Abstract

The filtered backpropagation (FBPP) algorithm, originally developed by Devaney [Ultrason. Imaging 4, 336 (1982)], has been widely used for reconstructing images in diffraction tomography. It is generally known that the FBPP algorithm requires scattered data from a full angular range of 2π for exact reconstruction of a generally complex-valued object function. However, we reveal that one needs scattered data only over the angular range 0ϕ3π/2 for exact reconstruction of a generally complex-valued object function. Using this insight, we develop and analyze a family of minimal-scan filtered backpropagation (MS-FBPP) algorithms, which, unlike the FBPP algorithm, use scattered data acquired from view angles over the range 0ϕ3π/2. We show analytically that these MS-FBPP algorithms are mathematically identical to the FBPP algorithm. We also perform computer simulation studies for validation, demonstration, and comparison of these MS-FBPP algorithms. The numerical results in these simulation studies corroborate our theoretical assertions.

© 1999 Optical Society of America

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References

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  1. M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
    [CrossRef]
  2. L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Hilger, London, 1991).
  3. R. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
    [CrossRef]
  4. G. Kino, “Acoustic imaging for nondestructive evaluation,” Proc. IEEE 67, 510–525 (1979).
    [CrossRef]
  5. A. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. IGR-22, 3–13 (1984).
    [CrossRef]
  6. E. Robinson, “Image reconstruction in exploration geophysics,” IEEE Trans. Sonics Ultrason. SU-31, 259–270 (1984).
    [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  8. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1994).
  9. A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  10. M. Kaveh, M. Soumekh, J. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
    [CrossRef]
  11. S. Pan, A. 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]
  12. A. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
    [CrossRef]
  13. A. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
    [CrossRef]
  14. X. Pan, “A unified reconstruction theory for diffraction tomography with considerations of noise control,” J. Opt. Soc. Am. A 15, 2312–2326 (1998).
    [CrossRef]
  15. D. Parker, “Optimal short scan convolution reconstruction for fanbeam CT,” Med. Phys. 9, 254–257 (1982).
    [CrossRef] [PubMed]
  16. G. Heidbreder, “Multiple scattering and the method of Rytov,” J. Opt. Soc. Am. 57, 1477–1479 (1967).
    [CrossRef]

1998 (1)

1995 (1)

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

1989 (1)

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

1986 (1)

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

1984 (3)

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

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

E. Robinson, “Image reconstruction in exploration geophysics,” IEEE Trans. Sonics Ultrason. SU-31, 259–270 (1984).
[CrossRef]

1983 (1)

S. Pan, A. 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]

1982 (2)

D. Parker, “Optimal short scan convolution reconstruction for fanbeam CT,” Med. Phys. 9, 254–257 (1982).
[CrossRef] [PubMed]

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

1979 (2)

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

G. Kino, “Acoustic imaging for nondestructive evaluation,” Proc. IEEE 67, 510–525 (1979).
[CrossRef]

1967 (1)

Andre, M. P.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Apivey, B. A.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Barrett, T. K.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1994).

Devaney, A.

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

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

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

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

Gelius, L.

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Hilger, London, 1991).

Greenleaf, J.

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

Heidbreder, G.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Johansen, I.

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Hilger, London, 1991).

Kak, A.

S. Pan, A. 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. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
[CrossRef]

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

Kino, G.

G. Kino, “Acoustic imaging for nondestructive evaluation,” Proc. IEEE 67, 510–525 (1979).
[CrossRef]

Martin, P. J.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Mueller, R.

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

Olson, L. K.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Otto, G. P.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Palmer, D. A.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

Pan, S.

S. Pan, A. 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]

Pan, X.

Parker, D.

D. Parker, “Optimal short scan convolution reconstruction for fanbeam CT,” Med. Phys. 9, 254–257 (1982).
[CrossRef] [PubMed]

Robinson, E.

E. Robinson, “Image reconstruction in exploration geophysics,” IEEE Trans. Sonics Ultrason. SU-31, 259–270 (1984).
[CrossRef]

Soumekh, M.

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

Sponheim, N.

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Hilger, London, 1991).

Stamnes, J.

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Hilger, London, 1991).

Wade, G.

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

Acoust. Imaging (1)

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Apivey, D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging,” Acoust. Imaging 21, 379–390 (1995).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

S. Pan, A. 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. Geosci. Remote Sens. (1)

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

IEEE Trans. Sonics Ultrason. (2)

E. Robinson, “Image reconstruction in exploration geophysics,” IEEE Trans. Sonics Ultrason. SU-31, 259–270 (1984).
[CrossRef]

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

Inverse Probl. (2)

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

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

J. Opt. Soc. Am. (1)

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

Med. Phys. (1)

D. Parker, “Optimal short scan convolution reconstruction for fanbeam CT,” Med. Phys. 9, 254–257 (1982).
[CrossRef] [PubMed]

Proc. IEEE (2)

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

G. Kino, “Acoustic imaging for nondestructive evaluation,” Proc. IEEE 67, 510–525 (1979).
[CrossRef]

Ultrason. Imaging (1)

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

Other (3)

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Hilger, London, 1991).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1994).

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Figures (9)

Fig. 1
Fig. 1

Classical scan configuration of 2D (transmission) DT. The incident plane wave propagates along the η axis, and the scattered data are measured along the line η=l.

Fig. 2
Fig. 2

The FDP theorem states that the 1D FT of the scattered data along the line η=l is equal to the 2D FT of the object function along a semicircle AOB with a radius of ν0 in its Fourier space.

Fig. 3
Fig. 3

As the measurement angle ϕ varies from 0 to 2π, the two segments OA and OB of the semicircle AOB generate two distinct coverages, (a) and (b), respectively, of the 2D Fourier space of the object function.

Fig. 4
Fig. 4

As ϕ varies from 0 to 3π/2, the two segments OA and OB yield two incomplete coverages (a) and (b), respectively, of the 2D Fourier space of the object function. Superimposing the two incomplete coverages in (a) and (b), one obtains (c) a complete coverage of the 2D Fourier space of the object function.

Fig. 5
Fig. 5

(a) Complete data space W that contains data from the view angles in [0, 2π]. (b) The subspace M that contains minimal complete data from the view angles in [0, 3π/2]. (c) The subspaces A, B, C, and D in the complete data space. The boundary between subspaces A and B is specified by the equation ϕ=π/2+2α. The boundary between subspaces B and C is specified by the equation ϕ=π+2α. The boundary between subspaces C and D is specified by the equation ϕ=3π/2.

Fig. 6
Fig. 6

(a) Real and (b) imaginary components of the image reconstructed from the complete data set by use of the FBPP algorithm.

Fig. 7
Fig. 7

(a) Real and (b) imaginary components of the image reconstructed from the minimal complete data set by use of the FBPP algorithm.

Fig. 8
Fig. 8

(a) Real and (b) imaginary components of the image reconstructed from the minimal complete data set by use of the MS-FBPP algorithm that is specified by Eq. (11).

Fig. 9
Fig. 9

(a) Real and (b) imaginary components of the image reconstructed from the minimal complete data set by use of the MS-FBPP algorithm that is specified by Eq. (12).

Equations (28)

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M(νm, ϕ)=Us(νm, ϕ) jν2π2ν02U0 exp(-j2πνl),
M(νm, ϕ)
=-- a(r)exp{-j2π[νmξ+(ν-ν0)η]}drif|νm|ν00if|νm|>ν0,
a(r, θ)=12 ϕ=02πνm=-ν0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ,
α=sgn(νm)arcsin12ν0 νm2-νμ2.
M(νm, ϕ)=M(-νm, ϕ+π-2α).
M(νm, ϕ)=w(νm, ϕ)M(νm, ϕ),
w(νm, ϕ)+w(-νm, ϕ+π-2α)=1
w(νm, ϕ)=1
w(νm, ϕ)=0
a(w)(r, θ)=ϕ=03π/2νm=-ν0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
a(w)(r, θ)=ϕ=02πνm=-ν0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
a(w)(r, θ)=a(r, θ)=12 ϕ=02πνm=-ν0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
w(νm, ϕ)=1/20ϕπ/2+2α1π/2+2αϕπ+2α1/2π+2αϕ3π/203π/2ϕ2π..
w(νm, ϕ)
=sin2π4 ϕ(π/4)-α,0ϕπ/2+2α1,π/2+2αϕπ+2αsin2π4 (3π/2)-ϕ(π/4)+α,π+2αϕ3π/20,3π/2ϕ2π..
a(w)(r, θ)=ϕ=03π/2νm=-ν0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ,
a(w)(r, θ)=T++T-,
T+=νm=0ν0ϕ=0π/2+2α+ϕ=π/2+2απ+2α+ϕ=π+2α3π/2×ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ,
T-=νm=-ν0ν0ϕ=0π/2+2α+ϕ=π/2+2απ+2α+ϕ=π+2α3π/2×ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
T-=νm=0ν0ϕ=0π/2-2α+ϕ=π/2-2απ-2α+ϕ=π-2α3π/2×ν0ν |νm|M(-νm, ϕ)×exp[-j2πνar cos(ϕ+α-θ)]dνmdϕ.
T-=νm=0ν0ϕ=π+2α3/2π+ϕ=3/2π2π+ϕ=0π/2+2α×ν0ν |νm|M(-νm, ϕ+π-2α)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
a(w)(r, θ)
=νm=0ν0ϕ=0π/2+2α[w(νm, ϕ)+w(-νm, ϕ+π-2α)]×ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ+νm=0ν0ϕ=π/2+2απ+2α[w(νm, ϕ)]ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ+νm=0ν0ϕ=π+2α3/2π[w(νm, ϕ)+w(-νm, ϕ+π-2α)]ν0ν |νm|M(νm, ϕ)×exp[j2πvar cos(ϕ-α-θ)]dνmdϕ+νm=0ν0ϕ=3/2π2π[w(-νm, ϕ+π-2α)]×ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
a(w)(r, θ)=ϕ=02πνm=0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
2a(r, θ)=ϕ=02πνm=0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ+ϕ=02πνm=-ν00 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
a(r, θ)=ϕ=02πνm=0ν0 ν0ν |νm|M(νm, ϕ)×exp[j2πνar cos(ϕ-α-θ)]dνmdϕ.
a(w)(r, θ)=a(r, θ).

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