Abstract

We compare the filtered backpropagation algorithm with the filtered backprojection algorithm for reconstructing the complex refractive-index distribution of semitransparent, cylindrical objects. Before reconstruction, the recorded scattered light is propagated back to the reconstruction area by inverse diffraction. Our comparison is based on computer-simulated data, and experimental optical data obtained from fibers with step-index, graded-index, and uniform-index distributions. The results show that both the filtered backpropagation algorithm and the filtered backprojection algorithm can produce accurate reconstructions of the complex refractive-index distribution as long as the weak-scattering approximation is valid. The good agreement between the results obtained from these two reconstruction algorithms indicates that the errors introduced by the assumption of straight-line propagation inside the object are negligible compared with those introduced by the weak-scattering approximation.

© 1995 Optical Society of America

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    [CrossRef]
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1995

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 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]

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]

1993

1992

1991

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]

1990

1989

G. W. Faris, H. Hertz, “Tunable differential interferometer for optical tomography,” Appl. Opt. 28, 4662–4667 (1989).
[CrossRef] [PubMed]

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

1988

1987

1985

R. Snyder, L. Hesselink, “High speed optical tomography for flow visualization,” Appl. Opt. 24, 4046–4051 (1985).
[CrossRef] [PubMed]

H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 54, 131–136 (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, 860–874 (1984).
[CrossRef]

1983

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

1982

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

1979

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

1970

Brenner, K.-H.

Devaney, A. J.

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, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, Vol. 18, H. Lee, G. Wade, eds. (Plenum, New York, 1991).

Faris, G. W.

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]

Hertz, H.

Hertz, H. M.

H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[CrossRef]

Hesselink, L.

Johansen, I.

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, Vol. 18, H. Lee, G. Wade, eds. (Plenum, New York, 1991).

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Kaveh, M.

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

Kawata, S.

Kobayashi, T.

Kuroiwa, Y.

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, 860–874 (1984).
[CrossRef]

Maleki, M. H.

Minami, S.

Mueller, R. K.

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

Nakamura, O.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. 5.

Noda, T.

Ooki, H.

Sasaki, O.

Singer, W.

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, 860–874 (1984).
[CrossRef]

Snyder, R.

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]

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, Vol. 18, H. Lee, G. Wade, eds. (Plenum, New York, 1991).

Stamnes, J. J.

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

T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” in Proceedings of the ICO Topical Meeting on Optics (International Commission on Optics, Kyoto, Japan, 1995), Opt. Rev. 2.

T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the quantitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).

T. C. Wedberg, J. J. Stamnes, “Quantitative microscopy of phase objects by optical diffraction tomography,” in Microscopy, Holography, and Interferometry in Biomedicine, A. F. Fercher, A. Lewis, H. Podbielska, H. Schneckenburger, T. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2083, 168–173 (1993).

Tatarski, V. T.

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

Wade, G.

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

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 complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 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]

T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the quantitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).

T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” in Proceedings of the ICO Topical Meeting on Optics (International Commission on Optics, Kyoto, Japan, 1995), Opt. Rev. 2.

T. C. Wedberg, J. J. Stamnes, “Quantitative microscopy of phase objects by optical diffraction tomography,” in Microscopy, Holography, and Interferometry in Biomedicine, A. F. Fercher, A. Lewis, H. Podbielska, H. Schneckenburger, T. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2083, 168–173 (1993).

Wedberg, W. C.

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

Wolf, E.

Appl. Opt.

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. 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, 860–874 (1984).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

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]

J. Microsc.

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[CrossRef]

Phys. Rev. Lett.

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[CrossRef] [PubMed]

Proc. IEEE

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

Pure Appl. Opt.

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

Ultrasonic Imaging

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

Other

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

T. C. Wedberg, J. J. Stamnes, “Analytical and numerical examination of the quantitative imaging properties of optical diffraction tomography,” J. Mod. Opt. (to be published).

T. C. Wedberg, J. J. Stamnes, “Quantitative microscopy of phase objects by optical diffraction tomography,” in Microscopy, Holography, and Interferometry in Biomedicine, A. F. Fercher, A. Lewis, H. Podbielska, H. Schneckenburger, T. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2083, 168–173 (1993).

T. C. Wedberg, J. J. Stamnes, “Quantitative imaging by optical diffraction tomography,” in Proceedings of the ICO Topical Meeting on Optics (International Commission on Optics, Kyoto, Japan, 1995), Opt. Rev. 2.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. 5.

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, Vol. 18, H. Lee, G. Wade, eds. (Plenum, New York, 1991).

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Figures (9)

Fig. 1
Fig. 1

Classical scan configuration.

Fig. 2
Fig. 2

Real part of the reconstructed refractive index obtained with (a) the hybrid FBP algorithm, (b) the hybrid filtered backprojection algorithm. Each reconstruction is based on data for one illumination angle.

Fig. 3
Fig. 3

Real part of the reconstructed refractive index obtained with the hybrid FBP algorithm and the hybrid filtered backprojection algorithm. Each reconstruction was based on 25 illumination angles.

Fig. 4
Fig. 4

(a) Real and (b) imaginary parts of the reconstructed refractive index of an object with a complex refractive index.

Fig. 5
Fig. 5

Real part of the reconstructed refractive index of an object with a graded index.

Fig. 6
Fig. 6

Average error in the real part of the reconstructed refractive index as a function of the refractive index of the object relative to that of the background medium for various object radii r 0.

Fig. 7
Fig. 7

Diametrical sections of the real part of the reconstructed refractive index of a mineral fiber obtained with the hybrid FBP algorithm and the hybrid filtered backprojection algorithm.

Fig. 8
Fig. 8

Intensity distribution recorded for the graded-index fiber for one illumination direction.

Fig. 9
Fig. 9

Diametrical sections of the real part of the reconstructed refractive index of a graded-index fiber.

Equations (11)

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O ( x , y ) = 1 - [ n ( r ) n 0 ] 2 ;             r = x e ^ x + y e ^ y .
U ( r ) = U 0 ( r ) exp [ ϕ ( r ) ] = U 0 ( r ) + U s ( r ) .
O lp ( r ) = 1 2 π - π π Π ϕ 0 ( x sin ϕ 0 - y cos ϕ 0 , x cos ϕ 0 + y sin ϕ 0 ) d ϕ 0 ,
Π ϕ 0 ( ξ , η ) = 1 2 π exp [ i k ( l 0 - η ) ] - k k Γ ˜ ϕ 0 ( κ ) κ × exp [ i γ ( η - l 0 ) ] exp ( i κ ξ ) d κ ,
Γ ˜ ϕ 0 ( κ ) = - Γ ϕ 0 ( ξ ) exp ( - i κ ξ ) d ξ ;             γ = ( k 2 - κ 2 ) 1 / 2 .
Γ ϕ 0 ( ξ ) = { i k ln [ U ( ξ ) U 0 ( ξ ) ] in the Rytov approximation i k U ( ξ ) - U 0 ( ξ ) U 0 ( ξ ) in the Born approximation .
Π ϕ 0 ( ξ , η ) = 1 2 π exp [ i k ( l 0 - η ) ] - k k Γ ˜ ϕ 0 ( κ ) κ exp ( i κ ξ ) d κ .
n ( r ) = { 1.02 , r 5 λ 1.01 , 5 λ < r 25 λ .
n ( r ) = { 0.98 + 0.04 1 + r r 0 r r 0 = 20 λ 1 otherwise .
ϕ ( r ) 2 k 2 O ( r ) .
Δ n re = { 1 N r r 0 [ n - n re ( r ) ] 2 } 1 / 2 ,

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