propagation (Oxford U., Press, Oxford, UK, 1994).
  12. V. S. Vladimirov, Generalized Functions in the Mathematical Physics (Mir, Moscow, 1997).
  13. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK1998).
  14. The scattered electric field can be determined by decomposing the total electric field Eas E=Es+Ei,where Eiis the incident field, and by solving for Ethe Helmholtz equation with a Dirichlet boundary condition, which corresponds to perfectly conducting objects. Note that such boundary condition corresponds also to the case of acoustic soft scatterers.4
  15. N. Morita, “The boundary-element method,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita ed. (Artech House, Boston, Mass.1990).
  16. L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, New York, 1996).
  17. A. D. Yaghjian, T. B. Hansen, A. J. Devaney, “Minimum source region for a given far-field pattern,” IEEE Trans. Antennas Propag. 45, 911–912 (1997).
    [CrossRef]
  18. I. D. King, T. E. Hodgetts, “An approximate treatment of scattering based on the delta boundary operator technique,” IMA J. Appl. Math. 64, 139–155 (2000).
    [CrossRef]
  19. L. Kantorovich, G. Akilov, Functional Analysis (Pergamon, New York, 1982).
  20. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  21. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  22. E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
    [CrossRef]
  23. R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
    [CrossRef]
  24. A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
    [CrossRef]
  25. R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
    [CrossRef]
  26. A. Liseno, R. Pierri, F. Soldovieri, “Depth resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
    [CrossRef]
  27. A†denotes the adjoint of A.
  28. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, (Nauka, Moscow, 1981).

2001 (4)

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

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

A. Liseno, R. Pierri, F. Soldovieri, “Depth resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

2000 (3)

I. D. King, T. E. Hodgetts, “An approximate treatment of scattering based on the delta boundary operator technique,” IMA J. Appl. Math. 64, 139–155 (2000).
[CrossRef]

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

R. Kress, “Numerical methods in inverse obstacle scattering,” Aust. New Zeal. Ind. Appl. Math. J. 42, C44–C67 (2000).

1999 (1)

Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
[CrossRef]

1998 (2)

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

1997 (2)

B. Borden, “Some issues in inverse synthetic aperture radar image reconstruction,” Inverse Probl. 13, 571–584 (1997).
[CrossRef]

A. D. Yaghjian, T. B. Hansen, A. J. Devaney, “Minimum source region for a given far-field pattern,” IEEE Trans. Antennas Propag. 45, 911–912 (1997).
[CrossRef]

1993 (2)

A. Kirsch, “The domain derivative and two applications in inverse scattering theory,” Inverse Probl. 9, 81–96 (1993).
[CrossRef]

E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
[CrossRef]

1979 (1)

R. D. Marger, N. Bleistein, “An examination of the limited aperture problem of physical optics inverse scattering,” IEEE Trans. Antennas Propag. AP-26, 695–699 (1979).

Akilov, G.

L. Kantorovich, G. Akilov, Functional Analysis (Pergamon, New York, 1982).

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK1998).

Bleistein, N.

R. D. Marger, N. Bleistein, “An examination of the limited aperture problem of physical optics inverse scattering,” IEEE Trans. Antennas Propag. AP-26, 695–699 (1979).

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK1998).

Borden, B.

B. Borden, “Some issues in inverse synthetic aperture radar image reconstruction,” Inverse Probl. 13, 571–584 (1997).
[CrossRef]

Brancaccio, A.

Brandfass, M.

K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).

Carin, L.

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

Chen, K. M.

Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
[CrossRef]

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Dai, Y.

Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
[CrossRef]

Damarla, R.

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

Devaney, A. J.

A. D. Yaghjian, T. B. Hansen, A. J. Devaney, “Minimum source region for a given far-field pattern,” IEEE Trans. Antennas Propag. 45, 911–912 (1997).
[CrossRef]

Dong, Y.

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

Fellinger, P.

K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).

Fomin, S. V.

N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, (Nauka, Moscow, 1981).

Geng, N.

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

Gurke, T.

K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).

Hansen, T. B.

A. D. Yaghjian, T. B. Hansen, A. J. Devaney, “Minimum source region for a given far-field pattern,” IEEE Trans. Antennas Propag. 45, 911–912 (1997).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Hodgetts, T. E.

I. D. King, T. E. Hodgetts, “An approximate treatment of scattering based on the delta boundary operator technique,” IMA J. Appl. Math. 64, 139–155 (2000).
[CrossRef]

Jones, D. S.

D. S. Jones, Methods in Electromagnetic Wave Propagation (Oxford U., Press, Oxford, UK, 1994).

Kantorovich, L.

L. Kantorovich, G. Akilov, Functional Analysis (Pergamon, New York, 1982).

King, I. D.

I. D. King, T. E. Hodgetts, “An approximate treatment of scattering based on the delta boundary operator technique,” IMA J. Appl. Math. 64, 139–155 (2000).
[CrossRef]

Kirsch, A.

A. Kirsch, “The domain derivative and two applications in inverse scattering theory,” Inverse Probl. 9, 81–96 (1993).
[CrossRef]

Kolmogorov, N.

N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, (Nauka, Moscow, 1981).

Kress, R.

R. Kress, “Numerical methods in inverse obstacle scattering,” Aust. New Zeal. Ind. Appl. Math. J. 42, C44–C67 (2000).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Kreutter, T.

K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).

Langenberg, K. J.

K. J. Langenberg, “Elastic wave inverse scattering as applied to nondestructive evaluation,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike eds. (Adam Hilger, Bristol, UK, 1992).

K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).

Lee, C. K.

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

Leone, G.

Liseno, A.

A. Liseno, R. Pierri, F. Soldovieri, “Depth resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

Marger, R. D.

R. D. Marger, N. Bleistein, “An examination of the limited aperture problem of physical optics inverse scattering,” IEEE Trans. Antennas Propag. AP-26, 695–699 (1979).

Morita, N.

N. Morita, “The boundary-element method,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita ed. (Artech House, Boston, Mass.1990).

Nyquist, D. P.

Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
[CrossRef]

Pierri, R.

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

A. Liseno, R. Pierri, F. Soldovieri, “Depth resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

Qing, A.

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

Rothwell, E. J.

Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
[CrossRef]

Scalas, E.

Schwartz, L.

L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, New York, 1996).

Soldovieri, F.

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

A. Liseno, R. Pierri, F. Soldovieri, “Depth resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

Solimene, R.

Sullivan, A.

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

Viano, G. A.

Vladimirov, V. S.

V. S. Vladimirov, Generalized Functions in the Mathematical Physics (Mir, Moscow, 1997).

Yaghjian, A. D.

A. D. Yaghjian, T. B. Hansen, A. J. Devaney, “Minimum source region for a given far-field pattern,” IEEE Trans. Antennas Propag. 45, 911–912 (1997).
[CrossRef]

Aust. New Zeal. Ind. Appl. Math. J. (1)

R. Kress, “Numerical methods in inverse obstacle scattering,” Aust. New Zeal. Ind. Appl. Math. J. 42, C44–C67 (2000).

IEEE Trans. Antennas Propag. (5)

A. Sullivan, R. Damarla, N. Geng, Y. Dong, L. Carin, “Ultrawide-band synthetic aperture radar for detection of unexploded ordnance: modeling and measurements,” IEEE Trans. Antennas Propag. 48, 1306–1315 (2000).
[CrossRef]

R. D. Marger, N. Bleistein, “An examination of the limited aperture problem of physical optics inverse scattering,” IEEE Trans. Antennas Propag. AP-26, 695–699 (1979).

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

Y. Dai, E. J. Rothwell, K. M. Chen, D. P. Nyquist, “Time-domain imaging of radar targets using sinogram restoration for limited-view reconstruction,” IEEE Trans. Antennas Propag. 47, 1323–1329 (1999).
[CrossRef]

A. D. Yaghjian, T. B. Hansen, A. J. Devaney, “Minimum source region for a given far-field pattern,” IEEE Trans. Antennas Propag. 45, 911–912 (1997).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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

IMA J. Appl. Math. (1)

I. D. King, T. E. Hodgetts, “An approximate treatment of scattering based on the delta boundary operator technique,” IMA J. Appl. Math. 64, 139–155 (2000).
[CrossRef]

Int. J. Electron. Commun. (AEÜ) (1)

A. Liseno, R. Pierri, F. Soldovieri, “Depth resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

Inverse Probl. (3)

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

B. Borden, “Some issues in inverse synthetic aperture radar image reconstruction,” Inverse Probl. 13, 571–584 (1997).
[CrossRef]

A. Kirsch, “The domain derivative and two applications in inverse scattering theory,” Inverse Probl. 9, 81–96 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Other (13)

A†denotes the adjoint of A.

N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, (Nauka, Moscow, 1981).

L. Kantorovich, G. Akilov, Functional Analysis (Pergamon, New York, 1982).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

K. J. Langenberg, “Elastic wave inverse scattering as applied to nondestructive evaluation,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike eds. (Adam Hilger, Bristol, UK, 1992).

K. J. Langenberg, M. Brandfass, P. Fellinger, T. Gurke, T. Kreutter, “A unified theory of multidimensional electromagnetic vector inverse scattering within the Kirchoff or Born approximation,” in Radar Target Imaging, W.-M. Boerner, H. Überall eds. (Springer-Verlag, Berlin, 1994).

D. S. Jones, Methods in Electromagnetic Wave Propagation (Oxford U., Press, Oxford, UK, 1994).

V. S. Vladimirov, Generalized Functions in the Mathematical Physics (Mir, Moscow, 1997).

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK1998).

The scattered electric field can be determined by decomposing the total electric field Eas E=Es+Ei,where Eiis the incident field, and by solving for Ethe Helmholtz equation with a Dirichlet boundary condition, which corresponds to perfectly conducting objects. Note that such boundary condition corresponds also to the case of acoustic soft scatterers.4

N. Morita, “The boundary-element method,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita ed. (Artech House, Boston, Mass.1990).

L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, New York, 1996).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Illuminated and shadow regions of the contour Γ for a given angle of incidence θi.

Fig. 3
Fig. 3

Case of a perfectly conducting plane.

Fig. 4
Fig. 4

Density of L2(O) in E(O).

Fig. 5
Fig. 5

Allowable spectral domain Suv.

Fig. 6
Fig. 6

Singular values of L for the considered measurement configuration.

Fig. 7
Fig. 7

Reconstruction of a cylinder with circular cross section and radius 1.75a. The positive part of -Re{Rγ} is depicted. Noiseless data.

Fig. 8
Fig. 8

Reconstruction of a cylinder with circular cross section and radius 1.75a. The positive part of -Re{Rγ} is depicted. Noisy data with 5-dB signal-to-noise ratio.

Fig. 9
Fig. 9

Reconstruction of a cylinder with circular cross section and radius 1.75a. -Re{Rγ} is depicted. Noiseless data.

Fig. 10
Fig. 10

Reconstruction of the bean-shaped cylinder.

Fig. 11
Fig. 11

Reconstruction of two cylinders with circular cross section and radii 0.8a separated by 0.8a.

Tables (1)

Tables Icon

Table 1 Relevant Symbols

Equations (56)

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

Es(k0, r)=-ωμ04ΓH0(2)(k0R)J(r)dΓ,
J=2Ji+j2nΓH0(2)(k0R)J(r)dΓonΓ,
JδΓ|φ=ΓJ(r)φ(r)dΓ,
Es(k0, r)=-ωμ04 JδΓ|H0(2)(k0R),
J=2Ji+j2n JδΓ|H0(2)(k0R)onΓ,
Es(k0, r)=-ωμ04OJH0(2)(k0R)dO,
J=2N×Hiı^z+j2 NOH0(2)(k0R)JdO,
Es=Ae[J],
J=2N×Hiı^z+NAs[J],
J=(I-NAs)-12N×Hiı^z,
Es=Ae[(I-NAs)-12N×Hiı^z],
J=2Ji-Γj2 nˆ(r-r)k0×H1(2)(k0R)R J(r)dΓonΓ.
JJPO=2JionΓi0onΓs,
-nˆ(r-r)k0H1(2)(k0R)R
nˆ(r-r) H1(2)(k0R)R|H1(2)(k0R)|,
 
JPO(r)=2nˆ×Hiı^zU(-nˆk^i)onΓ,
Hi(r)=1ζ0 (k^i×ı^z) exp(-jkir),
JPO(r)=-2ζ0 (nˆk^1) exp(-jkir)U(-nˆk^i)onΓ.
JPO(r)=-2ζ0 nxexp(-jk0x)U(-nx)onΓ.
Es(k0, r)=OK(k0, r; r)nxU(-nx)δΓdO,
L : γE(O)Lγ=γ|K(k0, r; r)L2(S)
Es(k0, r)=ΓiK(k0, r; r)nxU(-nx)dΓi,
φD(0)limk+O χk(r)φ(r)dO=γ|φ.
A : χL2(O)Aχ=OK(k0, r; r)χ(r)dOL2(S),
f,gL2(O)=O f(r)g*(r)dO,
h,lL2(S)=S h(k0, r)l*(k0, r)dS,
Aχ=n=0+σnχ,unL2(O)vn.
Lγ=n=0+σnγ|un*vn.
γ=n=0+γ|un*un+n=1dim[N(A)]γ|ψn*ψn.
Lγ=Es,
χ=n=0+1σn Esb, vnL2(S)un.
γ=n=0+1σn Es,vnL2(S)un
Es(u, v)=OnxU(-nx)δΓexp[-j(ux+vy)]dO,
Rγ=n=0N-11σn Es,vnL2(S)un,
(x(t), y(t))=(b cos t, b sin t),t0, π23π2, 2πb3cos(3t-π), 2b3+b3sin(3t-π),tπ2, 5π6b3cos(-3t+π), b3sin(-3t+π),t5π6, 7π6b3cos(3t+π), -2b3+b3sin(3t+π),t7π6, 3π2,
χ|φ=Oχ(x, y)φ(x, y)dOφD(O).
limk+dk|φ=d|φ,φD(0)
(Lγ)(k0, r)=limk+(Aχk)(k0, r),(k0, r)S.
AχkLγinL2(S).
Lγ=n=0+Lγ,vnL2(S)vn.
Lγ,vnL2(S)=limk+Aχk, vnL2(S)=limk+ σnχk, unL2(O)=σnγ|un*,
Lγ=n=0+σnγ|un*vn.
fN(x, y)=n=0N-1χk, unL2(O)un(x, y),(x, y)O.
γ|φ=limk+OlimN+ fN(x, y)φ(x, y)dO,
|fN(x, y)|c(x, y)O,N0.
γ|φ=limk+ limN+n=0N-1χk, unL2(O)un|φ.
γ|φ-n=0N-1γ|un*un|φ
|γ|φ-lk|+lk-n=0N-1χk,unL2(O)un|φ+n=0N-1|γ|un*-χk|un*||un|φ|
n=0+1σn δEs, vnL2(S)un
n=0+1σn2 |δEs,vnL2(S)|2<+,
Lγ=n=0+1σn Es, vnL2(S)un, K(k0, r; r)=n=0+Es, vnL2(S)vn,
γE(O)=supφD(O)1|γ|φ|=supφO|γ|φ|φD(O),
Esk-EsL2(S)=σk0,
γk=γ+1σk uk,γkE(O)k,
γk-γE(O)=1σk+.

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