Abstract

Diffraction tomography (DT) is an inversion scheme used to reconstruct the spatially variant refractive-index distribution of a scattering object. We developed computationally efficient algorithms for image reconstruction in three-dimensional (3D) DT. A unique and important aspect of these algorithms is that they involve only a series of two-dimensional reconstructions and thus greatly reduce the prohibitively large computational load required by conventional 3D reconstruction algorithms. We also investigated the noise characteristics of these algorithms and developed strategies that exploit the statistically complementary information inherent in the measured data to achieve a bias-free reduction of the reconstructed image variance. We performed numerical studies that corroborate our theoretical assertions.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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,” presented at the NATO Advanced Workshop on Inverse Problems in Scattering and Imaging, North Falmouth, Mass., April 14–19, 1991.
  3. R. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
    [CrossRef]
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  5. W. Chew, Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [CrossRef] [PubMed]
  6. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  7. A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  8. Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
    [CrossRef]
  9. M. Kaveh, M. Soumekh, J. Greenleaf, “Signal processing for diffraction tomography,” IEEE Trans. Sonics Ultrason. SU-31, 230–239 (1984).
    [CrossRef]
  10. 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]
  11. X. Pan, “A unified reconstruction theory for diffraction tomography with considerations of noise control,” J. Opt. Soc. Am. A 15, 2312–2326 (1998).
    [CrossRef]
  12. We are preparing a manuscript entitled “Investigation of the noise properties of an infinite class of reconstruction methods in diffraction tomography.”
  13. P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N.J., 1986).
  14. P. Grassin, B. Duchene, W. Tabbara, “Diffraction tomography: some applications and extension to 3-D ultrasound imaging,” in Mathematical Methods in Tomography (Springer-Verlag, Heidelberg, Germany, 1991), pp. 98–105.
  15. C. E. Metz, X. Pan, “A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT,” IEEE Trans. Med. Imaging 14, 643–658 (1995).
    [CrossRef] [PubMed]
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).
  17. D. Rouseff, R. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
    [CrossRef]
  18. D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
    [CrossRef]
  19. H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 219–286 (1984).
  20. N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng. 25, 627–635 (1986).
    [CrossRef]
  21. X. Pan, C. Metz, “Analytical approaches for image reconstruction in 3D SPECT,” in Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 103–116.
  22. M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  23. Z.-Q. Lu, Y.-Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. 43, 296–302 (1996).
    [CrossRef]

1998 (1)

1997 (1)

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

1996 (1)

Z.-Q. Lu, Y.-Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. 43, 296–302 (1996).
[CrossRef]

1995 (2)

C. E. Metz, X. Pan, “A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT,” IEEE Trans. Med. Imaging 14, 643–658 (1995).
[CrossRef] [PubMed]

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]

1991 (1)

D. Rouseff, R. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[CrossRef]

1990 (1)

W. Chew, Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

1986 (1)

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng. 25, 627–635 (1986).
[CrossRef]

1985 (1)

Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
[CrossRef]

1984 (3)

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

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

H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 219–286 (1984).

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 (1)

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

1979 (1)

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

1969 (1)

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

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]

Asakura, T.

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng. 25, 627–635 (1986).
[CrossRef]

Barrett, H. H.

H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 219–286 (1984).

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.

W. Chew, Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

Devaney, A.

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

Duchene, B.

P. Grassin, B. Duchene, W. Tabbara, “Diffraction tomography: some applications and extension to 3-D ultrasound imaging,” in Mathematical Methods in Tomography (Springer-Verlag, Heidelberg, Germany, 1991), pp. 98–105.

Fischer, D. G.

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

Gelius, L.

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” presented at the NATO Advanced Workshop on Inverse Problems in Scattering and Imaging, North Falmouth, Mass., April 14–19, 1991.

Grassin, P.

P. Grassin, B. Duchene, W. Tabbara, “Diffraction tomography: some applications and extension to 3-D ultrasound imaging,” in Mathematical Methods in Tomography (Springer-Verlag, Heidelberg, Germany, 1991), pp. 98–105.

Greenleaf, J.

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

Ingard, K. U.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N.J., 1986).

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,” presented at the NATO Advanced Workshop on Inverse Problems in Scattering and Imaging, North Falmouth, Mass., April 14–19, 1991.

Kadono, H.

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng. 25, 627–635 (1986).
[CrossRef]

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]

Kak, A. C.

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

Larsen, L.

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

Lu, Z.

Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
[CrossRef]

Lu, Z.-Q.

Z.-Q. Lu, Y.-Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. 43, 296–302 (1996).
[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]

Metz, C.

X. Pan, C. Metz, “Analytical approaches for image reconstruction in 3D SPECT,” in Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 103–116.

Metz, C. E.

C. E. Metz, X. Pan, “A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT,” IEEE Trans. Med. Imaging 14, 643–658 (1995).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N.J., 1986).

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.

X. Pan, “A unified reconstruction theory for diffraction tomography with considerations of noise control,” J. Opt. Soc. Am. A 15, 2312–2326 (1998).
[CrossRef]

C. E. Metz, X. Pan, “A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT,” IEEE Trans. Med. Imaging 14, 643–658 (1995).
[CrossRef] [PubMed]

X. Pan, C. Metz, “Analytical approaches for image reconstruction in 3D SPECT,” in Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 103–116.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

Porter, R.

D. Rouseff, R. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[CrossRef]

Rouseff, D.

D. Rouseff, R. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[CrossRef]

Slaney, M.

M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (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,” presented at the NATO Advanced Workshop on Inverse Problems in Scattering and Imaging, North Falmouth, Mass., April 14–19, 1991.

Stamnes, J.

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” presented at the NATO Advanced Workshop on Inverse Problems in Scattering and Imaging, North Falmouth, Mass., April 14–19, 1991.

Tabbara, W.

P. Grassin, B. Duchene, W. Tabbara, “Diffraction tomography: some applications and extension to 3-D ultrasound imaging,” in Mathematical Methods in Tomography (Springer-Verlag, Heidelberg, Germany, 1991), pp. 98–105.

Takai, N.

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng. 25, 627–635 (1986).
[CrossRef]

Wade, G.

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

Wang, Y.

W. Chew, Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

Wolf, E.

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

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

Zhang, Y.-Y.

Z.-Q. Lu, Y.-Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. 43, 296–302 (1996).
[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. Med. Imaging (2)

C. E. Metz, X. Pan, “A unified analysis of exact methods of inverting the 2-D exponential Radon transform, with implications for noise control in SPECT,” IEEE Trans. Med. Imaging 14, 643–658 (1995).
[CrossRef] [PubMed]

W. Chew, Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

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

IEEE Trans. Sonics Ultrason. (1)

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

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

Z.-Q. Lu, Y.-Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. 43, 296–302 (1996).
[CrossRef]

Inverse Probl. (1)

Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
[CrossRef]

J. Acoust. Soc. Am. (1)

D. Rouseff, R. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[CrossRef]

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

Opt. Commun. (2)

D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

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

Opt. Eng. (1)

N. Takai, H. Kadono, T. Asakura, “Statistical properties of the speckle phase in image and diffraction fields,” Opt. Eng. 25, 627–635 (1986).
[CrossRef]

Proc. IEEE (1)

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

Prog. Opt. (1)

H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 219–286 (1984).

Ultrason. Imaging (1)

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

Other (7)

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

L. Gelius, I. Johansen, N. Sponheim, J. Stamnes, “Diffraction tomography applications in medicine and seismics,” presented at the NATO Advanced Workshop on Inverse Problems in Scattering and Imaging, North Falmouth, Mass., April 14–19, 1991.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

We are preparing a manuscript entitled “Investigation of the noise properties of an infinite class of reconstruction methods in diffraction tomography.”

P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N.J., 1986).

P. Grassin, B. Duchene, W. Tabbara, “Diffraction tomography: some applications and extension to 3-D ultrasound imaging,” in Mathematical Methods in Tomography (Springer-Verlag, Heidelberg, Germany, 1991), pp. 98–105.

X. Pan, C. Metz, “Analytical approaches for image reconstruction in 3D SPECT,” in Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 103–116.

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 (6)

Fig. 1
Fig. 1

Classical scan geometry of 3D DT. A plane wave is incident at angle ϕ, and the scattered field is measured in the ξz plane located at η=l. The measurement angle ϕ is varied between 0 and 2π.

Fig. 2
Fig. 2

Block diagram describing the E-C reconstruction algorithms.

Fig. 3
Fig. 3

True images at slices through the (a) z=1 pixel, (b) z=3 pixel, and (c) z=5 pixel planes of the scattering object function and images at slices through the (d) z=1 pixel, (e) z=3 pixel, and (f) z=5 pixel planes, reconstructed by use of the 3D E-C algorithm specified by c=0.5 in Eq. (27).

Fig. 4
Fig. 4

Images at transverse slices through the z=0 plane of scattering object 2, reconstructed from noisy data by use of the 3D E-C algorithm specified by (a) c=0.5 and (b) c=1.0 in Eq. (27). (c) Image variances along vertical lines through the centers of the z=0 plane images reconstructed by use of the 3D E-C algorithm specified by c=0.5 (solid curve) and c=1.0 (dashed curve). These results corroborate our assertion that different E-C algorithms respond to noise distinctively.

Fig. 5
Fig. 5

GV calculated from 100 noisy images reconstructed by use of the 3D E-C algorithms specified by different values of c in Eq. (27). The data are normalized by the value of the GV obtained with c=0.5.

Fig. 6
Fig. 6

PV in three different transverse planes of scattering object 2, reconstructed from noisy data by use of the 3D E-C algorithms specified by different values of c in Eq. (27). The data are normalized by the value of the GV obtained with c=0.5.

Equations (41)

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

M(νm, νz, ϕ)=jν2π2ν02U0exp(-j2πνl)Us(νm, νz, ϕ),
M(νm, νz, ϕ)=--a(r)exp{-j2π[νmξ+νzz+(ν-ν0)η]}dr
M(νm, νz, ϕ)=jν2π2ν02U0exp[-j2π(ν-ν0)l]×Ψs(νm, νz, ϕ),
p(ξ, z, ϕ0)=η=-a(r, θ, z)dη,
Pk(νa, νz)=12πϕ=02πξ,z=-p(ξ, z, ϕ0)×exp(-j2πνaξ-j2πνzz-jkϕ0)dξdzdϕ0,
Pk(νa, νz)=(-j)kz=-exp(-j2πνzz)dz×θ=02πr=0a(r, θ, z)×exp(-jkθ)Jk(2πνar)r drdθ,
Mk(νm, νz)=12π02πM(νm, νz, ϕ)exp(-jkϕ)dϕ.
Mk(νm, νz)=z=-exp(-j2πνzz)dzθ=02πr=0a(r, θ, z)×12πϕ=02πexp{2πνμr sin(ϕ-θ)-j[kϕ+2πνmr cos(ϕ-θ)]}dϕr drdθ,
Mk(νm, νz)=(-j)kνm+νμνm2-νμ2k×z=-exp(-j2πνzz)dz×θ=02πr=0a(r, θ, z)exp(-jkθ)×Jk(2πrνm2-νμ2)r drdθ,
Pk(νa, νz)=[γ(νm, νz)]kMk(νm, νz),
νa2=νm2-νμ2,
γ(νm, νz)=νm2-νμ2νm+νμ.
νm1=-νm2=νm=(νa2+νz2)1-νa2+νz24ν02-νz21/2,
Pk(νa, νz)=[γ(νm1, νz)]kMk(νm1, νz)=[γ(νm, νz)]kMk(νm, νz)
Pk(νa, νz)=[γ(νm2, νz)]kMk(νm2, νz)=(-1)k[γ(νm, νz)]-kMk(-νm, νz).
Pk(ω)(νa, νz)=ωk(νm, νz)[γ(νm, νz)]kMk(νm, νz)+[1-ωk(νm, νz)]×(-1)k[γ(νm, νz)]-k×Mk(-νm, νz),
var{Pk(ω)(νa, νz)}=τ+τ-τ+τ-+τ-τ+-2(-1)k Re(ρ)×(R2+I2)-2τ-τ+-2(-1)k Re(ρ)×R-(-1)k Im(ρ)I+τ-τ+,
ρ=γ2kcov{Mk(νm, νz),Mk(-νm, νz)}τ+τ-,
Rop=τ--(-1)kτ+τ- Re(ρ)τ++τ--2(-1)kτ+τ- Re(ρ),
Iop=(-1)kτ+τ- Im(ρ)τ++τ--2(-1)kτ+τ- Re(ρ),
cov{us(ξ, z, ϕ),us(ξ, z, ϕ)}=σ2(ξ, z, ϕ)δ(ξ-ξ)δ(z-z)δ(ϕ-ϕ),
Rop=12,Iop=12(-1)k Im(ρ)1-(-1)k Re(ρ).
p(ξ, z, ϕ0)=k=-exp(jkϕ0)νa2+νz22ν02Pk(νa, νz)×exp[j2π(νaξ+νzz)]dνadνz.
a(r, θ, z)=-12π02πdϕ0×[p(ξ, z, ϕ0) *ξ F-1{|νa|}]ξ=r cos(θ-ϕ0),
a(r, θ, z)=2πk=-jkνz=-νa=0Pk(νa, νz)×exp(jkθ+j2πνzz)Jk(2πνar)νa dνadνz.
GV=θ=02πz=-r=0var{a(r, θ, z)}r drdzdθ=2πk=-νz=-νa=0var{Pk(νa, νz)}νa dνadνz.
PV(z)=θ=02πr=0var{a(r, θ, z)}r drdθ=2πk=-νa=0var{Pk(νa, z)}νa dνa,
p(Δu(r), Δu(i))=12πσiσrexp-12Δu(r)2σr2+Δu(i)2σi2,
ωk(νm, νz)=c,νm>00.5,νm=01-c,νm<0,
var{a(c)(r, θ, z)}=1M-1i=1Mai(c)(r, θ, z)2-1Mi=1Mai(c)(r, θ, z)2,
aFBPP(r, θ, z)=12ϕ=02πdϕ×νm2+νz2ν02ν0ν|νm|M(νm, νz, ϕ)×exp[j2π(νmξ-j2πνμη+νzz)]×dνmdνz,
aGFBPP(ω)(r)=12ϕ=02πdϕ×νm2+νz2ν02ν0ν|νm|M(ω)(νm, ϕ, νz)×exp[j2π(νmξ-jνμη+νzz)]dνmdνz,
M(ω)(νm, νz, ϕ)=k=-2ωk(νm, νz)Mk(νm, νz)exp(jkϕ)
ωk(νm, νz)+ωk(-νm, νz)=1.
a(ω)(r, θ, z)=2πk=-jkνz=-νa=0Pk(ω)(νa, νz)×exp(jkθ+j2πνzz)×Jk(2πνar)νa dνadνz,
a(ω)(r, θ, z)=2πk=-jkνz=-ν0ν0νa=02ν02-νz2{ωk(νm, νz)×[γ(νm, νz)]kMk(νm, νz)+ωk(-νm, νz)×(-1)k[γ(νm, νz)]-kMk(-νm, νz)}×exp(jkθ+j2πνzz)×Jk(2πνar)νa dνadνz.
a(ω)(r, θ, z)=2πk=-jk exp(jkθ)×νz=-ν0ν0νm=0ν02-νz2{ωk(νm, νz)×[γ(νm, νz)]kMk(νm, νz)+ωk(-νm, νz)×(-1)k[γ(νm, νz)]-k×Mk(-νm, νz)}exp(j2πνzz)×Jk(2πνm2-νμ2r)ν0ννm dνmdνz.
aGFBPP(ω)(r, θ, z)=k=-ϕ=02πdϕνm2+νz2ν02ν0ν|νm|ωk(νm, νz)Mk(νm, νz)×expj2πνmξ-jνμη+νzz+k2πϕdνmdνz
=k=-ϕ=02πdϕνz=-ν0ν0νm=0ν02-νze×ν0ννmωk(νm, νz)Mk(νm, νz)×expj2πνmξ-jνμη+νzz+k2πϕdνmdνz+k=-ϕ=02πdϕνz=-ν0ν0νm=-ν02-νz20ν0ν(-νm)×ωk(νm, νz)Mk(νm, νz)expj2πνmξ-jνμη+νzz+k2πϕdνmdνz
=2πk=-νz=-ν0ν0νm0ν02-νz2ν0ν×νmωk(νm, νz)Mk(νm, νz)exp(jνzz)×ϕ=02πexpj2πνmξ-jνnη+k2πϕdϕdνmdνz+2πk=-νz=-ν0ν0νm=0ν02-νz2ν0ννmωk(-νm, νz)×Mk(-νm, νz)exp(jνzz)×ϕ=02πexpj2π-νmξ-jνμη+k2πϕdϕdνmdνz.
aGFBPP(ω)(r, θ, z)=2πk=-jk exp(jkθ)νz=-ν0ν0νm=0ν02-νz2{ωk(νm, νz)×[γ(νm, νz)]kMk(νm, νz)+ωk(-νm, νz)×(-1)k[γ(νm, νz)]-kMk(-νm, νz)}×exp(j2πνzz)Jk(2πνm2-νμ2r)ν0ννm dνmdνz.

Metrics