Abstract

An iterative inverse-scattering approach to reconstruction of electrical parameter distributions of a three-dimensional object by using time-domain field data is presented. The approach is the extension of the forward–backward time-stepping algorithm previously proposed for a two-dimensional object. Numerical examples of simulation data are given to assess the effectiveness of the proposed approach.

© 2003 Optical Society of America

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  1. K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximations,” Jpn. J. Appl. Phys. 14, 1921–1927 (1975).
    [CrossRef]
  2. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [PubMed]
  3. 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]
  4. S. Caorsi, G. L. Gragnami, M. Pastorino, “Reconstruction of dielectric permittivity distributions in 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imaging 12, 232–239 (1993).
    [CrossRef]
  5. G. P. Otto, W. C. Chew, “Microwave inverse scattering—local shape function imaging for improved resolution of strong scatterers,” IEEE Trans. Microwave Theory Tech. 42, 137–141 (1994).
    [CrossRef]
  6. H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
    [CrossRef]
  7. C. S. Park, S. K. Park, J. W. Ra, “Moment method and iterative reconstruction of two-dimensional complex permittivity by using effective modes with multipole sources in the presence of noise,” Radio Sci. 31, 1877–1886 (1996).
    [CrossRef]
  8. A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
    [CrossRef]
  9. T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
    [CrossRef]
  10. P. M. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
    [CrossRef]
  11. O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
    [CrossRef]
  12. S. Y. Yang, H. K. Choi, J. W. Ra, “Reconstruction of a large and high-contrast penetrable object by using the genetic and Levenberg–Marquardt algorithm,” Microwave Opt. Technol. Lett. 16, 17–21 (1997).
    [CrossRef]
  13. M. Pastorino, A. Massa, S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas. 49, 573–578 (2000).
    [CrossRef]
  14. T. Isernia, V. Pascazio, R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (2001).
    [CrossRef]
  15. S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
    [CrossRef]
  16. M. Moghaddam, W. C. Chew, “Study of some practical issues in inversion with the Born iterative method using time-domain data,” IEEE Trans. Antennas Propag. 41, 177–184 (1993).
    [CrossRef]
  17. W. H. Weedon, W. C. Chew, “Time-domain inverse scattering using the local shape function (LSF) method,” Inverse Probl. 9, 551–564 (1993).
    [CrossRef]
  18. S. He, P. Fuks, G. W. Larson, “Optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
    [CrossRef]
  19. W. H. Yu, R. Mittra, “A nonlinear optimization technique for reconstructing dielectric scatterers with possible high contrasts,” Microwave Opt. Technol. Lett. 14, 268–271 (1997).
    [CrossRef]
  20. T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of electrical property distributions by a forward–backward time-stepping method,” J. Electromagn. Waves Appl. 14, 1609–1626 (2000).
    [CrossRef]
  21. M. Gustafsson, S. He, “An optimization approach to two-dimensional time domain electromagnetic inverse problems,” Radio Sci. 35, 525–536 (2000).
    [CrossRef]
  22. N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
    [CrossRef]
  23. S. Caorsi, G. L. Gragnani, M. Pastorino, “Redundant electromagnetic data for microwave imaging of three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
    [CrossRef]
  24. J.-H. Lin, W. C. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997).
    [CrossRef]
  25. S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
    [CrossRef] [PubMed]
  26. A. Abubakar, P. M. van den Berg, B. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).
  27. H. Harada, M. Tanaka, T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave Opt. Technol. Lett. 29, 332–336 (2001).
    [CrossRef]
  28. V. Hutson, J. S. Pym, Applications of Functional Analysis and Operator Theory (Academic, New York, 1980).
  29. T. Tanaka, N. Kuroki, T. Takenaka, “Filtered forward–backward time-stepping method applied to reconstruction of dielectric cylinders,” J. Electromagn. Waves Appl. 17, 253–270 (2003).
    [CrossRef]
  30. T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of an anisotropic cylindrical object by a forward–backward time-stepping method,” IEICE Trans. Electron. E84-C, 1910–1916 (2001).

2003 (1)

T. Tanaka, N. Kuroki, T. Takenaka, “Filtered forward–backward time-stepping method applied to reconstruction of dielectric cylinders,” J. Electromagn. Waves Appl. 17, 253–270 (2003).
[CrossRef]

2002 (1)

S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
[CrossRef]

2001 (3)

T. Isernia, V. Pascazio, R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (2001).
[CrossRef]

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of an anisotropic cylindrical object by a forward–backward time-stepping method,” IEICE Trans. Electron. E84-C, 1910–1916 (2001).

H. Harada, M. Tanaka, T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave Opt. Technol. Lett. 29, 332–336 (2001).
[CrossRef]

2000 (5)

A. Abubakar, P. M. van den Berg, B. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of electrical property distributions by a forward–backward time-stepping method,” J. Electromagn. Waves Appl. 14, 1609–1626 (2000).
[CrossRef]

M. Gustafsson, S. He, “An optimization approach to two-dimensional time domain electromagnetic inverse problems,” Radio Sci. 35, 525–536 (2000).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

M. Pastorino, A. Massa, S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas. 49, 573–578 (2000).
[CrossRef]

1999 (1)

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

1997 (6)

J.-H. Lin, W. C. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997).
[CrossRef]

W. H. Yu, R. Mittra, “A nonlinear optimization technique for reconstructing dielectric scatterers with possible high contrasts,” Microwave Opt. Technol. Lett. 14, 268–271 (1997).
[CrossRef]

S. Y. Yang, H. K. Choi, J. W. Ra, “Reconstruction of a large and high-contrast penetrable object by using the genetic and Levenberg–Marquardt algorithm,” Microwave Opt. Technol. Lett. 16, 17–21 (1997).
[CrossRef]

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

P. M. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

1996 (2)

C. S. Park, S. K. Park, J. W. Ra, “Moment method and iterative reconstruction of two-dimensional complex permittivity by using effective modes with multipole sources in the presence of noise,” Radio Sci. 31, 1877–1886 (1996).
[CrossRef]

S. He, P. Fuks, G. W. Larson, “Optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

1995 (1)

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

1994 (2)

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

S. Caorsi, G. L. Gragnani, M. Pastorino, “Redundant electromagnetic data for microwave imaging of three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

1993 (3)

S. Caorsi, G. L. Gragnami, M. Pastorino, “Reconstruction of dielectric permittivity distributions in 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imaging 12, 232–239 (1993).
[CrossRef]

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

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

1991 (1)

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

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[PubMed]

1975 (1)

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximations,” Jpn. J. Appl. Phys. 14, 1921–1927 (1975).
[CrossRef]

Abubakar, A.

A. Abubakar, P. M. van den Berg, B. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

Baranov, V. Y.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Borisov, V. Y.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Bucci, O. M.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

Bulyshev, A. E.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Caorsi, S.

S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
[CrossRef]

M. Pastorino, A. Massa, S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas. 49, 573–578 (2000).
[CrossRef]

S. Caorsi, G. L. Gragnani, M. Pastorino, “Redundant electromagnetic data for microwave imaging of three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

S. Caorsi, G. L. Gragnami, M. Pastorino, “Reconstruction of dielectric permittivity distributions in 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imaging 12, 232–239 (1993).
[CrossRef]

Chew, W. C.

J.-H. Lin, W. C. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997).
[CrossRef]

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

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

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

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

Choi, H. K.

S. Y. Yang, H. K. Choi, J. W. Ra, “Reconstruction of a large and high-contrast penetrable object by using the genetic and Levenberg–Marquardt algorithm,” Microwave Opt. Technol. Lett. 16, 17–21 (1997).
[CrossRef]

Crocco, L.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[PubMed]

Donelli, M.

S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
[CrossRef]

Franceschini, D.

S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
[CrossRef]

Franchois, A.

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

Fuks, P.

S. He, P. Fuks, G. W. Larson, “Optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

Gragnami, G. L.

S. Caorsi, G. L. Gragnami, M. Pastorino, “Reconstruction of dielectric permittivity distributions in 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imaging 12, 232–239 (1993).
[CrossRef]

Gragnani, G. L.

S. Caorsi, G. L. Gragnani, M. Pastorino, “Redundant electromagnetic data for microwave imaging of three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

Gustafsson, M.

M. Gustafsson, S. He, “An optimization approach to two-dimensional time domain electromagnetic inverse problems,” Radio Sci. 35, 525–536 (2000).
[CrossRef]

Harada, H.

H. Harada, M. Tanaka, T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave Opt. Technol. Lett. 29, 332–336 (2001).
[CrossRef]

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

He, S.

M. Gustafsson, S. He, “An optimization approach to two-dimensional time domain electromagnetic inverse problems,” Radio Sci. 35, 525–536 (2000).
[CrossRef]

S. He, P. Fuks, G. W. Larson, “Optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

Hugonin, J.

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

Hutson, V.

V. Hutson, J. S. Pym, Applications of Functional Analysis and Operator Theory (Academic, New York, 1980).

Isernia, T.

T. Isernia, V. Pascazio, R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (2001).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Iwata, K.

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximations,” Jpn. J. Appl. Phys. 14, 1921–1927 (1975).
[CrossRef]

Jia, H.

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of an anisotropic cylindrical object by a forward–backward time-stepping method,” IEICE Trans. Electron. E84-C, 1910–1916 (2001).

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of electrical property distributions by a forward–backward time-stepping method,” J. Electromagn. Waves Appl. 14, 1609–1626 (2000).
[CrossRef]

Joachimowicz, N.

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

Kleinman, R. E.

P. M. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

Kooij, B.

A. Abubakar, P. M. van den Berg, B. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

Kuroki, N.

T. Tanaka, N. Kuroki, T. Takenaka, “Filtered forward–backward time-stepping method applied to reconstruction of dielectric cylinders,” J. Electromagn. Waves Appl. 17, 253–270 (2003).
[CrossRef]

Larson, G. W.

S. He, P. Fuks, G. W. Larson, “Optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

Lin, J.-H.

Massa, A.

S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
[CrossRef]

M. Pastorino, A. Massa, S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas. 49, 573–578 (2000).
[CrossRef]

Mittra, R.

W. H. Yu, R. Mittra, “A nonlinear optimization technique for reconstructing dielectric scatterers with possible high contrasts,” Microwave Opt. Technol. Lett. 14, 268–271 (1997).
[CrossRef]

Moghaddam, M.

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

Nagata, R.

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximations,” Jpn. J. Appl. Phys. 14, 1921–1927 (1975).
[CrossRef]

Nazarov, A. G.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Otto, G. P.

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

Park, C. S.

C. S. Park, S. K. Park, J. W. Ra, “Moment method and iterative reconstruction of two-dimensional complex permittivity by using effective modes with multipole sources in the presence of noise,” Radio Sci. 31, 1877–1886 (1996).
[CrossRef]

Park, S. K.

C. S. Park, S. K. Park, J. W. Ra, “Moment method and iterative reconstruction of two-dimensional complex permittivity by using effective modes with multipole sources in the presence of noise,” Radio Sci. 31, 1877–1886 (1996).
[CrossRef]

Pascazio, V.

T. Isernia, V. Pascazio, R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (2001).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Pastorino, M.

M. Pastorino, A. Massa, S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas. 49, 573–578 (2000).
[CrossRef]

S. Caorsi, G. L. Gragnani, M. Pastorino, “Redundant electromagnetic data for microwave imaging of three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

S. Caorsi, G. L. Gragnami, M. Pastorino, “Reconstruction of dielectric permittivity distributions in 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imaging 12, 232–239 (1993).
[CrossRef]

Pavlovsky, A. V.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Pichot, C.

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

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

Pierri, R.

T. Isernia, V. Pascazio, R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (2001).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Posukh, V. G.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Pym, J. S.

V. Hutson, J. S. Pym, Applications of Functional Analysis and Operator Theory (Academic, New York, 1980).

Ra, J. W.

S. Y. Yang, H. K. Choi, J. W. Ra, “Reconstruction of a large and high-contrast penetrable object by using the genetic and Levenberg–Marquardt algorithm,” Microwave Opt. Technol. Lett. 16, 17–21 (1997).
[CrossRef]

C. S. Park, S. K. Park, J. W. Ra, “Moment method and iterative reconstruction of two-dimensional complex permittivity by using effective modes with multipole sources in the presence of noise,” Radio Sci. 31, 1877–1886 (1996).
[CrossRef]

Semenov, S. Y.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Simonova, G. I.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Sizov, Y. E.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Souvorov, A. E.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Starostin, A. N.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Svenson, R. H.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Takenaka, T.

T. Tanaka, N. Kuroki, T. Takenaka, “Filtered forward–backward time-stepping method applied to reconstruction of dielectric cylinders,” J. Electromagn. Waves Appl. 17, 253–270 (2003).
[CrossRef]

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of an anisotropic cylindrical object by a forward–backward time-stepping method,” IEICE Trans. Electron. E84-C, 1910–1916 (2001).

H. Harada, M. Tanaka, T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave Opt. Technol. Lett. 29, 332–336 (2001).
[CrossRef]

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of electrical property distributions by a forward–backward time-stepping method,” J. Electromagn. Waves Appl. 14, 1609–1626 (2000).
[CrossRef]

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Tanaka, M.

H. Harada, M. Tanaka, T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave Opt. Technol. Lett. 29, 332–336 (2001).
[CrossRef]

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Tanaka, T.

T. Tanaka, N. Kuroki, T. Takenaka, “Filtered forward–backward time-stepping method applied to reconstruction of dielectric cylinders,” J. Electromagn. Waves Appl. 17, 253–270 (2003).
[CrossRef]

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of an anisotropic cylindrical object by a forward–backward time-stepping method,” IEICE Trans. Electron. E84-C, 1910–1916 (2001).

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of electrical property distributions by a forward–backward time-stepping method,” J. Electromagn. Waves Appl. 14, 1609–1626 (2000).
[CrossRef]

Tatsis, G. P.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

van den Berg, P. M.

A. Abubakar, P. M. van den Berg, B. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

P. M. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

Voinov, B. A.

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

Wall, D. J. N.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Wang, Y. M.

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

Weedon, W. H.

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

Yang, S. Y.

S. Y. Yang, H. K. Choi, J. W. Ra, “Reconstruction of a large and high-contrast penetrable object by using the genetic and Levenberg–Marquardt algorithm,” Microwave Opt. Technol. Lett. 16, 17–21 (1997).
[CrossRef]

Yu, W. H.

W. H. Yu, R. Mittra, “A nonlinear optimization technique for reconstructing dielectric scatterers with possible high contrasts,” Microwave Opt. Technol. Lett. 14, 268–271 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (6)

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 45, 203–214 (1997).
[CrossRef]

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

S. He, P. Fuks, G. W. Larson, “Optimization approach to time-domain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag. 44, 1277–1282 (1996).
[CrossRef]

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

S. Caorsi, G. L. Gragnani, M. Pastorino, “Redundant electromagnetic data for microwave imaging of three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, V. Y. Baranov, “Three-dimensional microwave tomography: experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng. 46, 937–946 (1999).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sens. (3)

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (2001).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

M. Pastorino, A. Massa, S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas. 49, 573–578 (2000).
[CrossRef]

IEEE Trans. Med. Imaging (2)

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

S. Caorsi, G. L. Gragnami, M. Pastorino, “Reconstruction of dielectric permittivity distributions in 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imaging 12, 232–239 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

IEICE Trans. Electron. (2)

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of an anisotropic cylindrical object by a forward–backward time-stepping method,” IEICE Trans. Electron. E84-C, 1910–1916 (2001).

A. Abubakar, P. M. van den Berg, B. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

Inverse Probl. (2)

P. M. van den Berg, R. E. Kleinman, “A contrast source inversion method,” Inverse Probl. 13, 1607–1620 (1997).
[CrossRef]

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

J. Electromagn. Waves Appl. (2)

T. Takenaka, H. Jia, T. Tanaka, “Microwave imaging of electrical property distributions by a forward–backward time-stepping method,” J. Electromagn. Waves Appl. 14, 1609–1626 (2000).
[CrossRef]

T. Tanaka, N. Kuroki, T. Takenaka, “Filtered forward–backward time-stepping method applied to reconstruction of dielectric cylinders,” J. Electromagn. Waves Appl. 17, 253–270 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximations,” Jpn. J. Appl. Phys. 14, 1921–1927 (1975).
[CrossRef]

Microwave Opt. Technol. Lett. (4)

W. H. Yu, R. Mittra, “A nonlinear optimization technique for reconstructing dielectric scatterers with possible high contrasts,” Microwave Opt. Technol. Lett. 14, 268–271 (1997).
[CrossRef]

S. Caorsi, M. Donelli, D. Franceschini, A. Massa, “An iterative multiresolution approach for microwave imaging applications,” Microwave Opt. Technol. Lett. 32, 352–356 (2002).
[CrossRef]

S. Y. Yang, H. K. Choi, J. W. Ra, “Reconstruction of a large and high-contrast penetrable object by using the genetic and Levenberg–Marquardt algorithm,” Microwave Opt. Technol. Lett. 16, 17–21 (1997).
[CrossRef]

H. Harada, M. Tanaka, T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave Opt. Technol. Lett. 29, 332–336 (2001).
[CrossRef]

Radio Sci. (2)

M. Gustafsson, S. He, “An optimization approach to two-dimensional time domain electromagnetic inverse problems,” Radio Sci. 35, 525–536 (2000).
[CrossRef]

C. S. Park, S. K. Park, J. W. Ra, “Moment method and iterative reconstruction of two-dimensional complex permittivity by using effective modes with multipole sources in the presence of noise,” Radio Sci. 31, 1877–1886 (1996).
[CrossRef]

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[PubMed]

Other (1)

V. Hutson, J. S. Pym, Applications of Functional Analysis and Operator Theory (Academic, New York, 1980).

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Figures (6)

Fig. 1
Fig. 1

Three-dimensional object surrounded by a homogeneous medium under illumination of a short pulsed wave generated by an electric dipole.

Fig. 2
Fig. 2

Geometrical configuration of a transmitter and receivers. The arrow located at the center of a surface of a measurement cube represents an electric dipole. All six components of electromagnetic fields are measured at receiver points denoted by the circles.

Fig. 3
Fig. 3

Relative dielectric profiles for slices in the z axis of a 0.89λ reconstruction cube: (a) true object (a homogeneous dielectric cube of side 0.63λ), (b) reconstructed object at the 300th iteration.

Fig. 4
Fig. 4

Relative dielectric profiles for slices in the z axis of a 0.89λ reconstruction cube: (a) true object (an inhomogeneous dielectric cube of side 0.63λ), (b) reconstructed object at the 300th iteration.

Fig. 5
Fig. 5

Reconstructed relative dielectric profiles for slices in the z axis of a 0.89λ cubic object from noisy data with SNR=0 dB at the 300th iteration.

Fig. 6
Fig. 6

Relative dielectric profiles for slices in the z axis of a 1.33λ reconstruction cube: (a) true object (a homogeneous dielectric cube of side 0.95λ), (b) reconstructed object at the 300th iteration with application of low-pass filters.

Equations (39)

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

jm=J(t)δ(r-rmt)d,
Lvm=jm,
vm=(ExmEymEzmηHxmηHymηHzm)t,
jm=(ηjxmηjymηjzm000)t.
LA¯x+B¯y+C¯z-F¯(ct)-G¯,
A¯=00000000000-10000100000000010000-10000, B¯=000001000000000-10000-1000000000100000, C¯=0000-10000100000000010000-10000000000, F¯=εrI¯00μrI¯,G¯=ησI¯000
vm(r, 0)=0.
F(p)=0cTm=1Mn=1NKmn(t)|vm(p; rnr, t)-v˜m(rnr, t)|2d(ct),
F(p)δp=20cTm=1Mn=1Num(p; rnr, t)t×δvm(p; rnr, t)d(ct),
um(p; rnr, t)=Kmn(t)[vm(p; rnr, t)-v˜m(rnr, t)]
L(δvm)=δF¯(ct)+δG¯vm
F(p)δp=m=1Mn=1N2um(p; r, t)×δ(r-rnr)t,δvm(p; r, t),
(f, g)0cTf(r, t)tg(r, t)dvd(ct),
(L*a, b)=(a, Lb),
L*-x A¯t-y B¯t-z C¯t+(ct) F¯t-G¯t=-A¯x-B¯y-C¯z+F¯(ct)-G¯.
L*wm=n=1N2um(p; r, t)δ(r-rnr),
wm(p; r, T)=0.
F(p)δp=gε, δεr+gμ, δμr+gσ, δ(ησ),
a(r), b(r)Va(r)b(r)dv.
gε(r)=20cTm=1Mi=13wmi(p; r, t) vmi(p; r, t)(ct)d(ct),
gμ(r)=20cTm=1Mi=46wmi(p; r, t) vmi(p; r, t)(ct)d(ct),
gσ(r)=20cTm=1Mi=13wmi(p; r, t)vmi(p; r, t)d(ct),
J(t)=d3dt3exp[-α2(t-τ)2],
SNR=10 log10m=1Mn=1N0cT|v˜m(rnr, t)|2d(ct)m=1Mn=1N0cT|nmn(t)|2d(ct).
0cTV(L*wm)tδvmd(ct)dv=0cTVwmt(Lδvm)d(ct)dv.
0cTV(L*wm)tδvmd(ct)dv=0cTVwmtA¯x+B¯y+C¯z-F¯(ct)-G¯×δvmd(ct)dv.
(wmtA¯) δvmx=(wmtA¯δvm)x-(wmtA¯)x δvm,
VwmtA¯δvmxdv=zyx(wmtA¯vm)x-(wmtA¯)x δvmdxdydz.
VwmtA¯δvmxdv=zywmtA¯δvm|x=-x=+dydz-V(wmtA¯)x δvmdv.
lim|r|δvm(p; r, t)=0,for 0tT,
VwmtA¯δvmxdv=-V(wmtA¯)x δvmdv.
VwmtB¯δvmydv=-V(wmt)B¯y δvmdv,
VwmtC¯δvmzdv=-V(wmtC¯)z δvmdv.
0cTwmtF¯δvm(ct)d(ct)=0cT(wmtF¯δvm)(ct)d(ct)-0cT(wmtF¯)(ct) δvmd(ct).
0cTwmtF¯δvm(ct)d(ct)=wmtF¯δvm|t=0t=T-0cT(wmtF¯)(ct) δvmd(ct).
0cTwmtF¯δvm(ct)d(ct)=-0cT(wmtF¯)(ct) δvmd(ct).
0cTV(L*wm)tδvmd(ct)dv=0cTV-wmtx A¯-wmty B¯-wmtz C¯+wmt(ct)×F¯-wmtG¯δvmd(ct)dv;
(L*wm)t=-wmtx A¯-wmty B¯-wmtz C¯+wmt(ct)F¯-wmtG¯,
L*=-A¯x-B¯y-C¯z+F¯(ct)-G¯.

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