Abstract

The nonlinear (quadratic) distorted approximation of the inverse scattering of dielectric cylinders is investigated, with the aim of pointing out the influence of the background medium. We refer to a canonical geometry consisting of a radially symmetric dielectric cylinder illuminated at a single frequency. We discuss how the spatial variations of those unknown dielectric profile functions that can be reconstructed by a stable inversion procedure are related to the permittivity of the background cylinder. First, results for the linear distorted approximation, obtained by means of the singular-value decomposition, are recalled and compared with the Born approximation. It turns out that the distorted model provides a smoother behavior of the singular values, and thus the inversion is more sensitive to the presence of uncertainties in the data. Furthermore, a stable inversion procedure can reconstruct only a very limited class of unknowns in correspondence with fast spatial variations related to the background permittivity and the excitation frequency. On the other hand, the quadratic model improves the approximation in the distorted case. This can be traced not only to the higher allowable level of permittivity but mainly to the fact that the model makes it possible to reconstruct different spatial features as the solution space changes. Numerical results show that the quadratic inversion performs better than the linear one for the same amount of uncertainty in the data.

© 2001 Optical Society of America

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  1. R. Pierri, R. Persico, R. Bernini, “Information content of the Born field scattered by an embedded slab: multifrequency, multiview, and multifrequency–multiview cases,” J. Opt. Soc. Am. A 16, 2392–2399 (1999).
    [CrossRef]
  2. 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]
  3. 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]
  4. 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]
  5. P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997).
    [CrossRef]
  6. I. T. Rekanos, T. V. Yioultsis, T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microwave Theory Tech. 47, 336–344 (1999).
    [CrossRef]
  7. 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]
  8. N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
    [CrossRef] [PubMed]
  9. B. J. Kooij, P. M. van den Berg, “Non linear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1706–1712 (1998).
    [CrossRef]
  10. P. M. van den Berg, A. L. van Broekhoven, A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325–1344 (1999).
    [CrossRef]
  11. U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997).
    [CrossRef]
  12. 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]
  13. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).
  14. R. F. Remis, P. M. van den Berg, “On the equivalence of the Newton–Kantorovich and distorted Born methods,” Inverse Probl. 16, L1–L4 (2000).
    [CrossRef]
  15. 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]
  16. R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
    [CrossRef]
  17. A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
    [CrossRef]
  18. G. Leone, A. Brancaccio, R. Pierri, “Linear and quadratic inverse scattering for angularly varying circular cylinders,” J. Opt. Soc. Am. A 16, 2887–2895 (1999).
    [CrossRef]
  19. E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
    [CrossRef]
  20. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  21. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  22. 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.
  23. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
    [CrossRef]
  24. The behavior of the singular values of a linear integral operator follows a pattern similar to the decay of the Fourier coefficients of a function [F. Smithies, “The eigenvalues and singular values of integral equations,” Proc. London Math. Soc. 43, 255–279 (1937)]. If it is analytical, they decay exponentially fast to zero with the order; if the derivative of some order is discontinuous, they decay at a rate that is faster the higher the order for which the derivative exists and is continuous; if it is singular, but integrable, the rate is smoother.
  25. R. Pierri, A. Brancaccio, F. De Blasio, “Multifrequency dielectric profile inversion for a cylindrically stratified medium,” IEEE Trans. Geosci. Remote Sens. 38, 1716–1724 (2000).
    [CrossRef]
  26. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
    [CrossRef]
  27. H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer, numerical values,” IEEE Trans. Antennas Propag. AP-23, 723–725 (1975).
    [CrossRef]
  28. L. V. Kantorovic, G. P. Akilov, Analisi funzionale (Editori riuniti, Rome, 1980).
  29. R. Pierri, G. Leone, R. Persico, “A second order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000).
    [CrossRef]

2000 (3)

R. F. Remis, P. M. van den Berg, “On the equivalence of the Newton–Kantorovich and distorted Born methods,” Inverse Probl. 16, L1–L4 (2000).
[CrossRef]

R. Pierri, G. Leone, R. Persico, “A second order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000).
[CrossRef]

R. Pierri, A. Brancaccio, F. De Blasio, “Multifrequency dielectric profile inversion for a cylindrically stratified medium,” IEEE Trans. Geosci. Remote Sens. 38, 1716–1724 (2000).
[CrossRef]

1999 (5)

1998 (3)

N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
[CrossRef] [PubMed]

B. J. Kooij, P. M. van den Berg, “Non linear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1706–1712 (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 (4)

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

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]

U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997).
[CrossRef]

P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997).
[CrossRef]

1996 (1)

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

1995 (1)

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

1990 (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]

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]

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]

1975 (1)

H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer, numerical values,” IEEE Trans. Antennas Propag. AP-23, 723–725 (1975).
[CrossRef]

1964 (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

1937 (1)

The behavior of the singular values of a linear integral operator follows a pattern similar to the decay of the Fourier coefficients of a function [F. Smithies, “The eigenvalues and singular values of integral equations,” Proc. London Math. Soc. 43, 255–279 (1937)]. If it is analytical, they decay exponentially fast to zero with the order; if the derivative of some order is discontinuous, they decay at a rate that is faster the higher the order for which the derivative exists and is continuous; if it is singular, but integrable, the rate is smoother.

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).

Abubakar, A.

P. M. van den Berg, A. L. van Broekhoven, A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325–1344 (1999).
[CrossRef]

Akilov, G. P.

L. V. Kantorovic, G. P. Akilov, Analisi funzionale (Editori riuniti, Rome, 1980).

Barlaud, M.

P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997).
[CrossRef]

Bernini, R.

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 Electron Physics, P. W. Hawks, ed. (Academic, London, 1989), pp. 1–120.

Blanc-Feraud, L.

P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997).
[CrossRef]

Bolomey, J. Ch.

N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
[CrossRef] [PubMed]

Brancaccio, A.

Broquetas, A.

N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
[CrossRef] [PubMed]

Bussey, H. E.

H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer, numerical values,” IEEE Trans. Antennas Propag. AP-23, 723–725 (1975).
[CrossRef]

Chew, W.

Chew, W. C.

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 (IEEE Press, Piscataway, N.J., 1995).

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]

De Blasio, F.

R. Pierri, A. Brancaccio, F. De Blasio, “Multifrequency dielectric profile inversion for a cylindrically stratified medium,” IEEE Trans. Geosci. Remote Sens. 38, 1716–1724 (2000).
[CrossRef]

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]

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]

Isernia, T.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Joachimowicz, N.

N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
[CrossRef] [PubMed]

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]

Kantorovic, L. V.

L. V. Kantorovic, G. P. Akilov, Analisi funzionale (Editori riuniti, Rome, 1980).

Kooij, B. J.

B. J. Kooij, P. M. van den Berg, “Non linear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1706–1712 (1998).
[CrossRef]

Leone, G.

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.

Lobel, P.

P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997).
[CrossRef]

Mallorqui, J. J.

N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
[CrossRef] [PubMed]

Pascazio, V.

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

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.

Pichot, Ch.

P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (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]

Pierri, R.

R. Pierri, G. Leone, R. Persico, “A second order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000).
[CrossRef]

R. Pierri, A. Brancaccio, F. De Blasio, “Multifrequency dielectric profile inversion for a cylindrically stratified medium,” IEEE Trans. Geosci. Remote Sens. 38, 1716–1724 (2000).
[CrossRef]

G. Leone, A. Brancaccio, R. Pierri, “Linear and quadratic inverse scattering for angularly varying circular cylinders,” J. Opt. Soc. Am. A 16, 2887–2895 (1999).
[CrossRef]

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, R. Persico, R. Bernini, “Information content of the Born field scattered by an embedded slab: multifrequency, multiview, and multifrequency–multiview cases,” J. Opt. Soc. Am. A 16, 2392–2399 (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. Brancaccio, “Imaging of a rotationally symmetric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

Rekanos, I. T.

I. T. Rekanos, T. V. Yioultsis, T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microwave Theory Tech. 47, 336–344 (1999).
[CrossRef]

Remis, R. F.

R. F. Remis, P. M. van den Berg, “On the equivalence of the Newton–Kantorovich and distorted Born methods,” Inverse Probl. 16, L1–L4 (2000).
[CrossRef]

Richmond, J. H.

H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer, numerical values,” IEEE Trans. Antennas Propag. AP-23, 723–725 (1975).
[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]

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

Smithies, F.

The behavior of the singular values of a linear integral operator follows a pattern similar to the decay of the Fourier coefficients of a function [F. Smithies, “The eigenvalues and singular values of integral equations,” Proc. London Math. Soc. 43, 255–279 (1937)]. If it is analytical, they decay exponentially fast to zero with the order; if the derivative of some order is discontinuous, they decay at a rate that is faster the higher the order for which the derivative exists and is continuous; if it is singular, but integrable, the rate is smoother.

Soldovieri, F.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

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]

Tamarkin, J. D.

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

Tautenhahn, U.

U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997).
[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]

Tsiboukis, T. D.

I. T. Rekanos, T. V. Yioultsis, T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microwave Theory Tech. 47, 336–344 (1999).
[CrossRef]

van Broekhoven, A. L.

P. M. van den Berg, A. L. van Broekhoven, A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325–1344 (1999).
[CrossRef]

van den Berg, P. M.

R. F. Remis, P. M. van den Berg, “On the equivalence of the Newton–Kantorovich and distorted Born methods,” Inverse Probl. 16, L1–L4 (2000).
[CrossRef]

P. M. van den Berg, A. L. van Broekhoven, A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325–1344 (1999).
[CrossRef]

B. J. Kooij, P. M. van den Berg, “Non linear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1706–1712 (1998).
[CrossRef]

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]

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]

Yioultsis, T. V.

I. T. Rekanos, T. V. Yioultsis, T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microwave Theory Tech. 47, 336–344 (1999).
[CrossRef]

Acta Math. (1)

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

Bell Syst. Tech. J. (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer, numerical values,” IEEE Trans. Antennas Propag. AP-23, 723–725 (1975).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

R. Pierri, A. Brancaccio, F. De Blasio, “Multifrequency dielectric profile inversion for a cylindrically stratified medium,” IEEE Trans. Geosci. Remote Sens. 38, 1716–1724 (2000).
[CrossRef]

IEEE Trans. Med. Imaging (2)

N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998).
[CrossRef] [PubMed]

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. (2)

B. J. Kooij, P. M. van den Berg, “Non linear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1706–1712 (1998).
[CrossRef]

I. T. Rekanos, T. V. Yioultsis, T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microwave Theory Tech. 47, 336–344 (1999).
[CrossRef]

Inverse Probl. (5)

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]

P. M. van den Berg, A. L. van Broekhoven, A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325–1344 (1999).
[CrossRef]

U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997).
[CrossRef]

R. F. Remis, P. M. van den Berg, “On the equivalence of the Newton–Kantorovich and distorted Born methods,” Inverse Probl. 16, L1–L4 (2000).
[CrossRef]

P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

J. Opt. Soc. Am. A (7)

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. London Math. Soc. (1)

The behavior of the singular values of a linear integral operator follows a pattern similar to the decay of the Fourier coefficients of a function [F. Smithies, “The eigenvalues and singular values of integral equations,” Proc. London Math. Soc. 43, 255–279 (1937)]. If it is analytical, they decay exponentially fast to zero with the order; if the derivative of some order is discontinuous, they decay at a rate that is faster the higher the order for which the derivative exists and is continuous; if it is singular, but integrable, the rate is smoother.

Radio Sci. (1)

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Other (5)

L. V. Kantorovic, G. P. Akilov, Analisi funzionale (Editori riuniti, Rome, 1980).

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

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

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.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

First derivative of the real part of the kernel K for θ=0, k0a=15, k0b=20, and different values of the background permittivity.

Fig. 3
Fig. 3

Singular values of the DB operator for k0a=k0b=15 and different values of the background permittivity.

Fig. 4
Fig. 4

Contrast function for the validation of the direct quadratic approximation in the case k0a=5 and ε1=10.

Fig. 5
Fig. 5

Modulus of the exact, linear, and quadratic approximations of the field scattered by the contrast profile of Fig. 4.

Fig. 6
Fig. 6

Phase of the exact, linear, and quadratic approximations of the field scattered by the contrast profile of Fig. 4.

Fig. 7
Fig. 7

Reconstruction of a continuous contrast function (solid curve) by linear (dotted curve) and quadratic (dashed curve) inversion in the case k0a=15 and ε1=10.

Fig. 8
Fig. 8

Tubelike contrast function (solid line) and its reconstruction (dashed line) by (a) linear, (b) linear with threshold, and (c) quadratic inversion in the case k0a=15 and ε1=10.

Equations (45)

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χ(r)=ε(r)-εb(r).
Es(θ)=ω2μ00aχ(r)02πGe(r, θ, θ)E(r, θ)dθrdr,
E(r, θ)=Eb(r, θ)+ω2μ00aχ(r)02πGi(r, r, θ, θ)×E(r, θ)dθrdr.
E(r, θ)=Eb(r, θ)+ω2μ00aχ(r)02πGi(r, r, θ, θ)×Eb(r, θ)dθrdr.
Es(θ)=0aχ(r)K(r, θ)dr+0a0aχ(r)χ(r)Q(r, r, θ)drdr,
K(r, θ)
=ω2μ002πGe(r, θ, θ)Eb(r, θ)rdθ,
Q(r, r, θ)
=ω4μ0202πGe(r, θ, θ)×02πGi(r, r, θ, θ)Eb(r, θ)rrdθdθ
εb(r)=ε1ε0,r<bε0,r>b,
Eb(r, θ)=ncnJn(k1r)exp(jnθ),
cn=j-n(-2j)/(πk0b)(k1/k0)Hn(2)(k0b)Jn-1(k1b)-Jn(k1b)Hn-1(2)(k0b).
Ge(r, θ, θ)=-j4ncnJn(k1r)exp[jn(θ-θ)],
Gi(r, r, θ, θ)
=-j4n[Hn(2)(k1ρM)Jn(k1ρm)+anJn(k1r)Jn(k1r)]exp[jn(θ-θ)],
an=(k1/k0)Hn(2)(k0b)Hn-1(2)(k1b)-Hn(2)(k1b)Hn-1(2)(k0b)(k1/k0)Hn(2)(k0b)Jn-1(k1b)-Jn(k1b)Hn-1(2)(k0b).
K(r, θ)=-jπω2ω0r2ncn2Jn2(k1r)exp(jnθ),
Q(r, r, θ)=-π2ω4μ02ρmρM4ncn2Jn2(k1ρm)×[anJn2(k1ρM)+Hn(2)(k1ρM)Jn(k1ρM)]exp(jnθ).
anJn2(k1b)=-j ε1-1πn3ebk122.
K(r, θ)=rJ0[2k0rcos(θ/2)],
Es(2)(n)=Acn2L(χ)+A2cn2an[L(χ)]2+A2cn2L[χL1(χ)],
L1(χ)=rarχ(r)Hn(2)(k1r)Jn(k1r)dr.
Φ(χ)=Es(n)-{Acn2L(χ)+A2cn2an[L(χ)]2 +A2cn2L[χL1(χ)]}2,
χ(r)=m=1Mχmum(r),
L(χ)=m=1Mχm0arJn2(k1r)um(r)dr,
L[χL1(χ)]=m1=1Mm2=1Mχm1χm2×0aum1(r)Hn(2)(k1r)Jn(k1r)r×0rrJn2(k1r)um2(r)drdr,
err1=n=-NN|Es(n)-Es(1)(n)|2n=-NN|Es(n)|21/2,
err2=n=-NN|Es(n)-Es(2)(n)|2n=-NN|Es(n)|21/2,
Einc=-ωμ04 H0(2)(k1|r-r|).
Es=na¯nJn(k1r)exp(jnθ),
E=nb¯nHn(2)(k0r)exp(jnθ),
-ωμ04 Hn(2)(k1b)Jn(k1r)exp(-jnθ)+a¯nJn(k1b)
=b¯nHn(2)(k0b),
-ωμ04 k1Hn(2)(k1b)Jn(k1r)exp(-jnθ)+a¯nk1Jn(k1b)
=b¯nk0Hn(2)(k0b),
a¯n
=-ωμ04 Jn(k1r)exp(-jnθ)×(k1/k0)Hn(2)(k0b)Hn-1(2)(k1b)-Hn(2)(k1b)Hn-1(2)(k0b)(k1/k0)Hn(2)(k0b)Jn-1(k1b)-Jn(k1b)Hn-1(2)(k0b)
=-ωμ04 Jn(k1r)exp(-jnθ)an,
b¯n
=-ωμ04 Jn(k1r)exp(-jnθ)×(-2j)/(πk0b)(k1/k0)Hn(2)(k0b)Jn-1(k1b)-Jn(k1b)Hn-1(2)(k0b),
Gi(r, r, θ, θ)=1jωμ0 (Einc+Es),
Eb(r, θ)=n(j)-n[Jn(k0r)+dnHn(2)(k0r)]exp(jnθ),
dn=-(k1/k0)Jn(k0b)Jn-1(k1b)-Jn(k1b)Jn-1(k0b)(k1/k0)Hn(2)(k0b)Jn-1(k1b)-Jn(k1b)Hn-1(2)(k0b).
Ge(r, θ, θ)=-j4 Eb(r, θ-θ).
K(r, θ)=-jπω2μ0r2n(-1)n[Jn(k0r)+dnHn(2)×(k0r)]2exp(jnθ)

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