Abstract

A numerical algorithm for reconstruction of the permittivity of a three-dimensional penetrable object from scattering data is presented. The reconstruction algorithm is based on the local shape function method combined with the conjugate gradient method with fast Fourier transform. The nonlinearity that is due to multiple scattering is accounted for in an iterative minimization scheme. Numerical examples of simulation data are given.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Colton, “The inverse electromagnetic scattering problem for a perfectly conductingcylinder,” IEEE Trans. Antennas Propag. 29, 364–368 (1981).
    [CrossRef]
  2. C.-C. Chiu, Y.-W. Kiang, “Inverse scattering of a buried conducting cylinder,” Inverse Probl. 7, 187–202 (1991).
    [CrossRef]
  3. C.-C. Chiu, Y.-W. Kiang, “Microwave imaging of multiple conducting cylinders,” IEEE Trans. Antennas Propag. 40, 933–941 (1992).
    [CrossRef]
  4. P. Maponi, L. Misici, F. Zirilli, “An inverse problem for the three dimensional vector Helmholtz equationfor a perfectly conducting obstacle,” Comput. Math. Appl. 22, 137–146 (1991).
    [CrossRef]
  5. W. C. Chew, G. P. Otto, “Microwave imaging of multiple metallic cylinders using local shape functions,” presented at the IEEE-APS International Symposium, Chicago, Ill., July 18–25, 1992.
  6. W. C. Chew, G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shapefunctions,” IEEE Microwave Guided Wave Lett. 2, 284–286 (1992).
    [CrossRef]
  7. G. P. Otto, W. C. Chew, “Microwave inverse scattering-local shape function imaging for improvedresolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
    [CrossRef]
  8. W. H. Weedon, W. C. Chew, “Time-domain inverse scattering using local shape function (LSF) method,” Inverse Probl. 9, 551–564 (1993).
    [CrossRef]
  9. Y. M. Wang, W. C. Chew, “An inverse solution of two-dimensional electromagnetic inverse scatteringproblem,” Int. J. Imag. Syst. Tech. 1, 100–108 (1989).
    [CrossRef]
  10. W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using thedistorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [CrossRef]
  11. J.-Ch. Bolomey, Ch. Pichot, “Microwave tomography: from theory to practical imaging systems,” Int. J. Imag. Syst. Tech. 2, 144–156 (1990).
    [CrossRef]
  12. N. Joachimowicz, Ch. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagneticimaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
    [CrossRef]
  13. M. Moghaddam, W. C. Chew, “Comparison of the Born iterative method and Tarantola's method foran electromagnetic time-domain inverse problem,” Int. J. Imag. Syst. Tech. 3, 318–333 (1991).
    [CrossRef]
  14. M. Moghaddam, W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time domaindata,” IEEE Trans. Geosci. Remote Sensing 30, 147–156 (1992).
    [CrossRef]
  15. G. P. Otto, W. C. Chew, “Inverse scattering of Hz waves using local shape function imaging:a T-matrix formulation,” Int. J. Imaging Syst. Tech. 5, 22–27 (1994).
    [CrossRef]
  16. J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.
  17. J.-Ch. Bolomey, Ch. Durix, D. Lesselier, “Determination of conductivity profiles by time-domain reflectometry,” IEEE Trans. Antennas Propag. 27, 244–248 (1979).
    [CrossRef]
  18. Q.-H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sensing 32, 499–507 (1994).
    [CrossRef]
  19. W. C. Chew, Q.-H. Liu, “Inversion of induction tool measurements using the distorted Born iterativemethod and CG-FFHT,” IEEE Trans. Geosci. Remote Sensing 32, 878–884 (1994).
    [CrossRef]
  20. P. F. Polatin, K. Sarabandi, F. T. Ulaby, “An iterative inversion algorithm with application to the polarimetricradar response of vegetation canopies,” IEEE Trans. Geosci. Remote Sensing 32, 62–71 (1994).
    [CrossRef]
  21. R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
    [CrossRef]
  22. A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
    [CrossRef] [PubMed]
  23. A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111–112 (1982).
    [CrossRef] [PubMed]
  24. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–360 (1982).
    [CrossRef] [PubMed]
  25. M. Moghaddam, W. C. Chew, “Variable density linear acoustic inverse problem,” Ultrason. Imaging 15, 255–266 (1993).
    [CrossRef] [PubMed]
  26. D.-B. Lin, T.-H. Chu, “Bistaic frequency-swept microwave imaging principle, methodology andexperimental results,” IEEE Trans. Microwave Theory Tech. 41, 855–861 (1993).
    [CrossRef]
  27. P. Maponi, M. Recchioni, F. Zirilli, “Three-dimensional time harmonic electromagnetic inverse scattering—thereconstruction of the shape and the impedance of an obstacle,” Comput. Math. Appl. 31, 1–7 (1996).
    [CrossRef]
  28. H. Gan, W. C. Chew, “3D inhomogeneous inversion for microwave imaging using distorted born iterative method and BCG-FFT,” presented at the IEEE Antennas and Propagation Society International Symposium, Newport Beach, Calif., June 18–23, 1995.
  29. J.-H. Lin, W. C. Chew, “Ultrasonic imaging by local shape function method with CGFFT,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 956–969 (1996).
    [CrossRef]
  30. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
    [CrossRef]
  31. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990; reprinted by IEEE Press, Piscataway, N.J., 1995).
  32. C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994).
  33. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  34. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).
  35. J. A. Stratton, Electromagnetic Theorem (McGraw-Hill, New York, 1941).
  36. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1958).
  37. Y. M. Wang, W. C. Chew, “A recursive T-matrix approach for the solution of electromagnetic scatteringby many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1993).
    [CrossRef]
  38. W. C. Chew, “Recurrence relation for three dimensional scalar addition theory,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
    [CrossRef]
  39. W. C. Chew, Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
    [CrossRef]
  40. K. T. Kim, “The translation formula for vector multipole fields and the recurrencerelations for the translation coefficients of scalar and vector multipolefields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
    [CrossRef]
  41. R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
    [CrossRef]
  42. J.-H. Lin, W. C. Chew, “BiCG-FFT T-matrix method for solving for the scattering solution frominhomogeneous bodies,” IEEE Trans. Microwave Theory Tech. 44, 1150–1155 (1996).
    [CrossRef]
  43. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  44. W. C. Chew, C. C. Lu, “NEPAL—an algorithm for solving the volume integral equation,” Microwave Opt. Tech. Lett. 6, 185–188 (1993).
    [CrossRef]
  45. J.-H. Lin, W. C. Chew, “An application of nested equivalence principle algorithm (NEPAL) in matrix-vector multiplication of iterative algorithms,” in Radio Science Meeting Program and Abstracts, June 28–July 2, 1993, p. 317.
  46. D. Girard, “Practical optimal regularization of large linear systems,” Math. Modelling Num. Anal. 20, 75–87 (1986).
  47. Y. M. Qin, I. R. Ciric, “Method of selecting the regularization parameter for microwave imaging,” Electron. Lett. 30, 2028–2029 (1994).
    [CrossRef]

1996 (4)

P. Maponi, M. Recchioni, F. Zirilli, “Three-dimensional time harmonic electromagnetic inverse scattering—thereconstruction of the shape and the impedance of an obstacle,” Comput. Math. Appl. 31, 1–7 (1996).
[CrossRef]

J.-H. Lin, W. C. Chew, “Ultrasonic imaging by local shape function method with CGFFT,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 956–969 (1996).
[CrossRef]

K. T. Kim, “The translation formula for vector multipole fields and the recurrencerelations for the translation coefficients of scalar and vector multipolefields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
[CrossRef]

J.-H. Lin, W. C. Chew, “BiCG-FFT T-matrix method for solving for the scattering solution frominhomogeneous bodies,” IEEE Trans. Microwave Theory Tech. 44, 1150–1155 (1996).
[CrossRef]

1994 (6)

G. P. Otto, W. C. Chew, “Inverse scattering of Hz waves using local shape function imaging:a T-matrix formulation,” Int. J. Imaging Syst. Tech. 5, 22–27 (1994).
[CrossRef]

G. P. Otto, W. C. Chew, “Microwave inverse scattering-local shape function imaging for improvedresolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
[CrossRef]

Q.-H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sensing 32, 499–507 (1994).
[CrossRef]

W. C. Chew, Q.-H. Liu, “Inversion of induction tool measurements using the distorted Born iterativemethod and CG-FFHT,” IEEE Trans. Geosci. Remote Sensing 32, 878–884 (1994).
[CrossRef]

P. F. Polatin, K. Sarabandi, F. T. Ulaby, “An iterative inversion algorithm with application to the polarimetricradar response of vegetation canopies,” IEEE Trans. Geosci. Remote Sensing 32, 62–71 (1994).
[CrossRef]

Y. M. Qin, I. R. Ciric, “Method of selecting the regularization parameter for microwave imaging,” Electron. Lett. 30, 2028–2029 (1994).
[CrossRef]

1993 (6)

W. H. Weedon, W. C. Chew, “Time-domain inverse scattering using local shape function (LSF) method,” Inverse Probl. 9, 551–564 (1993).
[CrossRef]

M. Moghaddam, W. C. Chew, “Variable density linear acoustic inverse problem,” Ultrason. Imaging 15, 255–266 (1993).
[CrossRef] [PubMed]

D.-B. Lin, T.-H. Chu, “Bistaic frequency-swept microwave imaging principle, methodology andexperimental results,” IEEE Trans. Microwave Theory Tech. 41, 855–861 (1993).
[CrossRef]

W. C. Chew, C. C. Lu, “NEPAL—an algorithm for solving the volume integral equation,” Microwave Opt. Tech. Lett. 6, 185–188 (1993).
[CrossRef]

Y. M. Wang, W. C. Chew, “A recursive T-matrix approach for the solution of electromagnetic scatteringby many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1993).
[CrossRef]

W. C. Chew, Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
[CrossRef]

1992 (4)

W. C. Chew, “Recurrence relation for three dimensional scalar addition theory,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
[CrossRef]

C.-C. Chiu, Y.-W. Kiang, “Microwave imaging of multiple conducting cylinders,” IEEE Trans. Antennas Propag. 40, 933–941 (1992).
[CrossRef]

W. C. Chew, G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shapefunctions,” IEEE Microwave Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

M. Moghaddam, W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time domaindata,” IEEE Trans. Geosci. Remote Sensing 30, 147–156 (1992).
[CrossRef]

1991 (4)

N. Joachimowicz, Ch. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagneticimaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

M. Moghaddam, W. C. Chew, “Comparison of the Born iterative method and Tarantola's method foran electromagnetic time-domain inverse problem,” Int. J. Imag. Syst. Tech. 3, 318–333 (1991).
[CrossRef]

P. Maponi, L. Misici, F. Zirilli, “An inverse problem for the three dimensional vector Helmholtz equationfor a perfectly conducting obstacle,” Comput. Math. Appl. 22, 137–146 (1991).
[CrossRef]

C.-C. Chiu, Y.-W. Kiang, “Inverse scattering of a buried conducting cylinder,” Inverse Probl. 7, 187–202 (1991).
[CrossRef]

1990 (2)

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

J.-Ch. Bolomey, Ch. Pichot, “Microwave tomography: from theory to practical imaging systems,” Int. J. Imag. Syst. Tech. 2, 144–156 (1990).
[CrossRef]

1989 (1)

Y. M. Wang, W. C. Chew, “An inverse solution of two-dimensional electromagnetic inverse scatteringproblem,” Int. J. Imag. Syst. Tech. 1, 100–108 (1989).
[CrossRef]

1988 (1)

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

1986 (1)

D. Girard, “Practical optimal regularization of large linear systems,” Math. Modelling Num. Anal. 20, 75–87 (1986).

1982 (2)

A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111–112 (1982).
[CrossRef] [PubMed]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–360 (1982).
[CrossRef] [PubMed]

1981 (2)

A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
[CrossRef] [PubMed]

D. Colton, “The inverse electromagnetic scattering problem for a perfectly conductingcylinder,” IEEE Trans. Antennas Propag. 29, 364–368 (1981).
[CrossRef]

1979 (2)

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

J.-Ch. Bolomey, Ch. Durix, D. Lesselier, “Determination of conductivity profiles by time-domain reflectometry,” IEEE Trans. Antennas Propag. 27, 244–248 (1979).
[CrossRef]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
[CrossRef]

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Bolomey, J.-Ch.

J.-Ch. Bolomey, Ch. Pichot, “Microwave tomography: from theory to practical imaging systems,” Int. J. Imag. Syst. Tech. 2, 144–156 (1990).
[CrossRef]

J.-Ch. Bolomey, Ch. Durix, D. Lesselier, “Determination of conductivity profiles by time-domain reflectometry,” IEEE Trans. Antennas Propag. 27, 244–248 (1979).
[CrossRef]

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Broquetas, A.

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Chew, W. C.

J.-H. Lin, W. C. Chew, “Ultrasonic imaging by local shape function method with CGFFT,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 956–969 (1996).
[CrossRef]

J.-H. Lin, W. C. Chew, “BiCG-FFT T-matrix method for solving for the scattering solution frominhomogeneous bodies,” IEEE Trans. Microwave Theory Tech. 44, 1150–1155 (1996).
[CrossRef]

W. C. Chew, Q.-H. Liu, “Inversion of induction tool measurements using the distorted Born iterativemethod and CG-FFHT,” IEEE Trans. Geosci. Remote Sensing 32, 878–884 (1994).
[CrossRef]

G. P. Otto, W. C. Chew, “Inverse scattering of Hz waves using local shape function imaging:a T-matrix formulation,” Int. J. Imaging Syst. Tech. 5, 22–27 (1994).
[CrossRef]

G. P. Otto, W. C. Chew, “Microwave inverse scattering-local shape function imaging for improvedresolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
[CrossRef]

W. H. Weedon, W. C. Chew, “Time-domain inverse scattering using local shape function (LSF) method,” Inverse Probl. 9, 551–564 (1993).
[CrossRef]

M. Moghaddam, W. C. Chew, “Variable density linear acoustic inverse problem,” Ultrason. Imaging 15, 255–266 (1993).
[CrossRef] [PubMed]

W. C. Chew, C. C. Lu, “NEPAL—an algorithm for solving the volume integral equation,” Microwave Opt. Tech. Lett. 6, 185–188 (1993).
[CrossRef]

Y. M. Wang, W. C. Chew, “A recursive T-matrix approach for the solution of electromagnetic scatteringby many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1993).
[CrossRef]

W. C. Chew, Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
[CrossRef]

W. C. Chew, “Recurrence relation for three dimensional scalar addition theory,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
[CrossRef]

W. C. Chew, G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shapefunctions,” IEEE Microwave Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

M. Moghaddam, W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time domaindata,” IEEE Trans. Geosci. Remote Sensing 30, 147–156 (1992).
[CrossRef]

M. Moghaddam, W. C. Chew, “Comparison of the Born iterative method and Tarantola's method foran electromagnetic time-domain inverse problem,” Int. J. Imag. Syst. Tech. 3, 318–333 (1991).
[CrossRef]

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

Y. M. Wang, W. C. Chew, “An inverse solution of two-dimensional electromagnetic inverse scatteringproblem,” Int. J. Imag. Syst. Tech. 1, 100–108 (1989).
[CrossRef]

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990; reprinted by IEEE Press, Piscataway, N.J., 1995).

H. Gan, W. C. Chew, “3D inhomogeneous inversion for microwave imaging using distorted born iterative method and BCG-FFT,” presented at the IEEE Antennas and Propagation Society International Symposium, Newport Beach, Calif., June 18–23, 1995.

W. C. Chew, G. P. Otto, “Microwave imaging of multiple metallic cylinders using local shape functions,” presented at the IEEE-APS International Symposium, Chicago, Ill., July 18–25, 1992.

J.-H. Lin, W. C. Chew, “An application of nested equivalence principle algorithm (NEPAL) in matrix-vector multiplication of iterative algorithms,” in Radio Science Meeting Program and Abstracts, June 28–July 2, 1993, p. 317.

Chiu, C.-C.

C.-C. Chiu, Y.-W. Kiang, “Microwave imaging of multiple conducting cylinders,” IEEE Trans. Antennas Propag. 40, 933–941 (1992).
[CrossRef]

C.-C. Chiu, Y.-W. Kiang, “Inverse scattering of a buried conducting cylinder,” Inverse Probl. 7, 187–202 (1991).
[CrossRef]

Chu, T.-H.

D.-B. Lin, T.-H. Chu, “Bistaic frequency-swept microwave imaging principle, methodology andexperimental results,” IEEE Trans. Microwave Theory Tech. 41, 855–861 (1993).
[CrossRef]

Ciric, I. R.

Y. M. Qin, I. R. Ciric, “Method of selecting the regularization parameter for microwave imaging,” Electron. Lett. 30, 2028–2029 (1994).
[CrossRef]

Colton, D.

D. Colton, “The inverse electromagnetic scattering problem for a perfectly conductingcylinder,” IEEE Trans. Antennas Propag. 29, 364–368 (1981).
[CrossRef]

Devaney, A. J.

Durix, Ch.

J.-Ch. Bolomey, Ch. Durix, D. Lesselier, “Determination of conductivity profiles by time-domain reflectometry,” IEEE Trans. Antennas Propag. 27, 244–248 (1979).
[CrossRef]

Gan, H.

H. Gan, W. C. Chew, “3D inhomogeneous inversion for microwave imaging using distorted born iterative method and BCG-FFT,” presented at the IEEE Antennas and Propagation Society International Symposium, Newport Beach, Calif., June 18–23, 1995.

Girard, D.

D. Girard, “Practical optimal regularization of large linear systems,” Math. Modelling Num. Anal. 20, 75–87 (1986).

Harrington, R. F.

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

Hugonin, J.-P.

N. Joachimowicz, Ch. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagneticimaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Joachimowicz, N.

N. Joachimowicz, Ch. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagneticimaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Kaveh, M.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Kiang, Y.-W.

C.-C. Chiu, Y.-W. Kiang, “Microwave imaging of multiple conducting cylinders,” IEEE Trans. Antennas Propag. 40, 933–941 (1992).
[CrossRef]

C.-C. Chiu, Y.-W. Kiang, “Inverse scattering of a buried conducting cylinder,” Inverse Probl. 7, 187–202 (1991).
[CrossRef]

Kim, K. T.

K. T. Kim, “The translation formula for vector multipole fields and the recurrencerelations for the translation coefficients of scalar and vector multipolefields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
[CrossRef]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

Lesselier, D.

J.-Ch. Bolomey, Ch. Durix, D. Lesselier, “Determination of conductivity profiles by time-domain reflectometry,” IEEE Trans. Antennas Propag. 27, 244–248 (1979).
[CrossRef]

Lin, D.-B.

D.-B. Lin, T.-H. Chu, “Bistaic frequency-swept microwave imaging principle, methodology andexperimental results,” IEEE Trans. Microwave Theory Tech. 41, 855–861 (1993).
[CrossRef]

Lin, J.-H.

J.-H. Lin, W. C. Chew, “Ultrasonic imaging by local shape function method with CGFFT,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 956–969 (1996).
[CrossRef]

J.-H. Lin, W. C. Chew, “BiCG-FFT T-matrix method for solving for the scattering solution frominhomogeneous bodies,” IEEE Trans. Microwave Theory Tech. 44, 1150–1155 (1996).
[CrossRef]

J.-H. Lin, W. C. Chew, “An application of nested equivalence principle algorithm (NEPAL) in matrix-vector multiplication of iterative algorithms,” in Radio Science Meeting Program and Abstracts, June 28–July 2, 1993, p. 317.

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Liu, Q.-H.

Q.-H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sensing 32, 499–507 (1994).
[CrossRef]

W. C. Chew, Q.-H. Liu, “Inversion of induction tool measurements using the distorted Born iterativemethod and CG-FFHT,” IEEE Trans. Geosci. Remote Sensing 32, 878–884 (1994).
[CrossRef]

Lu, C. C.

W. C. Chew, C. C. Lu, “NEPAL—an algorithm for solving the volume integral equation,” Microwave Opt. Tech. Lett. 6, 185–188 (1993).
[CrossRef]

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Mallorqui, J. J.

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Maponi, P.

P. Maponi, M. Recchioni, F. Zirilli, “Three-dimensional time harmonic electromagnetic inverse scattering—thereconstruction of the shape and the impedance of an obstacle,” Comput. Math. Appl. 31, 1–7 (1996).
[CrossRef]

P. Maponi, L. Misici, F. Zirilli, “An inverse problem for the three dimensional vector Helmholtz equationfor a perfectly conducting obstacle,” Comput. Math. Appl. 22, 137–146 (1991).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1958).

Misici, L.

P. Maponi, L. Misici, F. Zirilli, “An inverse problem for the three dimensional vector Helmholtz equationfor a perfectly conducting obstacle,” Comput. Math. Appl. 22, 137–146 (1991).
[CrossRef]

Moghaddam, M.

M. Moghaddam, W. C. Chew, “Variable density linear acoustic inverse problem,” Ultrason. Imaging 15, 255–266 (1993).
[CrossRef] [PubMed]

M. Moghaddam, W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time domaindata,” IEEE Trans. Geosci. Remote Sensing 30, 147–156 (1992).
[CrossRef]

M. Moghaddam, W. C. Chew, “Comparison of the Born iterative method and Tarantola's method foran electromagnetic time-domain inverse problem,” Int. J. Imag. Syst. Tech. 3, 318–333 (1991).
[CrossRef]

Mueller, R. K.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Otto, G. P.

G. P. Otto, W. C. Chew, “Inverse scattering of Hz waves using local shape function imaging:a T-matrix formulation,” Int. J. Imaging Syst. Tech. 5, 22–27 (1994).
[CrossRef]

G. P. Otto, W. C. Chew, “Microwave inverse scattering-local shape function imaging for improvedresolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
[CrossRef]

W. C. Chew, G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shapefunctions,” IEEE Microwave Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

W. C. Chew, G. P. Otto, “Microwave imaging of multiple metallic cylinders using local shape functions,” presented at the IEEE-APS International Symposium, Chicago, Ill., July 18–25, 1992.

Pichot, Ch.

N. Joachimowicz, Ch. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagneticimaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

J.-Ch. Bolomey, Ch. Pichot, “Microwave tomography: from theory to practical imaging systems,” Int. J. Imag. Syst. Tech. 2, 144–156 (1990).
[CrossRef]

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Polatin, P. F.

P. F. Polatin, K. Sarabandi, F. T. Ulaby, “An iterative inversion algorithm with application to the polarimetricradar response of vegetation canopies,” IEEE Trans. Geosci. Remote Sensing 32, 62–71 (1994).
[CrossRef]

Qin, Y. M.

Y. M. Qin, I. R. Ciric, “Method of selecting the regularization parameter for microwave imaging,” Electron. Lett. 30, 2028–2029 (1994).
[CrossRef]

Recchioni, M.

P. Maponi, M. Recchioni, F. Zirilli, “Three-dimensional time harmonic electromagnetic inverse scattering—thereconstruction of the shape and the impedance of an obstacle,” Comput. Math. Appl. 31, 1–7 (1996).
[CrossRef]

Sarabandi, K.

P. F. Polatin, K. Sarabandi, F. T. Ulaby, “An iterative inversion algorithm with application to the polarimetricradar response of vegetation canopies,” IEEE Trans. Geosci. Remote Sensing 32, 62–71 (1994).
[CrossRef]

Stratton, J. A.

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

Tai, C. T.

C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994).

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Ulaby, F. T.

P. F. Polatin, K. Sarabandi, F. T. Ulaby, “An iterative inversion algorithm with application to the polarimetricradar response of vegetation canopies,” IEEE Trans. Geosci. Remote Sensing 32, 62–71 (1994).
[CrossRef]

Wade, G.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Wang, Y. M.

Y. M. Wang, W. C. Chew, “A recursive T-matrix approach for the solution of electromagnetic scatteringby many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1993).
[CrossRef]

W. C. Chew, Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
[CrossRef]

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

Y. M. Wang, W. C. Chew, “An inverse solution of two-dimensional electromagnetic inverse scatteringproblem,” Int. J. Imag. Syst. Tech. 1, 100–108 (1989).
[CrossRef]

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

Waterman, P. C.

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
[CrossRef]

Weedon, W. H.

W. H. Weedon, W. C. Chew, “Time-domain inverse scattering using local shape function (LSF) method,” Inverse Probl. 9, 551–564 (1993).
[CrossRef]

Wittmann, R. C.

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

Zirilli, F.

P. Maponi, M. Recchioni, F. Zirilli, “Three-dimensional time harmonic electromagnetic inverse scattering—thereconstruction of the shape and the impedance of an obstacle,” Comput. Math. Appl. 31, 1–7 (1996).
[CrossRef]

P. Maponi, L. Misici, F. Zirilli, “An inverse problem for the three dimensional vector Helmholtz equationfor a perfectly conducting obstacle,” Comput. Math. Appl. 22, 137–146 (1991).
[CrossRef]

Comput. Math. Appl. (2)

P. Maponi, L. Misici, F. Zirilli, “An inverse problem for the three dimensional vector Helmholtz equationfor a perfectly conducting obstacle,” Comput. Math. Appl. 22, 137–146 (1991).
[CrossRef]

P. Maponi, M. Recchioni, F. Zirilli, “Three-dimensional time harmonic electromagnetic inverse scattering—thereconstruction of the shape and the impedance of an obstacle,” Comput. Math. Appl. 31, 1–7 (1996).
[CrossRef]

Electron. Lett. (1)

Y. M. Qin, I. R. Ciric, “Method of selecting the regularization parameter for microwave imaging,” Electron. Lett. 30, 2028–2029 (1994).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

W. C. Chew, G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shapefunctions,” IEEE Microwave Guided Wave Lett. 2, 284–286 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (7)

C.-C. Chiu, Y.-W. Kiang, “Microwave imaging of multiple conducting cylinders,” IEEE Trans. Antennas Propag. 40, 933–941 (1992).
[CrossRef]

D. Colton, “The inverse electromagnetic scattering problem for a perfectly conductingcylinder,” IEEE Trans. Antennas Propag. 29, 364–368 (1981).
[CrossRef]

N. Joachimowicz, Ch. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagneticimaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

J.-Ch. Bolomey, Ch. Durix, D. Lesselier, “Determination of conductivity profiles by time-domain reflectometry,” IEEE Trans. Antennas Propag. 27, 244–248 (1979).
[CrossRef]

K. T. Kim, “The translation formula for vector multipole fields and the recurrencerelations for the translation coefficients of scalar and vector multipolefields,” IEEE Trans. Antennas Propag. 44, 1482–1487 (1996).
[CrossRef]

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

Y. M. Wang, W. C. Chew, “A recursive T-matrix approach for the solution of electromagnetic scatteringby many spheres,” IEEE Trans. Antennas Propag. 41, 1633–1639 (1993).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (4)

Q.-H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sensing 32, 499–507 (1994).
[CrossRef]

W. C. Chew, Q.-H. Liu, “Inversion of induction tool measurements using the distorted Born iterativemethod and CG-FFHT,” IEEE Trans. Geosci. Remote Sensing 32, 878–884 (1994).
[CrossRef]

P. F. Polatin, K. Sarabandi, F. T. Ulaby, “An iterative inversion algorithm with application to the polarimetricradar response of vegetation canopies,” IEEE Trans. Geosci. Remote Sensing 32, 62–71 (1994).
[CrossRef]

M. Moghaddam, W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time domaindata,” IEEE Trans. Geosci. Remote Sensing 30, 147–156 (1992).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

IEEE Trans. Microwave Theory Tech. (3)

G. P. Otto, W. C. Chew, “Microwave inverse scattering-local shape function imaging for improvedresolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
[CrossRef]

J.-H. Lin, W. C. Chew, “BiCG-FFT T-matrix method for solving for the scattering solution frominhomogeneous bodies,” IEEE Trans. Microwave Theory Tech. 44, 1150–1155 (1996).
[CrossRef]

D.-B. Lin, T.-H. Chu, “Bistaic frequency-swept microwave imaging principle, methodology andexperimental results,” IEEE Trans. Microwave Theory Tech. 41, 855–861 (1993).
[CrossRef]

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

J.-H. Lin, W. C. Chew, “Ultrasonic imaging by local shape function method with CGFFT,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 956–969 (1996).
[CrossRef]

Int. J. Imag. Syst. Tech. (3)

Y. M. Wang, W. C. Chew, “An inverse solution of two-dimensional electromagnetic inverse scatteringproblem,” Int. J. Imag. Syst. Tech. 1, 100–108 (1989).
[CrossRef]

J.-Ch. Bolomey, Ch. Pichot, “Microwave tomography: from theory to practical imaging systems,” Int. J. Imag. Syst. Tech. 2, 144–156 (1990).
[CrossRef]

M. Moghaddam, W. C. Chew, “Comparison of the Born iterative method and Tarantola's method foran electromagnetic time-domain inverse problem,” Int. J. Imag. Syst. Tech. 3, 318–333 (1991).
[CrossRef]

Int. J. Imaging Syst. Tech. (1)

G. P. Otto, W. C. Chew, “Inverse scattering of Hz waves using local shape function imaging:a T-matrix formulation,” Int. J. Imaging Syst. Tech. 5, 22–27 (1994).
[CrossRef]

Inverse Probl. (2)

W. H. Weedon, W. C. Chew, “Time-domain inverse scattering using local shape function (LSF) method,” Inverse Probl. 9, 551–564 (1993).
[CrossRef]

C.-C. Chiu, Y.-W. Kiang, “Inverse scattering of a buried conducting cylinder,” Inverse Probl. 7, 187–202 (1991).
[CrossRef]

J. Electromagn. Waves Appl. (2)

W. C. Chew, “Recurrence relation for three dimensional scalar addition theory,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
[CrossRef]

W. C. Chew, Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
[CrossRef]

Math. Modelling Num. Anal. (1)

D. Girard, “Practical optimal regularization of large linear systems,” Math. Modelling Num. Anal. 20, 75–87 (1986).

Microwave Opt. Tech. Lett. (1)

W. C. Chew, C. C. Lu, “NEPAL—an algorithm for solving the volume integral equation,” Microwave Opt. Tech. Lett. 6, 185–188 (1993).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (2)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–811 (1965).
[CrossRef]

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Ultrason. Imaging (2)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–360 (1982).
[CrossRef] [PubMed]

M. Moghaddam, W. C. Chew, “Variable density linear acoustic inverse problem,” Ultrason. Imaging 15, 255–266 (1993).
[CrossRef] [PubMed]

Other (11)

J.-H. Lin, C. C. Lu, Y. M. Wang, W. C. Chew, J. J. Mallorqui, A. Broquetas, Ch. Pichot, J.-Ch. Bolomey, “Processing microwave experimental data with the distorted Born iterative method of nonlinear inverse scattering,” in IEEE Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 500–503.

H. Gan, W. C. Chew, “3D inhomogeneous inversion for microwave imaging using distorted born iterative method and BCG-FFT,” presented at the IEEE Antennas and Propagation Society International Symposium, Newport Beach, Calif., June 18–23, 1995.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990; reprinted by IEEE Press, Piscataway, N.J., 1995).

C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994).

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

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

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

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1958).

W. C. Chew, G. P. Otto, “Microwave imaging of multiple metallic cylinders using local shape functions,” presented at the IEEE-APS International Symposium, Chicago, Ill., July 18–25, 1992.

J.-H. Lin, W. C. Chew, “An application of nested equivalence principle algorithm (NEPAL) in matrix-vector multiplication of iterative algorithms,” in Radio Science Meeting Program and Abstracts, June 28–July 2, 1993, p. 317.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) CG minimization scheme, (b) DBLIM.

Fig. 2
Fig. 2

Reconstruction of a 0.6λ cube divided into 15 × 15 × 15 cells with six plane-wave illuminations and 258 receivers: (a) slice images in the z axis, (b) cross-section profiles of (a). Solid curves, reconstructed profiles; dashed curves, true object.

Fig. 3
Fig. 3

Images and profiles (solid curves, reconstruction; dashed curves, true object) for the tenth slice in the z axis of an 1.0λ cube divided into 15 × 15 × 15 cells with 47 transmitters and 47 receivers located on an r = 1.5 λ sphere.

Equations (62)

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

E ( r ) = ψ t ( k 0 , r s ) · e s incident field + i = 1 N ψ t ( k 0 , r i ) · a i scattered field
[ ψ ( k ,   r ) ] lm = [ M lm ( k ,   r ) N lm ( k ,   r ) ] ,
l = 1 ,   2 ,   , m = - l , - l + 1 ,   ,   l ,
M lm ( k ,   r ) = × r ψ lm ( k ,   r ) ,
N lm ( k ,   r ) = 1 k   × M lm ( k ,   r ) .
ψ lm ( k ,   r ) = h l ( 1 ) ( kr ) Y lm ( θ ,   ϕ ) ,
Y lm ( θ ,   ϕ ) = ( - 1 ) m ( 2 l + 1 ) 4 π   ( l - m ) ! ( l + m ) ! 1 / 2 × P l m ( cos   θ ) exp ( im ϕ ) ,
Y l , - m ( θ ,   ϕ ) = ( - 1 ) m Y lm * ( θ ,   ϕ ) ,
Y lm ( θ ,   ϕ ) Y l m * d Ω = δ ll δ mm ,
M lm ( k ,   r ) = θ ˆ   im sin   θ   h l ( 1 ) ( kr ) Y lm ( θ ,   ϕ ) - ϕ ˆ h l ( 1 ) ( kr )   Y lm ( θ ,   ϕ ) θ ,
N lm ( k ,   r ) = r ˆ   l ( l + 1 ) kr   h l ( 1 ) ( kr ) Y lm ( θ ,   ϕ ) + θ ˆ   1 kr   r   [ rh l ( 1 ) ( kr ) ]   Y lm ( θ ,   ϕ ) θ + ϕ ˆ   im kr   sin   θ   r   [ rh l ( 1 ) ( kr ) ] Y lm ( θ ,   ϕ ) .
E ( r ) = R g ψ t ( k 0 ,   r j ) α ¯ js e s + R g ψ t ( k 0 ,   r j ) i = 1 i j N α ¯ ji a i + ψ t ( k 0 ,   r j ) a j ,
a j = T ¯ j ( 1 ) · [ α ¯ j s · e s + i = 1 i j N α ¯ j i · a i ] total impinging field amplitude at j th subscatterer ,             j = 1 , , N .
a j - T ¯ j ( 1 ) i = 1 i j N α ¯ ji a i = T ¯ j ( 1 ) α ¯ js e s
a - T ¯ A ¯ a = T ¯ s ,
[ T ¯ ] ij = δ ij T ¯ j ( 1 ) ,
[ A ¯ ] ij = α ¯ ij if j i 0 otherwise ,
[ a ] j = a j ,
[ s ] j = α ¯ js e s .
E sca ( r ) = Ψ t a = Ψ t [ I ¯ - D ¯ ( O ) A ¯ ] - 1 D ¯ ( O ) s ,
Φ sca = Ψ ¯ t a ,
S ( O ) = 1 2 [ Φ sca ( O ) - Φ meas sca 2 + γ O - O b 2 ] ,
c n - 1 = ( g n - g n - 1 ) g n g n - 1 g n - 1 ( h - 1 = 0 ) .
a n = g n h n h n H ¯ n h n
v = S O = F ¯ ( Φ sca - Φ meas sca ) + γ ( O - O b ) ,
H ¯ = v n O F ¯ F ¯ + γ I ¯ ,
F ¯ = Φ sca O = O   { Ψ ¯ t [ I ¯ - D ¯ ( O ) A ¯ ] - 1 D ¯ ( O ) s } .
F ¯ = Ψ ¯ t [ I ¯ - D ¯ ( O ) A ¯ ] - 1   D ¯ ( O ) O   ( A ¯ a + s ) ,
F ¯ = Ψ ¯ t [ I ¯ - D ¯ ( O ) A ¯ ] - 1   D ¯ ( O ) O [ I ¯ - A ¯ D ¯ ( O ) ] - 1 s ,
F ¯ ( Φ sca - Φ meas sca ) + γ ( O - O b ) = 0 .
Φ n sca Φ n - 1 sca + F ¯ n - 1 ( O n - O n - 1 ) ,
O n = O n - 1 + ( F ¯ n - 1 F ¯ n - 1 + γ I ¯ ) - 1 F ¯ n - 1 ( Φ meas sca - Φ n - 1 sca ) .
F ¯ n v = D ¯ ( O ) O   [ I ¯ - A ¯ D ¯ ( O ) ] - 1 s [ I ¯ - D ¯ ( O ) A ¯ ] - Ψ ¯ * v ,
M lm ( k ,   r ) = ν = 1 μ = - ν ν [ M ν μ ( k ,   r ) A ν μ , lm + N ν μ ( k ,   r ) B ν μ , lm ] ,
N lm ( k ,   r ) = ν = 1 μ = - ν ν [ M ν μ ( k ,   r ) B ν μ , lm + N ν μ ( k ,   r ) A ν μ , lm ] ,
ψ lm ( k ,   r ) = ν = 1 μ = - ν ν   Ψ ν μ ( r ) β ν μ , lm .
A ν μ , lm = β ν μ , lm + r   sin   θ   exp ( - i ϕ ) 2 ( ν + 1 ) × ( ν - μ + 2 ) ( ν - μ + 1 ) ( 2 ν + 1 ) ( 2 ν + 3 ) 1 / 2 β ν + 1 , μ - 1 , lm - r   sin   θ   exp ( - i ϕ ) 2 ν × ( ν + μ - 1 ) ( ν + μ ) ( 2 ν - 1 ) ( 2 ν + 1 ) 1 / 2 β ν - 1 , μ - 1 , lm - r   sin   θ   exp ( i ϕ ) 2 ( ν + 1 ) × ( ν + μ + 2 ) ( ν + μ + 1 ) ( 2 ν + 1 ) ( 2 ν + 3 ) 1 / 2 β ν + 1 , μ + 1 , lm + r   sin   θ   exp ( i ϕ ) 2 ν × ( ν - μ ) ( ν - μ - 1 ) ( 2 ν - 1 ) ( 2 ν + 1 ) 1 / 2 β ν - 1 , μ + 1 , lm + r   cos   θ   1 ν + 1 × ( ν + μ + 1 ) ( ν - μ + 1 ) ( 2 ν + 1 ) ( 2 ν + 3 ) 1 / 2 β ν + 1 , μ , lm + r   cos   θ   1 ν   ( ν + μ ) ( ν - μ ) ( 2 ν - 1 ) ( 2 ν + 1 ) 1 / 2 β ν - 1 , μ , lm ,
B ν μ , lm = r   cos   θ   i μ ν ( ν + 1 )   β ν μ , lm + ir   sin   θ 2 ν ( ν + 1 ) × { [ ( ν - μ ) ( ν + μ - 1 ) ] 1 / 2 × exp ( i ϕ ) β ν , μ + 1 , lm + [ ( ν + μ ) ( ν - μ - 1 ) ] 1 / 2 × exp ( - i ϕ ) β ν , μ - 1 , lm } .
a lm + β ν μ , l + 1 , m = - a lm - β ν μ , l - 1 , m + a ν - 1 , μ + β ν - 1 , μ , lm + a ν + 1 , μ - β ν + 1 , μ , lm ,
b ll + β ν μ , l + 1 , l + 1 = b ν - 1 , μ - 1 + β ν - 1 , μ - 1 , ll + b ν + 1 , μ - 1 - β ν + 1 , μ - 1 , ll ,
a lm + = - ( l + m + 1 ) ( l - m + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) 1 / 2 ,
a lm - = ( l + m ) ( l - m ) ( 2 l + 1 ) ( 2 l - 1 ) 1 / 2 ,
b lm + = ( l + m + 2 ) ( l + m + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) 1 / 2 ,
b lm - = ( l - m ) ( l - m - 1 ) ( 2 l + 1 ) ( 2 l - 1 ) 1 / 2 .
β 00 , lm = ( - 1 ) l β lm , 00 * = ( - 1 ) l 4 π Y lm * ( θ ,   ϕ ) j n ( kr ) .
A ν μ , l , - m = ( - 1 ) μ + m   exp [ - 2 i ( μ + m ) ϕ ] A ν , - μ , lm ,
A ν , - μ , l , - m = ( - 1 ) μ - m   exp [ 2 i ( μ - m ) ϕ ] A ν μ , lm ;
B ν μ , l , - m = ( - 1 ) μ + m + 1   exp [ - 2 i ( μ + m ) ϕ ] B ν , - μ , lm ,
B ν , - μ , l , - m = ( - 1 ) μ - m + 1   exp [ 2 i ( μ - m ) ϕ ] B ν μ , lm .
α ¯ ij = A 1 , - 1 , 1 , - 1 A 10 , 1 , - 1 A 11 , 1 , - 1 A 1 , - 1 , 10 A 10 , 10 A 11 , 10 A 1 , - 1 , 11 A 10 , 11 A 11 , 11 = β 00 , 00 - 1 2 5   β 20 , 00 - 3 2 5   β 21 , 00 - 3 10   β 22 , 00 3 2 5   β 2 , - 1 , 00 β 00 , 00 + 1 5   β 20 , 00 3 2 5   β 21 , 00 - 3 10   β 2 , - 2 , 00 - 3 2 5   β 2 , - 1 , 00 β 00 , 00 - 1 2 5   β 20 , 00 , ,
A 1 , - 1 , 1 , - 1 = A 11 , 11 = 4 π Y 0 , 0 ( θ ,   ϕ ) j 0 ( kr ) - 1 2 5   Y 2 , 0 ( θ ,   ϕ ) j 2 ( kr ) ,
A 10 , 1 , - 1 = - A 11 , 10 = 3 2 5   4 π Y 2 , - 1 ( θ ,   ϕ ) j 2 ( kr ) ,
A 11 , 1 , - 1 = - 3 / 10 4 π Y 2 , - 2 ( θ ,   ϕ ) j 2 ( kr ) ,
A 1 , - 1 , 10 = - A 10 , 11 = - 3 2 5   4 π Y 2 , 1 ( θ ,   ϕ ) j 2 ( kr ) ,
A 10 , 10 = 4 π Y 0 , 0 ( θ ,   ϕ ) j 0 ( kr ) + 1 5   Y 2 , 0 ( θ ,   ϕ ) j 2 ( kr ) ,
A 1 , - 1 , 11 = - 3 / 10 4 π Y 2 , - 2 ( θ ,   ϕ ) j 2 ( kr ) .
t lm TE = s μ b j ^ l ( k s a ) j ^ l ( k b a ) - b μ s j ^ l ( k s a ) j ^ l ( k b a ) b μ s h ^ l ( k b a ) j ^ l ( k s a ) - s μ b h ^ l ( k b a ) j ^ l ( k s a ) ,
t lm TM = b μ s j ^ l ( k s a ) j ^ l ( k b a ) - s μ b j ^ l ( k s a ) j ^ l ( k b a ) s μ b h ^ l ( k b a ) j ^ l ( k s a ) - b μ s h ^ l ( k b a ) j ^ l ( k s a ) ,
T ¯ lm = t lm TE 0 0 t lm TM .
t lm TE - i ( 1 - s / b ) ( 2 l + 3 ) [ 1 × 3 × ( 2 l + 1 ) ] 2   ( k b a ) ( 2 l + 3 )
as k b a 0 and μ s = μ b ,
t lm TM - i ( l + 1 ) ( 2 l + 1 ) ( 1 - s / b ) [ 1 × 3 × ( 2 l + 1 ) ] 2 [ l ( 1 + s / b ) + 1 ] × ( k b a ) ( 2 l + 1 ) as k b a 0 and μ s = μ b .

Metrics