Abstract

We compare two methods of reconstructing the complex index-of-refraction distribution of a scattering object from optical scattering data obtained in a set of scattering experiments employing incident monochromatic plane waves. The first method generates an approximate reconstruction directly from the far-field intensity, which is measured as a function of scattering angle for each incident plane wave. The second method uses an iterative phase-retrieval algorithm to extract the phase of the scattered field over any given plane from the measurement of the intensity of the total field over that plane and from a priori object-support information. The reconstruction is then performed from the scattered field that is so determined by using the filtered backpropagation algorithm of diffraction tomography. We compare the performance of the two procedures on computer-simulated and experimental scattering data.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.
  2. W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommelous, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Prob. 4, 305–331 (1988).
    [CrossRef]
  3. R. B. Pratt, M. H. Worthington, “The application of diffraction tomography to cross-hole seismic data,” Geophysics 53, 1284–1294 (1988).
    [CrossRef]
  4. M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
    [CrossRef]
  5. E. Wolf, “Determination of the amplitude and the phase of the scattered field by holography,”J. Opt. Soc. Am. 60, 18–20 (1970).
    [CrossRef]
  6. R. Snyder, L. Hesselink, “High speed optical tomography for flow visualization,” Appl. Opt. 24, 4046–4051 (1985).
    [CrossRef] [PubMed]
  7. M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]
  8. A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Prob. 2, 161–183 (1986).
    [CrossRef]
  9. V. T. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  10. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imag. 4, 336–350 (1982).
  11. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386 (1983).
    [CrossRef]
  12. See J. R. Shewell, E. Wolf, “Inverse diffraction and a new reciprocity theorem,”J. Opt. Soc. Am. 58, 1596–1603 (1968); G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,”J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  13. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  14. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  15. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  16. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  17. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  18. R. Rolleston, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
    [CrossRef] [PubMed]
  19. G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
    [CrossRef]
  20. L. Wang, B. Dong, G. Yang, “Phase retrieval from two intensity measurements in an optical system involving nonunitary transformation,” Appl. Opt. 29, 3422–3427 (1990).
    [CrossRef] [PubMed]
  21. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  22. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  23. M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).
    [CrossRef]
  24. L. S. Taylor, “The phase retrieval problem,”IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
    [CrossRef]
  25. R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
    [CrossRef]
  26. R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 Intl. Optical Computing Conf. I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
    [CrossRef]

1992 (1)

1990 (1)

1988 (2)

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

R. B. Pratt, M. H. Worthington, “The application of diffraction tomography to cross-hole seismic data,” Geophysics 53, 1284–1294 (1988).
[CrossRef]

1987 (1)

1986 (3)

R. Rolleston, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
[CrossRef] [PubMed]

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[CrossRef]

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Prob. 2, 161–183 (1986).
[CrossRef]

1985 (1)

1984 (2)

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).
[CrossRef]

1983 (1)

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

1982 (2)

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

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1981 (1)

L. S. Taylor, “The phase retrieval problem,”IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

1978 (1)

1974 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
[CrossRef]

1973 (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1970 (1)

1968 (1)

1963 (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Boucher, R. H.

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 Intl. Optical Computing Conf. I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
[CrossRef]

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
[CrossRef]

Chommelous, L.

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

Devaney, A. J.

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

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Prob. 2, 161–183 (1986).
[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 Imag. 4, 336–350 (1982).

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

Dong, B.

Duchene, B.

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

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

Fiddy, M. A.

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
[CrossRef]

Fienup, J. R.

George, N.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Greenway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
[CrossRef]

Hesselink, L.

Johansen, I.

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

Lesselier, D.

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

Levi, A.

Liu, G.

Maleki, M. H.

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).
[CrossRef]

Pichot, Ch.

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

Pratt, R. B.

R. B. Pratt, M. H. Worthington, “The application of diffraction tomography to cross-hole seismic data,” Geophysics 53, 1284–1294 (1988).
[CrossRef]

Rolleston, R.

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schatzberg, A.

Scott, P. D.

Shewell, J. R.

Snyder, R.

Sponheim, N.

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

Stark, H.

Tabbara, W.

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

Tatarski, V. T.

V. T. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Taylor, L. S.

L. S. Taylor, “The phase retrieval problem,”IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Wang, L.

Wolf, E.

Worthington, M. H.

R. B. Pratt, M. H. Worthington, “The application of diffraction tomography to cross-hole seismic data,” Geophysics 53, 1284–1294 (1988).
[CrossRef]

Yang, G.

Appl. Opt. (4)

Geophysics (1)

R. B. Pratt, M. H. Worthington, “The application of diffraction tomography to cross-hole seismic data,” Geophysics 53, 1284–1294 (1988).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. S. Taylor, “The phase retrieval problem,”IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

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

Inverse Prob. (2)

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

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Prob. 2, 161–183 (1986).
[CrossRef]

J. Math. Phys. (1)

M. Nieto-Vesperinas, “Inverse scattering problems: a study in terms of the zeros of entire functions,”J. Math. Phys. 25, 2109–2115 (1984).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Phys. D (3)

M. A. Fiddy, “Inversion of optical scattered field data,”J. Phys. D 19, 301–317 (1986).
[CrossRef]

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenway, G. Ross, “The application of dispersion relations (Hilbert transforms) to phase retrieval,”J. Phys. D 7, L65–L68 (1974).
[CrossRef]

Opt. Acta (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Ultrasonic Imag. (1)

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

Other (4)

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1990), Vol. 18.

V. T. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 Intl. Optical Computing Conf. I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
[CrossRef]

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 (10)

Fig. 1
Fig. 1

Geometry for DT scattering experiment. The incident wave travels along the η axis producing an angle φ0 with the x axis, and the total scattered field is measured on the measurement line η = l0.

Fig. 2
Fig. 2

Diagram illustrating the interative phase-retrieval algorithm using the measured intensity and the support of the object as constraints.

Fig. 3
Fig. 3

Schematic representation of the test object that we used in the simulation, which consisted of four cylindrical objects whose cross section on the xy plane is shown. Disks A–C have a purely real, and disk D a purely imaginary, index perturbation dn as shown.

Fig. 4
Fig. 4

Reconstruction of the test object for the measurement line at 4 mm with the use of (a) full simulated data, (b) the intensity-only algorithm, and (c) the phase-retrieval method. The top figures show the real part of the index perturbation, and the bottom figures show the imaginary part.

Fig. 5
Fig. 5

Cross-sectional cuts through the reconstructions in Fig. 4 along the line y = 0 for (a) the real part and (b) the imaginary part of the index perturbation. The phase-retrieved reconstruction (circles) lies on top of the full-data reconstruction (solid curve). Dashed curve, intensity-only algorithm.

Fig. 6
Fig. 6

Same as Fig. 4, except the measurement line is at 1 mm. (a) Full simulated data, (b) the intensity-only algorithm, and (c) the phase-retrieval method.

Fig. 7
Fig. 7

Cross-sectional cuts through the reconstructions in Fig. 6 along the line y = 0 for (a) the real part and (b) the imaginary part of the index perturbation. Circles, phase-retrieved reconstruction; solid curve, full-data reconstruction; dashed curves, intensity-only algorithm.

Fig. 8
Fig. 8

Normalized rms (a) amplitude and (b) phase errors of the phase-retrieval algorithm for the first view φ0 = 0, for the measurement line at 1 and 4 mm. Convergence at 1 mm (solid curve) requires more iterations than at 4 mm (dashed curve).

Fig. 9
Fig. 9

Phase of the total field on the measurement line at 7.6 mm that we obtained from the phase-retrieval algorithm with the use of the experimental fiber data.

Fig. 10
Fig. 10

Reconstruction of the radial profile of the fiber’s index perturbation that we obtained from experimental data with the use of the intensity-only algorithm (dashed curve) and the phase-retrieval algorithm (circles). The fiber’s assumed ideal profile (dashed–dotted curve) and reconstruction from simulated data (solid curve) are shown for reference.

Equations (16)

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

U ( r ) = exp { i k [ s 0 · r + δ W ( r ) ] } ,
n ( r ) = 1 + δ n ( r ) ,
δ W ( r ) = 2 i k exp ( - i k s 0 · r ) Σ d 2 r δ n ( r ) G ( r - r ) ,
G ( r - r ) = - i H 0 ( k r - r ) / 4
δ n ^ ( r ) = φ 0 [ - π , π ) B { δ W φ 0 ¯ } ( r ) ,
L ( ξ ; l 0 ) = exp ( i k l 0 ) I ( ξ ; l 0 ) ,
U ( ξ ; η 0 ) B { L ( ξ ; l 0 ) } ,
U ˜ ( κ ; η 0 ) = exp [ - i γ ( l 0 - η 0 ) ] U ˜ ( κ ; l 0 ) ,
γ = { k 2 - κ 2 if κ k i κ 2 - k 2 otherwise .
U 1 ( k ) ( ξ ) = R U 2 ( k ) ( ξ ) .
U 1 ( k ) ( ξ ) = { U 1 ( k ) ( ξ ) i f ξ U 1 inc otherwise ,
U 2 ( k ) ( ξ ) = P U 1 ( k ) ( ξ ) .
U 2 ( k + 1 ) ( ξ ) = I ( ξ ; l 0 ) exp [ i arg U 2 ( k ) ( ξ ) ] ,
U ( ξ ) = exp ( i k η ) + ψ ( ξ )
E k = { ξ [ I ( ξ ; l 0 ) - U 2 ( k ) ( ξ ) ] 2 ξ I ( ξ ; l 0 ) } 1 / 2 .
F k = { ξ [ arg U ( ξ ; l 0 ) - arg U 2 ( k ) ( ξ ) ] 2 ξ [ arg U ( ξ ; l 0 ) ] 2 } 1 / 2 ,

Metrics