Abstract

Diffraction tomography reconstructions of objects from limited transmitted field data sets are discussed together with theoretical analyses and results of numerical experiments. It is shown that limited data sets, representing only a small part of the complete data sets, can be used for reconstructions in diffraction tomography with satisfactory accuracy. We also find that, in diffraction tomography based on the hybrid filtered backpropagation and the first-Rytov approximation, the use of limited data sets can provide a larger range of validity than the use of complete data sets, the reason being that limited data sets pose less-severe phase-unwrapping problems.

© 2000 Optical Society of America

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  1. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  2. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [PubMed]
  3. 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, 1991), Vol. 18, pp. 401–411.
    [CrossRef]
  4. N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
    [CrossRef] [PubMed]
  5. L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
    [CrossRef]
  6. T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
    [CrossRef]
  7. T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent, birefringent fibres,” J. Microsc. (Oxford) 177, 53–67 (1995).
    [CrossRef]
  8. M. H. Maleki, A. J. Devaney, “Phase retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
    [CrossRef]
  9. T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
    [CrossRef]
  10. A. J. Devaney, “The limited view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
    [CrossRef]
  11. M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D 19, 301–317 (1986).
    [CrossRef]
  12. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.
  13. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE-22, 3–13 (1984).
    [CrossRef]
  14. H. Weyl, “Ausbrietung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
    [CrossRef]
  15. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sect. 5.2.
  16. L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 91–109.
  17. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Sect. 6.7.3.
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.
  19. B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first-Born and the first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  20. B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
    [CrossRef]

1998 (2)

B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first-Born and the first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
[CrossRef]

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

1995 (3)

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

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

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

1993 (1)

1991 (2)

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
[CrossRef] [PubMed]

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
[CrossRef]

1989 (1)

A. J. Devaney, “The limited view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

1986 (1)

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

1984 (1)

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

1982 (1)

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

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1919 (1)

H. Weyl, “Ausbrietung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Chen, B.

B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first-Born and the first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
[CrossRef]

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

Devaney, A. J.

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Phase retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
[CrossRef]

A. J. Devaney, “The limited view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

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

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

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, 1991), Vol. 18, pp. 401–411.
[CrossRef]

Fiddy, M. A.

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

Gelius, L.-J.

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
[CrossRef] [PubMed]

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
[CrossRef]

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 91–109.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

Johansen, I.

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
[CrossRef]

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
[CrossRef] [PubMed]

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, 1991), Vol. 18, pp. 401–411.
[CrossRef]

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Sect. 6.7.3.

Maleki, M. H.

Pedersen, H. M.

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Sect. 6.7.3.

Sponheim, N.

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
[CrossRef] [PubMed]

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
[CrossRef]

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, 1991), Vol. 18, pp. 401–411.
[CrossRef]

Stamnes, J. J.

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first-Born and the first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
[CrossRef]

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

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

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
[CrossRef]

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
[CrossRef] [PubMed]

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 91–109.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sect. 5.2.

Stamnes, K.

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.

Wedberg, T. C.

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

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

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Wedberg, W. C.

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

Weyl, H.

H. Weyl, “Ausbrietung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Ann. Phys. (Leipzig) (1)

H. Weyl, “Ausbrietung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Geosci. Remote Sens. (1)

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

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

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 370–379 (1991).
[CrossRef] [PubMed]

Inverse Probl. (1)

A. J. Devaney, “The limited view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

J. Acoust. Soc. Am. (1)

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “A generalized diffraction tomography algorithm,” J. Acoust. Soc. Am. 89, 523–528 (1991).
[CrossRef]

J. Microsc. (Oxford) (1)

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

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

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

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Pure Appl. Opt. (2)

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

B. Chen, J. J. Stamnes, A. J. Devaney, H. M. Pedersen, K. Stamnes, “Two-dimensional optical diffraction tomography for objects embedded in a random medium,” Pure Appl. Opt. 7, 1181–1199 (1998).
[CrossRef]

Ultrason. Imaging (1)

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

Other (6)

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, 1991), Vol. 18, pp. 401–411.
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sect. 5.2.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 91–109.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988), Sect. 6.7.3.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

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Figures (7)

Fig. 1
Fig. 1

Classical scan configuration of diffraction tomography.

Fig. 2
Fig. 2

Computed intensity of the total field transmitted through a circular cylinder with a radius of a = 10λ. The wavelength of the incident wave was λ = 0.6328 µm, and the refractive indices of the cylinder and the surrounding medium were 1.005 and 1.000, respectively. The distance between the object and the measurement line was l 0 = 800λ, and the sampling length was l s . (1) Complete data set, l s = 1024λ. (2) Limited data set 1, l s = 128λ. (3) Limited data set 2, l s = 64λ.

Fig. 3
Fig. 3

Reconstructions from Born data corresponding to Fig. 2: (a) based on the complete data set in Fig. 2, (b) based on the limited data set 1 in Fig. 2, (c) based on the limited data set 2 in Fig. 2.

Fig. 4
Fig. 4

Reconstructions from Rytov data corresponding to Fig. 2: (a) based on the complete data set in Fig. 2, (b) based on the limited data set 1 in Fig. 2, (c) based on the limited data set 2 in Fig. 2.

Fig. 5
Fig. 5

Computed intensity of the total field transmitted through a circular cylinder with a radius of a = 5λ. The wavelength of the incident wave was λ = 0.6328 µm, and the refractive indices of the cylinder and the surrounding medium were 1.055 and 1.000, respectively. The distance between the object and the measurement line was l 0 = 500λ, and the sampling length was l s . (1) Complete data set, l s = 1024λ. (2) Limited data set 1, l s = 96λ. (3) Limited data set 2, l s = 48λ.

Fig. 6
Fig. 6

Reconstructions from Rytov data corresponding to Fig. 5: (a) based on the complete data set in Fig. 5, (b) based on the limited data set 1 in Fig. 5, (c) based on the limited data set 2 in Fig. 5.

Fig. 7
Fig. 7

Illustration of the validity of conclusions for ODT with limited data sets in the general case.

Tables (3)

Tables Icon

Table 1 Reconstructions from Born Data Corresponding to Fig. 2

Tables Icon

Table 2 Reconstructions from Rytov Data Corresponding to Fig. 2

Tables Icon

Table 3 Reconstructions from Rytov Data Corresponding to Fig. 5

Equations (23)

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

ui=expikη,
u=ui+usc,
2+k2ur=k2Orur,
Or=1-n2rn02
2+k2uBscr=k2Oruir,
2+k2uirδψr=k2Or-δψ2uir.
2+k2uirδψRr=k2Oruir,
ur=uirexpδψRr.
2+k2Ωp=k2Oruir,
Ωprd= -i k24OruirH01k|rd-r|d2r,
H01k|rd-r|=1π-exp-iksˆ·r-rdkηdkξ,
2ik2 Ωprd; sˆ0=12π-- Orexp-iksˆ-sˆ0·rd2rexpiksˆ·rdkηdkξ.
uir=expiksˆ0·r,
2ik2 Ωpξ, l0; sˆ0=12π-O˜ksˆ-sˆ0expikηl0kη×expikξξdkξ.
D˜pkξ; η=l0; sˆ0=expikηl0kη Õksˆ-sˆ0,
D˜pkξ; η=l0; sˆ0=- Dpξ; η=l0; sˆ0×exp-ikξξdξ,
Dpξ; η=l0; sˆ0=2ik2 Ωpξ, l0; sˆ0,
O˜ksˆ-sˆ0=- Orexp-iksˆ-sˆ0·rd2r.
DpLξ; η=l0; sˆ0=Dpξ; η=l0; sˆ0for |ξ|L,0otherwise.
Gf=gx=- gxexp-i2πfxdx,
gx=-1Gf=12π- Gfexpi2πfxdf,
E%n=1N|r|a|nrer-n|2n-n021/2100%,
δψr=lnuruir=lnuruir+i Arguruir,

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