Abstract

The information content of the scattered field in the framework of the linear Born and distorted Born approximations for a one-dimensional lossless dielectric permittivity profile embedded in a lossless homogeneous half-space is analyzed. The number of degrees of freedom of the scattered field and the class of the retrievable profiles from a multifrequency, a multiview, and a multifrequency–multiview configuration are evaluated by analytical considerations and validated by numerical singular-value decomposition. The analysis stresses the effects of the background permittivity value on the degrees of freedom and on the class of retrievable profiles within the distorted Born approximation. In particular, the results show that, for high values of the half-space permittivity, the information content in the multiview approach at a fixed frequency becomes too poor to yield effective reconstructions, whereas a suitable multifrequency or multifrequency–multiview approach can provide better results.

© 1999 Optical Society of America

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References

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  1. D. Lesselier, B. Duchene, “Wavefield inversion of objects in stratified environments: from back-propagation schemes to full solutions,” in Review of Radio Science 1993–1996, R. Stone, ed. (Oxford U. Press, New York, 1996), pp. 235–268.
  2. R. Pierri, G. Leone, R. Bernini, R. Persico, “Tomografic inversion algorithms for permittivity reconstruction in subsurface prospection,” (presented at the 7th International Conference on Ground-Penetrating Radar, Lawrence, Kans., May 27–30, 1998.
  3. A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin-film optics,” Inverse Probl. 11, 251–270 (1995).
    [CrossRef]
  4. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  5. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).
  6. R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
    [CrossRef]
  7. I. Akduman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
    [CrossRef]
  8. T. J. Cui, C. H. Liang, “Direct profile inversion for a half-space weakly lossy medium,” J. Opt. Soc. Am. A 10, 1950–1952 (1993).
    [CrossRef]
  9. 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, Bristol, UK, 1992), pp. 23–46.
  10. T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
    [CrossRef]
  11. G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999).
    [CrossRef]
  12. L. Zeni, R. Bernini, R. Pierri, “Reconstruction of doping profiles in semiconductor materials using optical tomography,” Solid-State Electron. 43, 761–769 (1999).
    [CrossRef]
  13. T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
    [CrossRef]
  14. 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), Part I, pp. 173–194.
  15. A. G. Tijhuis, “Born-type reconstruction of material parameters of an inhomogeneous lossy dielectric slab from reflected-field data,” Wave Motion 11, 151–173 (1989).
    [CrossRef]
  16. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics, P. H. Hawkes, ed. (Academic, New York, 1989), Vol. 75, pp. 1–121.
  17. 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]
  18. B. Chen, J. J Stamnes, “Validity of diffraction tomography based on the first Born and first Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  19. 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]
  20. A. J. Devaney, “Current research topics in diffraction tomography,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, UK, 1992), pp. 47–58.
  21. M. Idemen, “On different possibilities offered by the Born approximation in inverse scattering problems,” Inverse Probl. 5, 1057–1074 (1989).
    [CrossRef]
  22. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. A 59, 799–804 (1969).
    [CrossRef]
  23. M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
    [CrossRef]
  24. R. Pierri, F. Soldovieri, “On the information content of the radiated fields in near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
    [CrossRef]
  25. A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science, Utrecht, The Netherlands, 1987).
  26. H. J. Landau, “Sampling, data transmission, and the Nyquist rate,” Proc. IEEE 55, 1701–1706 (1967).
    [CrossRef]
  27. D. Slepian, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  28. H. J. Landau, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. III. The dimension of the space of time- and band-limited signals,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
    [CrossRef]
  29. L. Landau, E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1969).
  30. R. Pierri, F. De Blasio, A. Brancaccio, “Multifrequency approach to inverse scattering: the linear and the quadratic models,” presented at the International Geoscience and Remote Sensing Symposium, Hamburg, Germany, June 28–July 2, 1999.

1999 (2)

G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999).
[CrossRef]

L. Zeni, R. Bernini, R. Pierri, “Reconstruction of doping profiles in semiconductor materials using optical tomography,” Solid-State Electron. 43, 761–769 (1999).
[CrossRef]

1998 (3)

1997 (2)

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

1995 (3)

I. Akduman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin-film optics,” Inverse Probl. 11, 251–270 (1995).
[CrossRef]

T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
[CrossRef]

1993 (1)

1989 (2)

A. G. Tijhuis, “Born-type reconstruction of material parameters of an inhomogeneous lossy dielectric slab from reflected-field data,” Wave Motion 11, 151–173 (1989).
[CrossRef]

M. Idemen, “On different possibilities offered by the Born approximation in inverse scattering problems,” Inverse Probl. 5, 1057–1074 (1989).
[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]

1980 (1)

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[CrossRef]

1969 (1)

G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. A 59, 799–804 (1969).
[CrossRef]

1967 (1)

H. J. Landau, “Sampling, data transmission, and the Nyquist rate,” Proc. IEEE 55, 1701–1706 (1967).
[CrossRef]

1962 (1)

H. J. Landau, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. III. The dimension of the space of time- and band-limited signals,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

1961 (1)

D. Slepian, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Akduman, I.

I. Akduman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

Bernini, R.

L. Zeni, R. Bernini, R. Pierri, “Reconstruction of doping profiles in semiconductor materials using optical tomography,” Solid-State Electron. 43, 761–769 (1999).
[CrossRef]

R. Pierri, G. Leone, R. Bernini, R. Persico, “Tomografic inversion algorithms for permittivity reconstruction in subsurface prospection,” (presented at the 7th International Conference on Ground-Penetrating Radar, Lawrence, Kans., May 27–30, 1998.

Bertero, M.

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[CrossRef]

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics, P. H. Hawkes, ed. (Academic, New York, 1989), Vol. 75, pp. 1–121.

Brancaccio, A.

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. De Blasio, A. Brancaccio, “Multifrequency approach to inverse scattering: the linear and the quadratic models,” presented at the International Geoscience and Remote Sensing Symposium, Hamburg, Germany, June 28–July 2, 1999.

Chen, B.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).

Colton, D.

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

Cui, T. J.

De Blasio, F.

R. Pierri, F. De Blasio, A. Brancaccio, “Multifrequency approach to inverse scattering: the linear and the quadratic models,” presented at the International Geoscience and Remote Sensing Symposium, Hamburg, Germany, June 28–July 2, 1999.

Devaney, A. J.

A. J. Devaney, “Current research topics in diffraction tomography,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, UK, 1992), pp. 47–58.

Dickens, T. A.

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

Duchene, B.

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

Fiddy, M. A.

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, Bristol, UK, 1992), pp. 23–46.

Idemen, M.

I. Akduman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

M. Idemen, “On different possibilities offered by the Born approximation in inverse scattering problems,” Inverse Probl. 5, 1057–1074 (1989).
[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]

Kleinman, R. E.

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), Part I, pp. 173–194.

Klibanov, M. V.

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin-film optics,” Inverse Probl. 11, 251–270 (1995).
[CrossRef]

Kress, R.

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

Landau, H. J.

H. J. Landau, “Sampling, data transmission, and the Nyquist rate,” Proc. IEEE 55, 1701–1706 (1967).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. III. The dimension of the space of time- and band-limited signals,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

Landau, L.

L. Landau, E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1969).

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.

G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (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, G. Leone, R. Bernini, R. Persico, “Tomografic inversion algorithms for permittivity reconstruction in subsurface prospection,” (presented at the 7th International Conference on Ground-Penetrating Radar, Lawrence, Kans., May 27–30, 1998.

Lesselier, D.

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

Liang, C. H.

Lifchitz, E.

L. Landau, E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1969).

Pasqualetti, F.

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[CrossRef]

Persico, R.

G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999).
[CrossRef]

R. Pierri, G. Leone, R. Bernini, R. Persico, “Tomografic inversion algorithms for permittivity reconstruction in subsurface prospection,” (presented at the 7th International Conference on Ground-Penetrating Radar, Lawrence, Kans., May 27–30, 1998.

Pierri, R.

G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999).
[CrossRef]

L. Zeni, R. Bernini, R. Pierri, “Reconstruction of doping profiles in semiconductor materials using optical tomography,” Solid-State Electron. 43, 761–769 (1999).
[CrossRef]

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in 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]

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

R. Pierri, G. Leone, R. Bernini, R. Persico, “Tomografic inversion algorithms for permittivity reconstruction in subsurface prospection,” (presented at the 7th International Conference on Ground-Penetrating Radar, Lawrence, Kans., May 27–30, 1998.

R. Pierri, F. De Blasio, A. Brancaccio, “Multifrequency approach to inverse scattering: the linear and the quadratic models,” presented at the International Geoscience and Remote Sensing Symposium, Hamburg, Germany, June 28–July 2, 1999.

Pollack, H. O.

H. J. Landau, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. III. The dimension of the space of time- and band-limited signals,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

D. Slepian, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Ronchi, L.

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[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]

Slepian, D.

D. Slepian, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Soldovieri, F.

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

Stamnes, J. J

Stamnes, J. J.

Tamburrino, A.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

Tijhuis, A. G.

A. G. Tijhuis, “Born-type reconstruction of material parameters of an inhomogeneous lossy dielectric slab from reflected-field data,” Wave Motion 11, 151–173 (1989).
[CrossRef]

A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science, Utrecht, The Netherlands, 1987).

Tikhonravov, A. V.

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin-film optics,” Inverse Probl. 11, 251–270 (1995).
[CrossRef]

Toraldo di Francia, G.

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[CrossRef]

G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. A 59, 799–804 (1969).
[CrossRef]

Van Den Berg, P. M.

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), Part I, pp. 173–194.

Viano, G. A.

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[CrossRef]

Wedberg, T. C.

Winbow, G. A.

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

Zeni, L.

L. Zeni, R. Bernini, R. Pierri, “Reconstruction of doping profiles in semiconductor materials using optical tomography,” Solid-State Electron. 43, 761–769 (1999).
[CrossRef]

Zuev, I. V.

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin-film optics,” Inverse Probl. 11, 251–270 (1995).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

D. Slepian, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wavefunctions, Fourier analysis and uncertainty. III. The dimension of the space of time- and band-limited signals,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

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)

M. Idemen, “On different possibilities offered by the Born approximation in inverse scattering problems,” Inverse Probl. 5, 1057–1074 (1989).
[CrossRef]

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

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin-film optics,” Inverse Probl. 11, 251–270 (1995).
[CrossRef]

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

I. Akduman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

J. Acoust. Soc. Am. (1)

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980).
[CrossRef]

Proc. IEEE (1)

H. J. Landau, “Sampling, data transmission, and the Nyquist rate,” Proc. IEEE 55, 1701–1706 (1967).
[CrossRef]

Solid-State Electron. (1)

L. Zeni, R. Bernini, R. Pierri, “Reconstruction of doping profiles in semiconductor materials using optical tomography,” Solid-State Electron. 43, 761–769 (1999).
[CrossRef]

Wave Motion (1)

A. G. Tijhuis, “Born-type reconstruction of material parameters of an inhomogeneous lossy dielectric slab from reflected-field data,” Wave Motion 11, 151–173 (1989).
[CrossRef]

Other (11)

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics, P. H. Hawkes, ed. (Academic, New York, 1989), Vol. 75, pp. 1–121.

A. J. Devaney, “Current research topics in diffraction tomography,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, UK, 1992), pp. 47–58.

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), Part I, pp. 173–194.

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, Bristol, UK, 1992), pp. 23–46.

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

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).

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

R. Pierri, G. Leone, R. Bernini, R. Persico, “Tomografic inversion algorithms for permittivity reconstruction in subsurface prospection,” (presented at the 7th International Conference on Ground-Penetrating Radar, Lawrence, Kans., May 27–30, 1998.

A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science, Utrecht, The Netherlands, 1987).

L. Landau, E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1969).

R. Pierri, F. De Blasio, A. Brancaccio, “Multifrequency approach to inverse scattering: the linear and the quadratic models,” presented at the International Geoscience and Remote Sensing Symposium, Hamburg, Germany, June 28–July 2, 1999.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Behavior of the singular values in the multifrequency case for three values of εb. The wavelength ranges from 0.2d to d, and θi=0.

Fig. 3
Fig. 3

Harmonic content of the first 2N singular functions in the multifrequency case in distorted and nondistorted cases. The wavelength ranges from 0.2d to d, and θi=0.

Fig. 4
Fig. 4

Behavior of the singular values in the multiview case for three values of εb. The wavelength is equal to 0.2d.

Fig. 5
Fig. 5

Harmonic content of the first 2N singular functions in the multiview case in a nondistorted and two distorted cases. The wavelength is equal to 0.2d.

Fig. 6
Fig. 6

Illustration of the reduction of the effective view angle set as a result of refraction.

Fig. 7
Fig. 7

Behavior of the singular values in the multifrequency–multiview case for two values of εb. The wavelength ranges from 0.2d to d.

Fig. 8
Fig. 8

Harmonic content of the first 2N singular functions in the multifrequency–multiview case in a nondistorted and two distorted cases. The wavelength ranges from 0.2d to d.

Equations (26)

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

Es(z, λ)=kb20dG(z, z, λ)χ(z)E(z, λ)dz,
χ(z)=ε(z)εb-1z00z<0
G(z, z, λ)=T21i2kz exp(-ikzz)exp[ik1 cos(θi)z],
z<0,z>0
kz={kb2-[k1sin(θi)]2}1/2,
T21=2kzk1 cos(θi)+kz.
E(z, λ)=Ei(z, λ)+kb20dG(z, z, λ)χ(z)E(z, λ)dz.
E(z, λ)Ei(z, λ)=T12 exp(-ikzz),
T12=2k1 cos(θi)k1 cos(θi)+kz.
E˜(k¯)=Es(k¯, z)-iT12T21kb2k¯ exp(ik1z)=0d exp(-ik¯z)χ(z)dz,
k¯=2kb.
Δ=πΩ=2πd.
I=4πεbλmax, 4πεbλmin.
N4πεb(1/λmin-1/λmax)Δ=2dεb(1/λmin-1/λmax).
E˜(k¯)=Es(k¯, z)-iT12T21kb2k¯ exp[ik1 cos(θi)z]=0d exp(-ik¯z)χ(z)dz,
k¯=2kz.
I=4πεb-ε1λ, 4πεbλ .
N(4π/λ)(εb-εb-ε1)Δ=2dλ (εb-εb-ε1).
I0=[0, 2k1],
Nmax2dε1λ.
θ1=arcsin(ε1/εb)<π/2
θ2=arccos1-εbε1+εb-ε1ε1.
2(k02ε1-ε1k02 sin2 θ2)1/2
=2[k02ε1(cos2 θ2)]1/2=2k0ε1-2k0εb+2k0εb-ε1.
I=4πεb-ε1 1λmax, 4πεb 1λmin
N4πεbλmin-εb-ε1λmaxΔ=2dεbλmin-εb-ε1λmax.

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