Abstract

The problem of reconstructing dielectric permittivity from scattered field data is dealt with for scalar two-dimensional geometry at a fixed frequency by use of a linearized approximation about a chosen reference permittivity profile. To investigate the capabilities and limits of linear inversion algorithms, we analyze the class of retrievable profiles with reference to some canonical geometries for which either analytical or numerical details can be worked through easily. The tool for such an analysis consists of the singular-value decomposition of the relevant scattering operators. For a constant reference permittivity function, the different behavior of linear inversion algorithms with respect to either radial or angular variations of the permittivity profiles is pointed out. In the last-named case the general situation of a multiview radiation is accounted for, and, unlike for the Born approximation, profiles that cannot be reconstructed by linear inversion comprise slowly varying functions. Moreover, the effect of an angularly varying reference profile is examined for a thin circular shell, permitting the possibility of reconstruction of rapidly varying angular profiles by linear inversion. Numerical results of linear inversions that confirm the predictions are shown.

© 1999 Optical Society of America

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References

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  1. J. Ch. Bolomey, Ch. Pichot, “Some applications of diffraction tomography to electromagnetics—the particular case of microwaves,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London1992), pp. 319–344.
  2. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  3. D. Lesselier, B. Duchene, “Wavefield inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science 1993–1996, R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996), pp. 235–268.
  4. A. Roger, F. Chapel, “Iterative methods for inverse problems,” in Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, T. K. Sarkar, ed. (Elsevier, Amsterdam, 1991), pp. 423–454.
  5. Y. Chen, “Inverse scattering via Heisenberg’s uncertainty principle,” Inverse Probl. 13, 253–282 (1997).
    [CrossRef]
  6. R. Pierri, A. Brancaccio, “Imaging of a dielectric cylinder: a quadratic approach in the rotational symmetric case,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
    [CrossRef]
  7. R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a quadratic model,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
    [CrossRef]
  8. R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
    [CrossRef]
  9. R. G. Keys, A. B. Weglein, “Generalised linear inversion and the first Born theory for acoustic media,” J. Math. Phys. 24, 1444–1449 (1983).
    [CrossRef]
  10. M. A. Fiddy, “Linearized and approximate methods for inversion of scattered field data,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London, 1992), pp. 23–46.
  11. W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [CrossRef] [PubMed]
  12. D. O. Batrakov, N. P. Zhuck, “Solution of a general inverse scattering problem using the distorted Born approximation and iterative technique,” Inverse Probl. 10, 39–54 (1994).
    [CrossRef]
  13. C. Torres-Verdin, T. M. Habashy, “A two-step linear inversion of two-dimensional electrical conductivity,” IEEE Trans. Antennas Propag. 43, 405–415 (1995).
    [CrossRef]
  14. M. S. Zhadanov, S. Fang, “Three-dimensional quasi-linear electromagnetic inversion,” Radio Sci. 31, 741–754 (1996).
    [CrossRef]
  15. S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
    [CrossRef]
  16. P. Chaturvedi, R. G. Plumb, “Electromagnetic imaging of underground targets using constrained optimization,” IEEE Trans. Geosci. Remote Sens. 33, 551–561 (1995).
    [CrossRef]
  17. H. Gan, R. Ludwig, P. L. Levin, “Nonlinear diffractive inverse scattering for multiple scattering in inhomogeneous acoustic background media,” J. Acoust. Soc. Am. 97, 764–776 (1995).
    [CrossRef]
  18. N. Joachimowitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
    [CrossRef]
  19. J.-H. Lin, W. C. 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]
  20. J. A. Scales, P. Docherty, A. Gersztenkorn, “Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer,” Inverse Probl. 6, 115–131 (1990).
    [CrossRef]
  21. R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse source problems,” J. Opt. Soc. Am. 72, 1707–1713 (1982).
    [CrossRef]
  22. 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]
  23. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  24. J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
    [CrossRef]
  25. E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
    [CrossRef]
  26. W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995).
  27. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  28. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

1999 (1)

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

1998 (1)

1997 (4)

1996 (1)

M. S. Zhadanov, S. Fang, “Three-dimensional quasi-linear electromagnetic inversion,” Radio Sci. 31, 741–754 (1996).
[CrossRef]

1995 (4)

P. Chaturvedi, R. G. Plumb, “Electromagnetic imaging of underground targets using constrained optimization,” IEEE Trans. Geosci. Remote Sens. 33, 551–561 (1995).
[CrossRef]

H. Gan, R. Ludwig, P. L. Levin, “Nonlinear diffractive inverse scattering for multiple scattering in inhomogeneous acoustic background media,” J. Acoust. Soc. Am. 97, 764–776 (1995).
[CrossRef]

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

C. Torres-Verdin, T. M. Habashy, “A two-step linear inversion of two-dimensional electrical conductivity,” IEEE Trans. Antennas Propag. 43, 405–415 (1995).
[CrossRef]

1994 (2)

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

D. O. Batrakov, N. P. Zhuck, “Solution of a general inverse scattering problem using the distorted Born approximation and iterative technique,” Inverse Probl. 10, 39–54 (1994).
[CrossRef]

1991 (1)

N. Joachimowitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

1990 (2)

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

J. A. Scales, P. Docherty, A. Gersztenkorn, “Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer,” Inverse Probl. 6, 115–131 (1990).
[CrossRef]

1984 (1)

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

1983 (1)

R. G. Keys, A. B. Weglein, “Generalised linear inversion and the first Born theory for acoustic media,” J. Math. Phys. 24, 1444–1449 (1983).
[CrossRef]

1982 (1)

1931 (1)

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Abramowitz, M.

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

Batrakov, D. O.

D. O. Batrakov, N. P. Zhuck, “Solution of a general inverse scattering problem using the distorted Born approximation and iterative technique,” Inverse Probl. 10, 39–54 (1994).
[CrossRef]

Bolomey, J. Ch.

J. Ch. Bolomey, Ch. Pichot, “Some applications of diffraction tomography to electromagnetics—the particular case of microwaves,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London1992), pp. 319–344.

Brancaccio, A.

Chapel, F.

A. Roger, F. Chapel, “Iterative methods for inverse problems,” in Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, T. K. Sarkar, ed. (Elsevier, Amsterdam, 1991), pp. 423–454.

Chaturvedi, P.

P. Chaturvedi, R. G. Plumb, “Electromagnetic imaging of underground targets using constrained optimization,” IEEE Trans. Geosci. Remote Sens. 33, 551–561 (1995).
[CrossRef]

Chen, Y.

Y. Chen, “Inverse scattering via Heisenberg’s uncertainty principle,” Inverse Probl. 13, 253–282 (1997).
[CrossRef]

Chew, W. C.

J.-H. Lin, W. C. 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]

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995).

Colton, D.

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

Devaney, A. J.

Docherty, P.

J. A. Scales, P. Docherty, A. Gersztenkorn, “Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer,” Inverse Probl. 6, 115–131 (1990).
[CrossRef]

Duchene, B.

D. Lesselier, B. Duchene, “Wavefield inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science 1993–1996, R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996), pp. 235–268.

Fang, S.

M. S. Zhadanov, S. Fang, “Three-dimensional quasi-linear electromagnetic inversion,” Radio Sci. 31, 741–754 (1996).
[CrossRef]

Fiddy, M. A.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

M. A. Fiddy, “Linearized and approximate methods for inversion of scattered field data,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London, 1992), pp. 23–46.

Gan, H.

H. Gan, R. Ludwig, P. L. Levin, “Nonlinear diffractive inverse scattering for multiple scattering in inhomogeneous acoustic background media,” J. Acoust. Soc. Am. 97, 764–776 (1995).
[CrossRef]

Gersztenkorn, A.

J. A. Scales, P. Docherty, A. Gersztenkorn, “Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer,” Inverse Probl. 6, 115–131 (1990).
[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]

Habashy, T. M.

C. Torres-Verdin, T. M. Habashy, “A two-step linear inversion of two-dimensional electrical conductivity,” IEEE Trans. Antennas Propag. 43, 405–415 (1995).
[CrossRef]

Harrington, R. F.

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

Hille, E.

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Hugonin, J. P.

N. Joachimowitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Joachimowitz, N.

N. Joachimowitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Kak, A. C.

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

Keys, R. G.

R. G. Keys, A. B. Weglein, “Generalised linear inversion and the first Born theory for acoustic media,” J. Math. Phys. 24, 1444–1449 (1983).
[CrossRef]

Klibanov, M.

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

Kress, R.

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

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations 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 quadratic model,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[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]

Lesselier, D.

D. Lesselier, B. Duchene, “Wavefield inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science 1993–1996, R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996), pp. 235–268.

Levin, P. L.

H. Gan, R. Ludwig, P. L. Levin, “Nonlinear diffractive inverse scattering for multiple scattering in inhomogeneous acoustic background media,” J. Acoust. Soc. Am. 97, 764–776 (1995).
[CrossRef]

Lin, F. C.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

Lin, J.-H.

Ludwig, R.

H. Gan, R. Ludwig, P. L. Levin, “Nonlinear diffractive inverse scattering for multiple scattering in inhomogeneous acoustic background media,” J. Acoust. Soc. Am. 97, 764–776 (1995).
[CrossRef]

McGahan, R. V.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

Morris, J. B.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

Pichot, C.

N. Joachimowitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Pichot, Ch.

J. Ch. Bolomey, Ch. Pichot, “Some applications of diffraction tomography to electromagnetics—the particular case of microwaves,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London1992), pp. 319–344.

Pierri, R.

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a quadratic model,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[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, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
[CrossRef]

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

Plumb, R. G.

P. Chaturvedi, R. G. Plumb, “Electromagnetic imaging of underground targets using constrained optimization,” IEEE Trans. Geosci. Remote Sens. 33, 551–561 (1995).
[CrossRef]

Pommet, D. A.

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

Porter, R. P.

Roger, A.

A. Roger, F. Chapel, “Iterative methods for inverse problems,” in Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, T. K. Sarkar, ed. (Elsevier, Amsterdam, 1991), pp. 423–454.

Scales, J. A.

J. A. Scales, P. Docherty, A. Gersztenkorn, “Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer,” Inverse Probl. 6, 115–131 (1990).
[CrossRef]

Slaney, M.

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

Stegun, I.

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

Tamarkin, J. D.

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Tamburrino, A.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
[CrossRef]

Torres-Verdin, C.

C. Torres-Verdin, T. M. Habashy, “A two-step linear inversion of two-dimensional electrical conductivity,” IEEE Trans. Antennas Propag. 43, 405–415 (1995).
[CrossRef]

Wang, Y. M.

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

Weglein, A. B.

R. G. Keys, A. B. Weglein, “Generalised linear inversion and the first Born theory for acoustic media,” J. Math. Phys. 24, 1444–1449 (1983).
[CrossRef]

Zhadanov, M. S.

M. S. Zhadanov, S. Fang, “Three-dimensional quasi-linear electromagnetic inversion,” Radio Sci. 31, 741–754 (1996).
[CrossRef]

Zhuck, N. P.

D. O. Batrakov, N. P. Zhuck, “Solution of a general inverse scattering problem using the distorted Born approximation and iterative technique,” Inverse Probl. 10, 39–54 (1994).
[CrossRef]

Acta Math. (1)

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

J. B. Morris, F. C. Lin, D. A. Pommet, R. V. McGahan, M. A. Fiddy, “A homomorphic filtering method for imaging strongly scattering penetrable objects,” IEEE Trans. Antennas Propag. 43, 1029–1035 (1995).
[CrossRef]

C. Torres-Verdin, T. M. Habashy, “A two-step linear inversion of two-dimensional electrical conductivity,” IEEE Trans. Antennas Propag. 43, 405–415 (1995).
[CrossRef]

N. Joachimowitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

P. Chaturvedi, R. G. Plumb, “Electromagnetic imaging of underground targets using constrained optimization,” IEEE Trans. Geosci. Remote Sens. 33, 551–561 (1995).
[CrossRef]

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

IEEE Trans. Med. Imaging (1)

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

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

Inverse Probl. (5)

J. A. Scales, P. Docherty, A. Gersztenkorn, “Regularisation of nonlinear inverse problems: imaging the near-surface weathering layer,” Inverse Probl. 6, 115–131 (1990).
[CrossRef]

D. O. Batrakov, N. P. Zhuck, “Solution of a general inverse scattering problem using the distorted Born approximation and iterative technique,” Inverse Probl. 10, 39–54 (1994).
[CrossRef]

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
[CrossRef]

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

Y. Chen, “Inverse scattering via Heisenberg’s uncertainty principle,” Inverse Probl. 13, 253–282 (1997).
[CrossRef]

J. Acoust. Soc. Am. (1)

H. Gan, R. Ludwig, P. L. Levin, “Nonlinear diffractive inverse scattering for multiple scattering in inhomogeneous acoustic background media,” J. Acoust. Soc. Am. 97, 764–776 (1995).
[CrossRef]

J. Math. Phys. (1)

R. G. Keys, A. B. Weglein, “Generalised linear inversion and the first Born theory for acoustic media,” J. Math. Phys. 24, 1444–1449 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Radio Sci. (1)

M. S. Zhadanov, S. Fang, “Three-dimensional quasi-linear electromagnetic inversion,” Radio Sci. 31, 741–754 (1996).
[CrossRef]

Other (8)

M. A. Fiddy, “Linearized and approximate methods for inversion of scattered field data,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London, 1992), pp. 23–46.

J. Ch. Bolomey, Ch. Pichot, “Some applications of diffraction tomography to electromagnetics—the particular case of microwaves,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, London1992), pp. 319–344.

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

D. Lesselier, B. Duchene, “Wavefield inversion of objects in stratified environments: from backpropagation schemes to full solutions,” in Review of Radio Science 1993–1996, R. Stone, ed. (Oxford U. Press, Oxford, UK, 1996), pp. 235–268.

A. Roger, F. Chapel, “Iterative methods for inverse problems,” in Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, T. K. Sarkar, ed. (Elsevier, Amsterdam, 1991), pp. 423–454.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995).

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

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

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Figures (12)

Fig. 1
Fig. 1

Relevant geometry.

Fig. 2
Fig. 2

Singular values of operator Ar, normalized to the maximum, for β0a=10 and three values of r (circles, r=5; crosses, r=10; stars, r=20).

Fig. 3
Fig. 3

Singular values of operator Ar, normalized to the maximum, for r=10 and three values of β0a (circles, β0a=5; crosses, β0a=10; stars, β0a=20).

Fig. 4
Fig. 4

First three singular functions of the operator Ar in the space of contrast function for β0a=10 and r=10 (first singular function, dotted curve; second singular function, dashed curved; third singular function, solid curve).

Fig. 5
Fig. 5

Reconstruction by linear inversion of a rapidly varying radial profile (solid curve, reference function; dashed curve, inversion by DBA; dotted curve, inversion by the Born approximation).

Fig. 6
Fig. 6

Reconstruction by linear inversion of a slowly varying radial profile (solid curve, reference function; dashed curve, inversion by DBA; dotted curve, inversion by the Born approximation).

Fig. 7
Fig. 7

Singular values of operator Aθ, normalized to the maximum, for β0a=10 and three values of r (circles, r=5; crosses, r=10; stars, r=20).

Fig. 8
Fig. 8

Singular values of operator Aθ, normalized to the maximum, for r=10 and three values of β0a (circles, β0a=5; crosses, β0a=10; stars, β0a=20).

Fig. 9
Fig. 9

Reconstruction by linear inversion of a slowly varying angular profile (solid curve, reference function; dashed curve, reconstructed function).

Fig. 10
Fig. 10

Reconstruction by linear inversion of a rapidly varying angular profile (solid curve, reference function; dashed curve, reconstructed function).

Fig. 11
Fig. 11

Singular values of operator A , normalized to the maximum, for a varying-background contrast function.

Fig. 12
Fig. 12

Harmonic content of the reconstructed function of a rapidly varying function for a variable background (the true contrast function has only the 35th-harmonic coefficient with a 0.5 level).

Equations (35)

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

χ(r)=(r)-ref (r)ref (r)rD0rD.
Ai( f )=DG(r, r)f(r)dr,rD,
Ae( f )=DG(r, r)f(r)dr,rΣ,
E=Einc+Ai(χE),
Es=Ae(χE).
E=(I-Aiχ)-1Einc,
Es=Ae[χ(I-Aiχ)-1Einc]=F(χ).
EEinc
EsAe(χEinc)=A(χ).
ref (r)=0rDr0rD,
Einc(r, θ, θinc)=n j-ncn×exp[-jn(θ-θinc)]Jn(β0rr),
cn=-j2 /(πβ0a)[rJn-1(β0rr)Hn(2)(β0a)-Jn(β0rr)Hn-1(2)(β0a)]-1,
cnJn(β0ra)=(2j/πβ0a)/Hn(β0a)×[rJn-1(β0ra)/Jn(β0ra)-Hn-1(β0a)/Hn(β0a)]-1(2/πβ0a)/Yn(β0a)[Jn-1(β0a)/Jn(β0a)-Yn-1(β0a)/Yn(β0a)]-1Jn(β0a),
Es(θ, θinc)=β02rn=-NNj-ncn exp( jnθ)m=-NNj-mcm×exp( jmθinc)02π0aχ(r, θ)×Jn(β0rr)Jm(β0rr)×exp[-j(n+m)θ]rdrdθ,
Es(θ)=Ar(χ)=2πβ02rn=-NN(-)ncn2
×exp( jnθ)0aχ(r)Jn2(β0rr)rdr.
πβ0r 0aχ(r)cos2(β0rr+nπ/4)dr
=π2β0r 0aχ(r)[1+cos(2β0rr+nπ/2)]dr.
uν(θ)=exp( jνθ)2π,
σν2=(2π)2β04r2n=-NN|cncν-nIn,ν-n|2,
ref (r)=0[1+χν cos(νθ)]rD,rD0rD,rD ,
G(θ, r, θ)=β02rn=-NN j-ncnJn(rβ0r)
×exp[ jn(θ-θ)],
ES(θ, θinc)=0ardr02πG(θ, r, θ)×Einc(r, θ, θinc)χ(θ)dθ=Aθ (χ),
Aθ(χ)=β02rn=-NNm=-NNcmcn j-(n+m) exp(-jmθinc)×exp(-jnθ)Inm02πχ(θ)×exp[ j(m+n)θ]dθ,
Inm=0aJn(β0rr)Jm(β0rr)rdr.
Aθ+(ES)=β02rn=-NNm=-NNj(m+n)cn*cm*×exp[-j(n+m)θ]Inm*×02π02πES(θ, θinc)×exp( jnθ)exp( jmθinc)dθdθ,
Aθ+Aθ(χ)=4π2β04r2n=-NNm=-NN|cmcn|2×exp[-j(m+n)θ]|Inm|202πχ(θ)×exp[ j(n+m)θ]dθ.
E(θ)=n=-en exp( jnθ).
E(θ)=Einc(θ, θinc)+DG(θ, θ)E(θ)χ(θ)ρdρdθ,
G(θ, θ)=-jβ024 H02(β0|r-r|)=-jβ024 μ=-+ exp[ jμ(θ-θ)]×Hμ(β0a)Jμ(β0a),
Einc(θ, θinc)m=-jmJm(β0a)exp[ jm(θ-θinc)].
χ(θ)=χν cos(νθ),
em=jmJm(β0a)exp(-jmθinc)-jπβ02aΔa2×Hm(β0a)Jm(β0a)χν(eν+m+em-ν).

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