Abstract

A quadratic approximation of the electromagnetic scattering equation is used to solve the inverse problem of the reconstruction of the dielectric permittivity function of a rotationally symmetric cylindrical object embedded in a homogeneous medium and illuminated by an incident plane wave at fixed frequency. The problem is formulated as the minimization of a properly defined functional. Because the operator involved in the inversion is quadratic, it is possible to discuss and avoid the presence of local minima (a well-known problem in nonlinear inversion). Theoretical considerations and numerical examples show the advantages of using a quadratic model in comparison with a linear one.

© 1997 Optical Society of America

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  1. M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered data,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]
  2. T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
    [CrossRef]
  3. T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional refractive indexdistribution in semi-transparent, birifrangent fibres,” J. Microsc. 177, 53–67 (1995).
    [CrossRef]
  4. W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applicationsin microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988).
    [CrossRef]
  5. J. Ch. Bolomey, “New concepts for microwave sensing”, in AdvancedMicrowave and Millimeter-Wave Detectors , S. S. Udpa, H. C. Han, eds., Proc. SPIE2275, 2–10 (1994).
    [CrossRef]
  6. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
    [CrossRef]
  7. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  8. I. Akduman, “An inverse scattering problem related to buried cylindrical bodiesilluminated by Gaussian beams,” Inverse Probl. 10, 213–226 (1994).
    [CrossRef]
  9. M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensitymeasurements,” Opt. Eng. 33, 3243–3253 (1994).
    [CrossRef]
  10. T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties ofoptical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
    [CrossRef]
  11. 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]
  12. M. M. Moghaddam, W. C. Chew, “Study of some practical issues in inversion with the Born iterativemethod using time-domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
    [CrossRef]
  13. S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problemsat fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
    [CrossRef]
  14. 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).
  15. A. Brancaccio, V. Pascazio, R. Pierri, “Reconstruction of dielectric profiles: a new approach as a quadratic inverse problem,” presented at the Institute of Electrical and Electronics Engineers–International Union of Radio Science Meeting on Microwaves in Medicine, Rome, October 1993.
  16. 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]
  17. A. Addivinola, A Brancaccio, G. Leone, R. Pierri, “Microwave tomography with a quadratic model: numerical results for the cylindrical case,” presented at the International Conference on Electromagnetic for Advanced Applications, Torino, Italy, September 1995.
  18. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
    [CrossRef]
  19. J. A. Stratton, Electromagnetic Theory (McGraw Hill, New York, 1941).
  20. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  21. O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
    [CrossRef]
  22. R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse sourceproblems,” J. Opt. Soc. Am. A 7, 1707–1713 (1990).
  23. I.e., space of the compact support continuous functions, with continuous derivatives until the third order.
  24. D. Colton, L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagneticwaves,” Arch. Ration. Mech. Anal. 119, 59–70 (1992).
    [CrossRef]
  25. M. Azimi, K. C. Kak, “Distortion in diffraction tomography caused by multiple scattering,” IEEE Trans. Med. Imaging MI-2, 176–195 (1983).
    [CrossRef]
  26. L. V. Kantarovic, G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977) [Analisi Funzionale (Editori Riuniti, Rome, 1980)].
  27. T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
    [CrossRef]
  28. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).
  29. H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylinder multilayer, numerical values,” IEEE Trans. Antennas Propag.723–725, 1975.
    [CrossRef]
  30. I. S. Gradshsteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).
  31. R. E. Kleinman, G. F. Roach, P. M. van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Am. A 7, 890–897 (1990).
    [CrossRef]

1995 (5)

T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional refractive indexdistribution in semi-transparent, birifrangent fibres,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

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

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

1994 (3)

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

I. Akduman, “An inverse scattering problem related to buried cylindrical bodiesilluminated by Gaussian beams,” Inverse Probl. 10, 213–226 (1994).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensitymeasurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

1993 (1)

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

1992 (2)

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered data,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

D. Colton, L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagneticwaves,” Arch. Ration. Mech. Anal. 119, 59–70 (1992).
[CrossRef]

1990 (2)

R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse sourceproblems,” J. Opt. Soc. Am. A 7, 1707–1713 (1990).

R. E. Kleinman, G. F. Roach, P. M. van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Am. A 7, 890–897 (1990).
[CrossRef]

1989 (1)

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

1988 (1)

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applicationsin 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]

1983 (2)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[CrossRef]

M. Azimi, K. C. Kak, “Distortion in diffraction tomography caused by multiple scattering,” IEEE Trans. Med. Imaging MI-2, 176–195 (1983).
[CrossRef]

1975 (1)

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

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).

Addivinola, A.

A. Addivinola, A Brancaccio, G. Leone, R. Pierri, “Microwave tomography with a quadratic model: numerical results for the cylindrical case,” presented at the International Conference on Electromagnetic for Advanced Applications, Torino, Italy, September 1995.

Akduman, I.

I. Akduman, “An inverse scattering problem related to buried cylindrical bodiesilluminated by Gaussian beams,” Inverse Probl. 10, 213–226 (1994).
[CrossRef]

Akilov, G. P.

L. V. Kantarovic, G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977) [Analisi Funzionale (Editori Riuniti, Rome, 1980)].

Azimi, M.

M. Azimi, K. C. Kak, “Distortion in diffraction tomography caused by multiple scattering,” IEEE Trans. Med. Imaging MI-2, 176–195 (1983).
[CrossRef]

Bolomey, J. Ch.

J. Ch. Bolomey, “New concepts for microwave sensing”, in AdvancedMicrowave and Millimeter-Wave Detectors , S. S. Udpa, H. C. Han, eds., Proc. SPIE2275, 2–10 (1994).
[CrossRef]

Brancaccio, A

A. Addivinola, A Brancaccio, G. Leone, R. Pierri, “Microwave tomography with a quadratic model: numerical results for the cylindrical case,” presented at the International Conference on Electromagnetic for Advanced Applications, Torino, Italy, September 1995.

Brancaccio, A.

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]

A. Brancaccio, V. Pascazio, R. Pierri, “Reconstruction of dielectric profiles: a new approach as a quadratic inverse problem,” presented at the Institute of Electrical and Electronics Engineers–International Union of Radio Science Meeting on Microwaves in Medicine, Rome, October 1993.

Bucci, O. M.

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Bussey, H. E.

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

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).

Chew, W. C.

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

Chommeloux, L.

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

Colton, D.

D. Colton, L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagneticwaves,” Arch. Ration. Mech. Anal. 119, 59–70 (1992).
[CrossRef]

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

Devaney, A. J.

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensitymeasurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered data,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse sourceproblems,” J. Opt. Soc. Am. A 7, 1707–1713 (1990).

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[CrossRef]

Duchene, B.

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

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Franceschetti, G.

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Gradshsteyn, I. S.

I. S. Gradshsteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Gutman, S.

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

Isernia, T.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

Joachimowitz, N.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applicationsin 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]

Kak, K. C.

M. Azimi, K. C. Kak, “Distortion in diffraction tomography caused by multiple scattering,” IEEE Trans. Med. Imaging MI-2, 176–195 (1983).
[CrossRef]

Kantarovic, L. V.

L. V. Kantarovic, G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977) [Analisi Funzionale (Editori Riuniti, Rome, 1980)].

Kleinman, R. E.

Klibanov, M.

S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problemsat 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, “Limitation of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Leone, G.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

A. Addivinola, A Brancaccio, G. Leone, R. Pierri, “Microwave tomography with a quadratic model: numerical results for the cylindrical case,” presented at the International Conference on Electromagnetic for Advanced Applications, Torino, Italy, September 1995.

Lesselier, D.

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

Maleki, M. H.

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensitymeasurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered data,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

Moghaddam, M. M.

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

Paivarinta, L.

D. Colton, L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagneticwaves,” Arch. Ration. Mech. Anal. 119, 59–70 (1992).
[CrossRef]

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]

A. Brancaccio, V. Pascazio, R. Pierri, “Reconstruction of dielectric profiles: a new approach as a quadratic inverse problem,” presented at the Institute of Electrical and Electronics Engineers–International Union of Radio Science Meeting on Microwaves in Medicine, Rome, October 1993.

Pichot, Ch.

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

Pierri, R.

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]

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

A. Brancaccio, V. Pascazio, R. Pierri, “Reconstruction of dielectric profiles: a new approach as a quadratic inverse problem,” presented at the Institute of Electrical and Electronics Engineers–International Union of Radio Science Meeting on Microwaves in Medicine, Rome, October 1993.

A. Addivinola, A Brancaccio, G. Leone, R. Pierri, “Microwave tomography with a quadratic model: numerical results for the cylindrical case,” presented at the International Conference on Electromagnetic for Advanced Applications, Torino, Italy, September 1995.

Porter, R. P.

R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse sourceproblems,” J. Opt. Soc. Am. A 7, 1707–1713 (1990).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Richmond, J. H.

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

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

Roach, G. F.

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).

Ryzhik, I. M.

I. S. Gradshsteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Schatzberg, A.

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]

Stamnes, J. J.

Stegun, I.

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

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw Hill, New York, 1941).

Tabbara, W.

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

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

van den Berg, P. M.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Wedberg, T. C.

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

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional refractive indexdistribution in semi-transparent, birifrangent fibres,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

Wedberg, W. C.

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional refractive indexdistribution in semi-transparent, birifrangent fibres,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

Arch. Ration. Mech. Anal. (1)

D. Colton, L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagneticwaves,” Arch. Ration. Mech. Anal. 119, 59–70 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

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

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

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

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

IEEE Trans. Biomed. Eng. (1)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
[CrossRef]

IEEE Trans. Med. Imaging (1)

M. Azimi, K. C. Kak, “Distortion in diffraction tomography caused by multiple scattering,” IEEE Trans. Med. Imaging MI-2, 176–195 (1983).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (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]

Inverse Probl. (4)

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

I. Akduman, “An inverse scattering problem related to buried cylindrical bodiesilluminated by Gaussian beams,” Inverse Probl. 10, 213–226 (1994).
[CrossRef]

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

S. Gutman, M. Klibanov, “Iterative method for multi-dimensional inverse scattering problemsat fixed frequencies,” Inverse Probl. 10, 573–599 (1994).
[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. Microsc. (1)

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional refractive indexdistribution in semi-transparent, birifrangent fibres,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensitymeasurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

Opt. Rev. (1)

T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

Other (11)

J. Ch. Bolomey, “New concepts for microwave sensing”, in AdvancedMicrowave and Millimeter-Wave Detectors , S. S. Udpa, H. C. Han, eds., Proc. SPIE2275, 2–10 (1994).
[CrossRef]

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

A. Addivinola, A Brancaccio, G. Leone, R. Pierri, “Microwave tomography with a quadratic model: numerical results for the cylindrical case,” presented at the International Conference on Electromagnetic for Advanced Applications, Torino, Italy, September 1995.

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).

A. Brancaccio, V. Pascazio, R. Pierri, “Reconstruction of dielectric profiles: a new approach as a quadratic inverse problem,” presented at the Institute of Electrical and Electronics Engineers–International Union of Radio Science Meeting on Microwaves in Medicine, Rome, October 1993.

I. S. Gradshsteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

I.e., space of the compact support continuous functions, with continuous derivatives until the third order.

J. A. Stratton, Electromagnetic Theory (McGraw Hill, New York, 1941).

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

L. V. Kantarovic, G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977) [Analisi Funzionale (Editori Riuniti, Rome, 1980)].

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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Physical interpretation of (a) the linear and (b) the quadratic approximations of the scattering operator.

Fig. 3
Fig. 3

Functional along the direction tχtrue+(1-t)χlocal for 10 real unknowns and (a) 5 complex data, (b) 6 complex data, (c) 10 complex data.

Fig. 4
Fig. 4

(a) Actual profile, (b) first-order reconstructed profile, (c) second-order reconstructed profile.

Fig. 5
Fig. 5

Dotted line, first-order reconstructed profile; solid line, actual profile.

Fig. 6
Fig. 6

Dotted line, second-order reconstructed profile; solid line, actual profile.

Fig. 7
Fig. 7

(a) Actual profile, (b) first-order reconstructed profile, (c) second-order reconstructed profile.

Fig. 8
Fig. 8

Second-order reconstructed profile for the example in Fig. 4(a) in presence of noisy data.    

Equations (36)

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E(r, θ)=Ei(r, θ)+k202π0RGi(r, r, θ, θ)χ(r)×E(r, θ)rdrdθ
Es(r, θ)=k202π0RGe(r, r, θ, θ)χ(r)×E(r, θ)rdrdθ
χ(r)=r(r)b-1
Es(θ)=n=-NNan exp(-jnθ).
χ(r)=ncnfn(kr),nI,
Es=A(χ),
A(χ)=k202π0RGe(r, r, θ, θ)χ(r)×Ei(r, θ)rdrdθ.
Es=A(χ)+B(χ, χ),
B(χ, χ)=k202π0RGe(r, r, θ, θ)χ(r)×02π0RGi(r, r, θ, θ)χ(r)×Ei(r, θ)rdrdθrdrdθ.
A(χ)=n=-NN exp(-jnθ)pIcpψ1(n, p),
ψ1(n, p)=-j2 (2π/kr)1/2 exp(jπ/4-jkr)×0kRxfp(x)Jn2(x)dx,
B(χ, χ)=n=-NN exp(-jnθ)pIqIcpcqψ2(n, p, q),
ψ2(n, p, q)=-π exp(jπ/4-jkr)4 (2π/kr)1/2×0kRxfp(x)Jn(x)×Hn(2)(x)0xxfq(x)Jn2(x)dx+Jn(x)xkRxfq(x)×Hn(2)(x)Jn(x)dxdx,
ϕ(χ)=F(χ)-E˜s2,
χm+1=χm+λTmϕm,
ϕ(χm+λξm)=aλ4+bλ3+cλ2+dλ+e,
a=B(ξm, ξm)2,
b=2 ReA(ξm)+B(ξm, χm)+B(χm, ξm), B(ξm, ξm),
c=2 ReF(χm)-E˜s, B(ξm, ξm)+A(ξm)+B(ξm, χm)+B(χm, ξm)2,
d=2 ReF(χm)-E˜s, A(ξm)+B(ξm, χm)+B(χm, ξm),
e=ϕ(χm),
4aλ3+3bλ2+2cλ+d=0.
b2ac=4[ReA(ξ)+B(ξ, χm)+B(χm, ξ),B(ξ, ξ)]2B(ξ, ξ)2A(ξ)+B(ξ, χm)+B(χm, ξ)2329.
ϕ(c)=n=-NNa˜n-pI cpψ1(n, p)-pIqI cpcqψ2(n, p, q)2,
fp(x)=rectx-(p+1/2)ΔxΔx,p=0, , P-1,
ψ1(n, p)=-j2 pΔx(p+1)ΔxxJn2(x)dx,
ψ2(n, p, p)=-π4 pΔx(p+1)ΔxxJn(x)×Hn(2)(x)pΔxxxJn2(x)dx+Jn(x)x(p+1)Δxx×Hn(2)(x)Jn(x)dxdx,
ψ2(n, p, q)=-π4 pΔx(p+1)ΔxxJn2(x)dxqΔx(q+1)Δx×xHn(2)(x)Jn(x)dx,
forp<q,
ψ2(n, p, q)=-π4 qΔx(q+1)ΔxxJn2(x)dxpΔx(p+1)ΔxxHn(2)×(x)Jn(x)dx,forp>q,
e12=ma˜m-p=0P-1c^pψ1(m, p)2m|a˜m|2,
e22=ma˜m-p=0P-1c^pψ1(m, p)-p=0P-1q=0P-1c^pc^qψ2(m, p, q)2m|a˜m|2,
re12=p=0P-1|c^p-cp(1)|2p=0P-1|c^p|2,
re22=p=0P-1|c^p-cp(2)|2p=0P-1|c^p|2,
A(χ)=-j k24 2krπ1/2 expjπ4-kr×02π0R exp[jkr cos(θ-θ)]χ(r)×exp(-jkr cos θ)rdrdθ=-j4 2krπ1/2 expjπ4-kr×02π0kRm(j)mJm(x)×exp[-jm(θ-θ)]pIcpfp(x)×n(-j)nJn(x)exp(-jnθ)xdxdθ=-j4 2krπ1/2 expjπ4-kr×m(j)m exp(-jmθ)pIcpn(-j)n×02π exp[j(m-n)θ]dθ×0kRJm(x)fp(x)Jn(x)xdx=-j2 2πkr1/2expjπ4-kr×m exp(-jmθ)pcp×0kRfp(x)Jm2(x)xdx.
B(χ, χ)=-116 2krπ1/2 expjπ4-kr×02π0kRm(j)mJm(x)×exp[-jm(θ-θ)]pIcpfp(x)×02π0kRsJs[min(x, x)]×Hs(2)[max(x, x)]qIcqfq(x)×n(-j)nJn(x)exp(-jnθ)×xdxdθxdxdθ=-14 π2πkr1/2 expjπ4-kr×mexp(-jmθ)pIcpqIcq×0kRJm(x)fp(x)0kRJm[min(x, x)]×Hm(2)[max(x, x)]fq(x<2>)Jm(x)xdxxdx=-14 π2πkr1/2 expjπ4-kr×m exp(-jmθ)pcpqcq×0kRJm(x)fp(x)0kRJm[min(x, x)]×Hm(2)[max(x, x)]fq(x)×Jm(x)xdxxdx,

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