Abstract

An ultrasonics diffraction tomography method is discussed that images cylindrical inhomogeneous fluid targets located in a fluid environment. This method is based on an exact model of the interaction between a compressional plane wave and the target (multiple scattering is not neglected). First the theoretical background is summarized. Then numerical simulations illustrate the behavior of the imaging procedure. These simulations are conducted under conditions close to those of experiments that we perform concurrently. Emphasis is put on the retrieval of large but weakly refractive two-layer circular shells, into which can be inserted a high-speed core. The influence of the conditions of the image formation and the discrepancy between images derived from exact scattered pressures and Born’s approximated ones are especially investigated. Finally, procedures of reconstruction of the target’s sound velocity and attenuation are presented and discussed using synthetically generated data.

© 1985 Optical Society of America

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  1. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).
    [PubMed]
  2. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
    [CrossRef]
  3. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Electron. GE-22, 3–13 (1984).
    [CrossRef]
  4. J. E. Greenleaf, “Computerized tomography with ultrasound,” Proc. IEEE 71, 330–337 (1983).
    [CrossRef]
  5. M. Kaveh, M. Soumekh, J. E. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
    [CrossRef]
  6. S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
    [CrossRef]
  7. M. Azimi, A. C. Kak, “Distortion in diffraction tomography caused by multiple scattering,” IEEE Trans. Med. Imaging MI-2, 176–195 (1983).
    [CrossRef]
  8. D. Nahamoo, S. X. Pan, A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. SU-31, 218–229 (1984).
    [CrossRef]
  9. C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. SU-31, 195–217 (1984).
    [CrossRef]
  10. M. F. Adams, A. P. Anderson, “Synthetic aperture tomographic (SAT) imaging for microwave diagnostics,” Proc. Inst. Electr. Eng. Part H 129, 83–88 (1982).
  11. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
    [CrossRef]
  12. M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).
  13. J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
    [CrossRef]
  14. Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
    [CrossRef]
  15. B. Duchêne, W. Tabbara, “Tomographic ultrasonore par diffraction,” Rev. Phys. Appl. 20, 299–304 (1985).
    [CrossRef]
  16. B. Duchêne, D. Lesselier, W. Tabbara, “Contribution to quantitative ultrasound diffraction tomography,” in Proceedings of the IEEE Symposium on Ultrasonics (Institute of Electrical and Electronics Engineers, New York, 1984).
  17. S. A. Johnson et al., “Inverse scattering solutions by a sinc basis, moment method,” —Parts I, II, Ultrasonic Imaging 5, 361–375, 376–392 (1983);Part III, Ultrasonic Imaging 6, 103–116 (1984).
    [CrossRef]
  18. P. C. Sabatier, “Theoretical considerations for inverse scattering,” Radio Sci. 18, 1–18 (1983).
    [CrossRef]
  19. D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964).
  20. B. Duchêne, W. Tabbara, “A geometrical optics method for assessing an inverse scattering problem for blood vessels—Part I: a multistatic single frequency approach,” IEEE Trans. Sonics Ultrason. SU-30, 13–19 (1983).
    [CrossRef]
  21. P. M. van den Berg, “Iterative computational techniques in scattering based upon the integrated square error criterion,” IEEE Trans. Antennas Propag. AP-32, 1063–1071 (1984).
    [CrossRef]

1985

Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
[CrossRef]

B. Duchêne, W. Tabbara, “Tomographic ultrasonore par diffraction,” Rev. Phys. Appl. 20, 299–304 (1985).
[CrossRef]

1984

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[CrossRef]

P. M. van den Berg, “Iterative computational techniques in scattering based upon the integrated square error criterion,” IEEE Trans. Antennas Propag. AP-32, 1063–1071 (1984).
[CrossRef]

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Electron. GE-22, 3–13 (1984).
[CrossRef]

M. Kaveh, M. Soumekh, J. E. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
[CrossRef]

D. Nahamoo, S. X. Pan, A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. SU-31, 218–229 (1984).
[CrossRef]

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. SU-31, 195–217 (1984).
[CrossRef]

1983

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

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

J. E. Greenleaf, “Computerized tomography with ultrasound,” Proc. IEEE 71, 330–337 (1983).
[CrossRef]

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

S. A. Johnson et al., “Inverse scattering solutions by a sinc basis, moment method,” —Parts I, II, Ultrasonic Imaging 5, 361–375, 376–392 (1983);Part III, Ultrasonic Imaging 6, 103–116 (1984).
[CrossRef]

P. C. Sabatier, “Theoretical considerations for inverse scattering,” Radio Sci. 18, 1–18 (1983).
[CrossRef]

B. Duchêne, W. Tabbara, “A geometrical optics method for assessing an inverse scattering problem for blood vessels—Part I: a multistatic single frequency approach,” IEEE Trans. Sonics Ultrason. SU-30, 13–19 (1983).
[CrossRef]

1982

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

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

M. F. Adams, A. P. Anderson, “Synthetic aperture tomographic (SAT) imaging for microwave diagnostics,” Proc. Inst. Electr. Eng. Part H 129, 83–88 (1982).

Adams, M. F.

M. F. Adams, A. P. Anderson, “Synthetic aperture tomographic (SAT) imaging for microwave diagnostics,” Proc. Inst. Electr. Eng. Part H 129, 83–88 (1982).

Anderson, A. P.

M. F. Adams, A. P. Anderson, “Synthetic aperture tomographic (SAT) imaging for microwave diagnostics,” Proc. Inst. Electr. Eng. Part H 129, 83–88 (1982).

Azimi, M.

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

Baribaud, M.

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

Bolomey, J. Ch.

Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
[CrossRef]

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Devaney, A. J.

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Electron. GE-22, 3–13 (1984).
[CrossRef]

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

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

Dubois, F.

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

Duchêne, B.

B. Duchêne, W. Tabbara, “Tomographic ultrasonore par diffraction,” Rev. Phys. Appl. 20, 299–304 (1985).
[CrossRef]

B. Duchêne, W. Tabbara, “A geometrical optics method for assessing an inverse scattering problem for blood vessels—Part I: a multistatic single frequency approach,” IEEE Trans. Sonics Ultrason. SU-30, 13–19 (1983).
[CrossRef]

B. Duchêne, D. Lesselier, W. Tabbara, “Contribution to quantitative ultrasound diffraction tomography,” in Proceedings of the IEEE Symposium on Ultrasonics (Institute of Electrical and Electronics Engineers, New York, 1984).

Floyrac, R.

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

Greenleaf, J. E.

M. Kaveh, M. Soumekh, J. E. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
[CrossRef]

J. E. Greenleaf, “Computerized tomography with ultrasound,” Proc. IEEE 71, 330–337 (1983).
[CrossRef]

Izadnegahdar, A.

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Jofre, L.

Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
[CrossRef]

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Johnson, S. A.

S. A. Johnson et al., “Inverse scattering solutions by a sinc basis, moment method,” —Parts I, II, Ultrasonic Imaging 5, 361–375, 376–392 (1983);Part III, Ultrasonic Imaging 6, 103–116 (1984).
[CrossRef]

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964).

Kak, A. C.

D. Nahamoo, S. X. Pan, A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. SU-31, 218–229 (1984).
[CrossRef]

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[CrossRef]

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

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

Kaveh, M.

M. Kaveh, M. Soumekh, J. E. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
[CrossRef]

Kom, M.

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[CrossRef]

Lee, H.

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. SU-31, 195–217 (1984).
[CrossRef]

Lesselier, D.

B. Duchêne, D. Lesselier, W. Tabbara, “Contribution to quantitative ultrasound diffraction tomography,” in Proceedings of the IEEE Symposium on Ultrasonics (Institute of Electrical and Electronics Engineers, New York, 1984).

Nahamoo, D.

D. Nahamoo, S. X. Pan, A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. SU-31, 218–229 (1984).
[CrossRef]

Pan, S. X.

D. Nahamoo, S. X. Pan, A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. SU-31, 218–229 (1984).
[CrossRef]

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

Peronnet, G.

Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
[CrossRef]

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Pichot, Ch.

Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
[CrossRef]

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Sabatier, P. C.

P. C. Sabatier, “Theoretical considerations for inverse scattering,” Radio Sci. 18, 1–18 (1983).
[CrossRef]

Schueler, C. F.

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. SU-31, 195–217 (1984).
[CrossRef]

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[CrossRef]

Solaimani, M.

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Soumekh, M.

M. Kaveh, M. Soumekh, J. E. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
[CrossRef]

Tabbara, W.

B. Duchêne, W. Tabbara, “Tomographic ultrasonore par diffraction,” Rev. Phys. Appl. 20, 299–304 (1985).
[CrossRef]

B. Duchêne, W. Tabbara, “A geometrical optics method for assessing an inverse scattering problem for blood vessels—Part I: a multistatic single frequency approach,” IEEE Trans. Sonics Ultrason. SU-30, 13–19 (1983).
[CrossRef]

B. Duchêne, D. Lesselier, W. Tabbara, “Contribution to quantitative ultrasound diffraction tomography,” in Proceedings of the IEEE Symposium on Ultrasonics (Institute of Electrical and Electronics Engineers, New York, 1984).

van den Berg, P. M.

P. M. van den Berg, “Iterative computational techniques in scattering based upon the integrated square error criterion,” IEEE Trans. Antennas Propag. AP-32, 1063–1071 (1984).
[CrossRef]

Wade, G.

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. SU-31, 195–217 (1984).
[CrossRef]

Wang, S.

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

IEEE Trans. Acoust. Speech Signal Process.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1262–1275 (1983).
[CrossRef]

IEEE Trans. Antennas Propag.

Ch. Pichot, L. Jofre, G. Peronnet, J. Ch. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag. AP-33, 416–425 (1985).
[CrossRef]

P. M. van den Berg, “Iterative computational techniques in scattering based upon the integrated square error criterion,” IEEE Trans. Antennas Propag. AP-32, 1063–1071 (1984).
[CrossRef]

IEEE Trans. Biomed. Eng.

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

IEEE Trans. Geosci. Electron.

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Electron. GE-22, 3–13 (1984).
[CrossRef]

IEEE Trans. Med. Imaging

M. Azimi, A. 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.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[CrossRef]

J. Ch. Bolomey, A. Izadnegahdar, L. Jofre, Ch. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

IEEE Trans. Sonics Ultrason.

D. Nahamoo, S. X. Pan, A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason. SU-31, 218–229 (1984).
[CrossRef]

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. SU-31, 195–217 (1984).
[CrossRef]

M. Kaveh, M. Soumekh, J. E. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
[CrossRef]

B. Duchêne, W. Tabbara, “A geometrical optics method for assessing an inverse scattering problem for blood vessels—Part I: a multistatic single frequency approach,” IEEE Trans. Sonics Ultrason. SU-30, 13–19 (1983).
[CrossRef]

Proc. IEEE

J. E. Greenleaf, “Computerized tomography with ultrasound,” Proc. IEEE 71, 330–337 (1983).
[CrossRef]

Proc. Inst. Electr. Eng. Part H

M. F. Adams, A. P. Anderson, “Synthetic aperture tomographic (SAT) imaging for microwave diagnostics,” Proc. Inst. Electr. Eng. Part H 129, 83–88 (1982).

M. Baribaud, F. Dubois, R. Floyrac, M. Kom, S. Wang, “Tomographic image reconstruction of biological objects from coherent microwave diffraction data,” Proc. Inst. Electr. Eng. Part H 129, 356–359 (1982).

Radio Sci.

P. C. Sabatier, “Theoretical considerations for inverse scattering,” Radio Sci. 18, 1–18 (1983).
[CrossRef]

Rev. Phys. Appl.

B. Duchêne, W. Tabbara, “Tomographic ultrasonore par diffraction,” Rev. Phys. Appl. 20, 299–304 (1985).
[CrossRef]

Ultrasonic Imaging

S. A. Johnson et al., “Inverse scattering solutions by a sinc basis, moment method,” —Parts I, II, Ultrasonic Imaging 5, 361–375, 376–392 (1983);Part III, Ultrasonic Imaging 6, 103–116 (1984).
[CrossRef]

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

Other

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964).

B. Duchêne, D. Lesselier, W. Tabbara, “Contribution to quantitative ultrasound diffraction tomography,” in Proceedings of the IEEE Symposium on Ultrasonics (Institute of Electrical and Electronics Engineers, New York, 1984).

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Figures (12)

Fig. 1
Fig. 1

Geometry of the problem in the (x, y) plane (A) and in the spectral plane (B).

Fig. 2
Fig. 2

Experimental images of a rubber pipe built from eight equally spaced views using the single transducer setup: nearest-neighbor (A) and bilinear interpolation (B).

Fig. 3
Fig. 3

Amplitudes of the exact and the Born-approximated scattered pressures (ppi)/pi on the probing line for the low-contrast shell (a = 1.5 mm, ca = 1540 m/sec; b = 2.5 mm, cb = 1560 m/sec): —, exact;— —, approximate pressure; the high-contrast shell (ca = 1470 m/sec; same a, b, and cb as above): —, · — ·, exact;– – –approximate pressure, the latter with a high-speed core included (a′ = 0.3 mm, ca′ = 2600 m/sec):× ×, exact; ● ●, approximate pressure. Wavelength λ0 = 0.735 mm in the outer medium (co = 1470 m/sec). Distance line ; target center d = 11 mm.

Fig. 4
Fig. 4

Same as Fig. 3, for the phases of the normalized scattered pressure on the probing line.

Fig. 5
Fig. 5

Image of the low-contrast shell as a function of the number N of views using the nearest-neighbor interpolation.

Fig. 6
Fig. 6

Same as Fig. 5 for the high-contrast shell.

Fig. 7
Fig. 7

Same as Fig. 5 for the high-speed core shell.

Fig. 8
Fig. 8

Images of the low-contrast shell (A), the high-contrast shell (B), and the high-speed core shell (C) using the nearest-neighbor interpolation with N = 80 views (top) and the bilinear one (bottom).

Fig. 9
Fig. 9

Images of the low-contrast shell (A), the high-contrast shell (B), and the high-speed core shell (C) built from exact scattered fields (top) and Born-approximated ones (bottom) using the nearest-neighbor interpolation with N = 64 views.

Fig. 10
Fig. 10

Speed profiles retrieved from the sources induced by a single illumination within two-layer square targets (half-sides b and a = 36/5, speeds cb, = 1560 and ca = 1540 m/sec, attenuation α = 60 dB/m) embedded in a homogeneous lossless space (c0 = 1470 m/sec, λ0 = 0.735 mm). Sources sampled each λ0/7.5 with T = 2 (top) or 1 (bottom) significant digits. First iterations (Born): b0 = 2 (× ×) or 1.33 (● ●). Final profiles b/λ0 = 2 (—) or 1.33 (— — —). The latter do not differ from the exact ones for T = 3 digits.

Fig. 11
Fig. 11

Same as Fig. 10 with a set of 48 illuminations (at equally spaced angles of incidence). The first iterations are meaningless and are not represented for b = 2λ0.

Fig. 12
Fig. 12

Similar to Figs. 10 and 11 for the attenuation profiles with a single illumination (top) or 48 (bottom), T = 3 significant digits. b/λ0 = 2 (—) or 1.33 (— — —). The first iterations are meaningless and are not represented.

Equations (14)

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

p θ ( i ) ( x , y ) = p i ( x , y ) + D [ k 2 ( x , y ) k 0 2 ] p θ ( i ) ( x , y ) G ( x , y , x , y ) d x d y , G ( x , y , x , y ) = j 4 H 0 ( 2 ) { k 0 [ ( x x ) 2 + ( y y ) 2 ] 1 / 2 } ,
ψ θ ( i ) ( l ) = p θ ( i ) ( x , y ) p i ( x , y ) p i ( x , y ) , l = x cos θ ( i ) + y sin θ ( i ) , ( x , y ) ;
φ D θ ( i ) ( x , y ) = [ k 2 ( x , y ) k 0 2 ] p θ ( i ) ( x , y ) p i ( x , y ) , ( x , y ) D .
ψ θ ( i ) ( l ) = 1 p i ( x , y ) D φ D θ ( i ) ( x , y ) p i ( x , y ) × G ( x , y , x , y ) d x d y
G ( x , y , x , y ) = j 4 π + exp [ j ( k 0 2 α 2 ) 1 / 2 ] | y y | k 0 2 α 2 × exp [ j α ( x x ) ] d α , Im [ ( k 0 2 α 2 ) 1 / 2 ] 0
ψ ̂ ( α ) θ ( i ) = + ψ θ ( i ) ( l ) e j α l d l ,
φ ̂ D θ ( i ) ( α , β ) = D φ D θ ( i ) ( x , y ) exp [ j ( α x + β y ) ] d x d y ,
φ ̂ D θ ( i ) ( ν , μ ) = j 2 ( k 0 2 α 2 ) 1 / 2 exp [ j γ ( α ) ] ψ ̂ θ ( i ) ( α ) , ν = α cos θ ( i ) γ ( α ) sin θ ( i ) , μ = α sin θ ( i ) + γ ( α ) cos θ ( i ) ,
γ ( α ) = k 0 + ( k 0 2 α 2 ) 1 / 2 ,
φ D ( x , y ) 1 4 π 2 k 0 + k 0 φ ˇ D ( α , β ) exp [ j ( α x + β y ) ] d α d β ,
φ ̂ D θ ( i ) ( 0 , 0 ) = D [ k 2 ( x , y ) k 0 2 ] p θ ( i ) ( x , y ) p i ( x , y ) d x d y ,
ξ ( x , y ) = φ D ( x , y ) p i ( x , y ) [ k 2 ( x , y ) k 0 2 ] p ( x , y ) ,
k n + 1 2 k 0 2 = ξ / p n , p n + 1 = p i + D * ( k n + 1 2 k 0 2 ) p n + 1 G d 2 D ,
p * ( x , y ) = p i ( x , y ) + D * ξ ( x , y ) G ( x , y , x , y ) d x d y , ( x , y ) D * .

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