Abstract

We discuss the problem of determining the class of dielectric profile functions that can be reconstructed from scattered electrical field data depending on the model to be inverted. We focus our attention on a cylindrical object with a circular cross section whose permittivity varies only along the angular coordinate. First we examine the linear Born approximation of the relationship between the permittivity function and the field scattered by the cylinder. We provide an analytical answer to this problem by singular-value decomposition of the relevant operator. We find that only slowly varying profiles can be recovered, both in the single-view and in the multiview case. Next we examine a quadratic approximation of the same relationship, which consists in adding a second-order term to the linear term. The effect of changing the model on the class of unknown functions that can be reconstructed is shown by means of both analytical arguments and numerical simulations. The result is that now the model includes more rapidly varying profiles.

© 1999 Optical Society of America

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References

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  1. D. Colton, “A survey of selected topics in inverse electromagnetic scattering theory,” in Inverse Problems in Wave Propagation, G. Chavent, G. Papanicolaou, P. Sacks, W. Symes, eds. (Springer-Verlag, New York, 1997), pp. 105–127.
  2. C. De Mol, “A critical survey of regularized inversion methods,” in Inverse Problems in Scattering and Imaging, M. Bertero, R. Pike, eds. (Hilger, Bristol, UK, 1992), pp. 345–370.
  3. W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
    [CrossRef]
  4. M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  5. B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  6. A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
    [CrossRef]
  7. M. Moghaddam, W. Chew, “Study of some practical issues in inversion with the Born iterative method using time domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
    [CrossRef]
  8. J. Lin, W. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997).
    [CrossRef]
  9. O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
    [CrossRef]
  10. S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
    [CrossRef]
  11. A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
    [CrossRef]
  12. H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
    [CrossRef]
  13. W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).
  14. R. E. Kleinman, P. M. van den Berg, “An extended range modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
    [CrossRef]
  15. P. M. van den Berg, R. E. Kleinman, “Gradient methods in inverse acoustic and electromagnetic scattering,” in Large-scale Optimization with Applications, L. T. Bigler, T. F. Coleman, A. R. Conn, F. N. Santosa, eds. (Springer-Verlag, New York, 1997), pp. 173–194.
  16. B. J. Kooij, P. M. van den Berg, “Nonlinear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1704–1712 (1998).
    [CrossRef]
  17. T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
    [CrossRef]
  18. P. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
    [CrossRef]
  19. R. Pierri, G. Leone, “On local minima in non-linear inversion,” presented at the Progress in Electromagnetic Research Symposium, PIERS ’97, Cambridge, Mass., July 1997.
  20. R. Pierri, A. Tamburrino, “On the local minima problem in the conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1578 (1997).
    [CrossRef]
  21. R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
    [CrossRef]
  22. R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
    [CrossRef]
  23. R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse source problems,” J. Opt. Soc. Am. 72, 1707–1713 (1982).
    [CrossRef]
  24. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawks, ed. (Academic, London, 1989), pp. 1–120.
  25. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  26. L. B. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).
  27. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
    [CrossRef]

1999 (1)

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

1998 (4)

B. J. Kooij, P. M. van den Berg, “Nonlinear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1704–1712 (1998).
[CrossRef]

B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
[CrossRef]

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

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

1997 (7)

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

P. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

R. Pierri, A. Tamburrino, “On the local minima problem in the conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1578 (1997).
[CrossRef]

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

J. Lin, W. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997).
[CrossRef]

1995 (1)

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

1994 (1)

S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
[CrossRef]

1993 (2)

M. Moghaddam, W. Chew, “Study of some practical issues in inversion with the Born iterative method using time domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “An extended range modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

1988 (1)

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

1984 (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

1982 (1)

1965 (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Bertero, M.

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawks, ed. (Academic, London, 1989), pp. 1–120.

Brancaccio, A.

Chen, B.

Chew, W.

J. Lin, W. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997).
[CrossRef]

M. Moghaddam, W. Chew, “Study of some practical issues in inversion with the Born iterative method using time domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
[CrossRef]

Chew, W. C.

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

Chommeloux, L.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

Colton, D.

D. Colton, “A survey of selected topics in inverse electromagnetic scattering theory,” in Inverse Problems in Wave Propagation, G. Chavent, G. Papanicolaou, P. Sacks, W. Symes, eds. (Springer-Verlag, New York, 1997), pp. 105–127.

De Mol, C.

C. De Mol, “A critical survey of regularized inversion methods,” in Inverse Problems in Scattering and Imaging, M. Bertero, R. Pike, eds. (Hilger, Bristol, UK, 1992), pp. 345–370.

Devaney, A. J.

Duchene, B.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

Ebbini, E. S.

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

Franchois, A.

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

Gutman, S.

S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
[CrossRef]

Haddadin, O. S.

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

Harada, H.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Isernia, T.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Jin, J.

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

Joachimowitz, N.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Kleinman, R. E.

P. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “An extended range modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

P. M. van den Berg, R. E. Kleinman, “Gradient methods in inverse acoustic and electromagnetic scattering,” in Large-scale Optimization with Applications, L. T. Bigler, T. F. Coleman, A. R. Conn, F. N. Santosa, eds. (Springer-Verlag, New York, 1997), pp. 173–194.

Klibanov, M.

S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
[CrossRef]

Kooij, B. J.

B. J. Kooij, P. M. van den Berg, “Nonlinear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1704–1712 (1998).
[CrossRef]

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Leone, G.

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

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

R. Pierri, G. Leone, “On local minima in non-linear inversion,” presented at the Progress in Electromagnetic Research Symposium, PIERS ’97, Cambridge, Mass., July 1997.

Lesselier, D.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

Lin, J.

Lu, C.

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

Michielssen, E.

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

Moghaddam, M.

M. Moghaddam, W. Chew, “Study of some practical issues in inversion with the Born iterative method using time domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
[CrossRef]

Pascazio, V.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Pichot, C.

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

Pichot, Ch.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

Pierri, R.

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

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

R. Pierri, A. Tamburrino, “On the local minima problem in the conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1578 (1997).
[CrossRef]

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

R. Pierri, G. Leone, “On local minima in non-linear inversion,” presented at the Progress in Electromagnetic Research Symposium, PIERS ’97, Cambridge, Mass., July 1997.

Porter, R. P.

Rall, L. B.

L. B. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Song, J. M.

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

Stamnes, J. J.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Tabbara, W.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

Takenaka, T.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Tamburrino, A.

R. Pierri, A. Tamburrino, “On the local minima problem in the conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1578 (1997).
[CrossRef]

Tanaka, M.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

van den Berg, P.

P. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

van den Berg, P. M.

B. J. Kooij, P. M. van den Berg, “Nonlinear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1704–1712 (1998).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “An extended range modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

P. M. van den Berg, R. E. Kleinman, “Gradient methods in inverse acoustic and electromagnetic scattering,” in Large-scale Optimization with Applications, L. T. Bigler, T. F. Coleman, A. R. Conn, F. N. Santosa, eds. (Springer-Verlag, New York, 1997), pp. 173–194.

Wall, D. J. N.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (5)

M. Moghaddam, W. Chew, “Study of some practical issues in inversion with the Born iterative method using time domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
[CrossRef]

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

W. C. Chew, J. Jin, C. Lu, E. Michielssen, J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag. 45, 535–543 (1997).

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

B. J. Kooij, P. M. van den Berg, “Nonlinear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1704–1712 (1998).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

Inverse Probl. (4)

S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
[CrossRef]

P. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
[CrossRef]

R. Pierri, A. Tamburrino, “On the local minima problem in the conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1578 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Radio Sci. (1)

R. E. Kleinman, P. M. van den Berg, “An extended range modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[CrossRef]

Other (7)

P. M. van den Berg, R. E. Kleinman, “Gradient methods in inverse acoustic and electromagnetic scattering,” in Large-scale Optimization with Applications, L. T. Bigler, T. F. Coleman, A. R. Conn, F. N. Santosa, eds. (Springer-Verlag, New York, 1997), pp. 173–194.

R. Pierri, G. Leone, “On local minima in non-linear inversion,” presented at the Progress in Electromagnetic Research Symposium, PIERS ’97, Cambridge, Mass., July 1997.

D. Colton, “A survey of selected topics in inverse electromagnetic scattering theory,” in Inverse Problems in Wave Propagation, G. Chavent, G. Papanicolaou, P. Sacks, W. Symes, eds. (Springer-Verlag, New York, 1997), pp. 105–127.

C. De Mol, “A critical survey of regularized inversion methods,” in Inverse Problems in Scattering and Imaging, M. Bertero, R. Pike, eds. (Hilger, Bristol, UK, 1992), pp. 345–370.

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawks, ed. (Academic, London, 1989), pp. 1–120.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

L. B. Rall, Computational Solution of Nonlinear Operator Equations (Wiley, New York, 1969).

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Figures (12)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

(a) Pictorial view of the equivalent current distribution inside a dielectric object with rectangular cross section under backward plane-wave incidence. (b) Plot of the field scattered by the profile shown in (a) under first-order (solid curve) and second-order (dotted curve) approximations.

Fig. 3
Fig. 3

Geometry of a cylindrical object with circular cross section.

Fig. 4
Fig. 4

Singular values of the first-order model approximation versus the index ([βa]=10).

Fig. 5
Fig. 5

Reference contrast for a rapidly angularly varying profile.

Fig. 6
Fig. 6

Modulus of the field scattered by the profile of Fig. 5: first-order (solid curve) and second-order (dashed curve) approximations.

Fig. 7
Fig. 7

Phase of the field scattered by the profile of Fig. 5: first-order (solid curve) and second-order (dashed curve) approximations.

Fig. 8
Fig. 8

Contrast function of Fig. 5 reconstructed under the first-order Born approximation.

Fig. 9
Fig. 9

Contrast function of Fig. 5 reconstructed under the second-order Born approximation.

Fig. 10
Fig. 10

Gray-level plot of the reference contrast function for two objects embedded in a constant object ([βa]=10).

Fig. 11
Fig. 11

Gray-level plot of the contrast function of Fig. 5 reconstructed under the first-order Born approximation.

Fig. 12
Fig. 12

Gray-level plot of the contrast function of Fig. 5 reconstructed under the second-order Born approximation.

Equations (29)

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

Es(1)=Ae(χEinc),
Es(2)=Ae(χEinc)+Ae[χAi(χEinc)],
χ(y)=sin(Py2π/λ),
χEinc=sin(Py2π/λ).
χAi(χEinc)χ2Einc=1-cos(2Py2π/λ).
Es(1)(θ, θinc)=β202π0a exp[jβr cos(θ-θ)]χ(θ)×exp[jβr cos(θinc-θ)]rdrdθ,
-j4 2πβR expjπ4-βR.
λn=2π2π h=-NN|ah,n-h|21/2,
un(θ)=12π exp(jnθ),
νn(θ, θinc)=h=-NN ah,n-h exp(jhθ)exp[j(n-h)θinc]2π h=-NN|ah,n-h|2,
anm=(j)n+m0βaJn(x)Jm(x)xdx.
χ(θ)=l=-LLcl exp(jlθ),
Es(2)(θ, θinc)
=Es(1)(θ, θinc)-jβ24 02π0a exp[jβr cos(θ-θ)]×χ(θ)02π0aH0(2){β[r2+r2-2rr×cos(θ-θ)]1/2}×χ(θ)exp[jβr cos(θi-θ)]rdrdθrdrdθ.
Es(2)=Es(1)+n=-NNm=-NNq=-QQbnmqcn+qcm-q×exp(jnθ)exp(jmθinc),
bnmq=-jπ2(j)n+m0βa0βaJn(x)Jm(x)Jq[min(x, x)]×Hq(2)[max(x, x)]xxdxdx,
A(c)=n=-NNm=-NNanmcn+m exp(jnθ)exp(jmθinc)
forθ=θ1,.,θS,θinc=θinc1,,θincV,
B(c, c)=n=-NNm=-NNq=-QQbnmq exp(jnθ)
×exp(jmθinc)cn-qcm+q
forθ=θ1,,θS,θinc=θinc1,,θincV,
A(c)+B(c, c)=Es,
ϕ(c)=Es-[A(c)+B(c, c)]2.
cl=0|l|2510-2 sin[(π/3)(|l|-30+2)](π/3)(|l|-30+2)26|l|30.
L(χ)=nm anm02πχ(θ)exp[-j(n+m)θ]dθ×exp(jnθ)exp(jmθinc),
L+(Es)=nmanm*02π exp(-jmθinc)02πEs(θ, θinc)×exp(-jnθ)dθdθinc exp[+j(n+m)θ],
L+Lexp(jνθ)2π=12π nmanm*02π exp(-jmθinc)×02πnan,ν-n2π exp(jnθ)×exp[j(ν-n)θinc]×exp(-jnθ)dθdθinc×exp[+j(n+m)θ]=n(2π)3|an,ν-n|2 exp(jνθ)2π=λν2 exp(jνθ)2π.
Jn(x)12πn ex2nn,Hn(2)(x)j 1πn ex2n-n,
|bnmq|14 0βa0βa|Jn(x)Jm(x)Jq[min(x, x)]×Hq(2)[max(x, x)]|xxdxdx1162nmπ2 e2nne2mm 1q ×0βa0βa(x)n(x)mmin(x, x)max(x, x)qxxdxdx1162nmπ2 e2nne2mm 1q ×0βa0βa(x)n+1(x)m+1dxdx=1162nmπ2 e2nne2mm (βa)n+m+4(n+2)(m+2) 1q.

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