Abstract

In diffraction tomography, the spatial distribution of the scattering object is reconstructed from the measured scattered data. For a scattering object that is illuminated with plane-wave radiation, under the condition of weak scattering one can invoke the Born (or the Rytov) approximation to linearize the equation for the scattered field (or the scattered phase) and derive a relationship between the scattered field (or the scattered phase) and the distribution of the scattering object. Reconstruction methods such as the Fourier domain interpolation methods and the filtered backpropagation method have been developed previously. However, the underlying relationship among and the noise properties of these methods are not evident. We introduce the concepts of ideal and modified sinograms. Analysis of the relationships between, and the noise properties of the two sinograms reveals infinite classes of methods for image reconstruction in diffraction tomography that include the previously proposed methods as special members. The methods in these classes are mathematically identical, but they respond to noise and numerical errors differently.

© 1998 Optical Society of America

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  1. R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
    [CrossRef]
  2. R. K. Mueller, M. Kaveh, R. D. Iverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Holography, vol. 8, A. F. Metherell, ed. (Plenum, New York, 1980), pp. 615–628.
  3. M. Slaney, A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A. J. Devaney, ed. (SPIE, Bellingham, Wash., 1983), pp. 2–19.
  4. C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. 31, 195–217 (1984).
    [CrossRef]
  5. J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
    [CrossRef]
  6. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GRS-22, 3–13 (1984).
    [CrossRef]
  7. R. S. Wu, M. N. Toksöz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysics 52, 11–25 (1987).
    [CrossRef]
  8. B. Duchêne, D. Lesselier, W. Tabbara, “Experimental investigation of a diffraction tomography technique in fluid ultrasonics,” IEEE Trans. Biomed. Eng. 35, 437–444 (1988).
  9. L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in Proceedings of NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Bristol, Philadelphia, 1992).
  10. J. F. Greenleaf, “3D and tomographic ultrasound,” Medical CT and Ultrasound: Current Technology and Applications, L. W. Goldman, J. B. Fowlkes, ed. (American Association of Physicists in Medicine, College Park, Md., 1995), pp. 267–284.
  11. T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
    [CrossRef]
  12. H. S. Janée, J. P. Jones, M. P. André, “Analysis of scatter fields in diffraction tomography experiments,” Acoust. Imaging 22, 21–26 (1996).
    [CrossRef]
  13. M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
    [CrossRef]
  14. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  15. H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, N.Y., 1984), vol. XXI, pp. 219–286.
  16. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  17. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1969).
  18. K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms,” J. Opt. Soc. Am. 60, 133–135 (1970).
    [CrossRef]
  19. K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
    [CrossRef]
  20. D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
    [CrossRef] [PubMed]
  21. F. Natterer, F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
    [CrossRef]
  22. A. Ishimaru, Wave Propagation and Scattering in Random Medium Vol. 2 (Academic, New York, 1978).
  23. K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms using Born and Rytov’s approximation,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 383 (1975).
    [CrossRef]
  24. S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262–1275 (1983).
    [CrossRef]
  25. A. J. Devaney, “A filtered backpropagation algorithm for diffraction images,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  26. D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
    [CrossRef]
  27. G. A. Tsihrintzis, A. J. Devaney, “Application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography,” IEEE Trans. Med. Imaging 12, 545–554 (1993).
    [CrossRef] [PubMed]
  28. G. A. Tsihrintzis, A. J. Devaney, “Stochastic diffraction tomography: theory and computer simulation,” Signal Process. 30, 49–64 (1993).
    [CrossRef]
  29. D. G. Fischer, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
    [CrossRef]
  30. T. F. Budinger, G. T. Gullberg, R. H. Huesman, “Emission computed tomography,” in Topics in Applied Physics: Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 147–246.
  31. R. J. Jaszczak, R. E. Coleman, C. B. Lim, F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci. 28, 69–80 (1981).
    [CrossRef]
  32. O. J. Tretiak, C. E. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39, 341–354 (1980).
    [CrossRef]
  33. G. T. Gullberg, T. F. Budinger, “The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography,” IEEE Trans. Biomed. Eng. 8, 142–157 (1981).
    [CrossRef]
  34. C. E. Metz, X. Pan, “A unified analysis of exact methods of inverting the 2D exponential Radon transform, with implications for noise control in SPECT,” IEEE Trans. Med. Imaging 14, 643–658 (1995).
    [CrossRef]
  35. X. Pan, C. E. Metz, “Analysis of noise properties of a class of exact methods of inverting the 2D exponential Radon transform,” IEEE Trans. Med. Imaging 14, 659–668 (1995).
  36. S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  37. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).
  38. M. L. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983).
  39. M. Anastasio, M. Kupinski, X. Pan, “Noise properties ofreconstructed images in ultrasound diffraction tomogra- phy,” in IEEE 1997 Nuclear Science Symposium and Medical Imaging Conference Record (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), vol. 2, pp. 1561–1565.
  40. Z. Lu, Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 296–302 (1996).
    [CrossRef]
  41. A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
    [CrossRef] [PubMed]

1997 (2)

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

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

1996 (3)

H. S. Janée, J. P. Jones, M. P. André, “Analysis of scatter fields in diffraction tomography experiments,” Acoust. Imaging 22, 21–26 (1996).
[CrossRef]

M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

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

1995 (3)

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

X. Pan, C. E. Metz, “Analysis of noise properties of a class of exact methods of inverting the 2D exponential Radon transform,” IEEE Trans. Med. Imaging 14, 659–668 (1995).

F. Natterer, F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

1993 (2)

G. A. Tsihrintzis, A. J. Devaney, “Application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography,” IEEE Trans. Med. Imaging 12, 545–554 (1993).
[CrossRef] [PubMed]

G. A. Tsihrintzis, A. J. Devaney, “Stochastic diffraction tomography: theory and computer simulation,” Signal Process. 30, 49–64 (1993).
[CrossRef]

1992 (1)

D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
[CrossRef] [PubMed]

1991 (2)

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

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

1988 (1)

B. Duchêne, D. Lesselier, W. Tabbara, “Experimental investigation of a diffraction tomography technique in fluid ultrasonics,” IEEE Trans. Biomed. Eng. 35, 437–444 (1988).

1987 (1)

R. S. Wu, M. N. Toksöz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysics 52, 11–25 (1987).
[CrossRef]

1985 (1)

A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[CrossRef] [PubMed]

1984 (2)

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

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. 31, 195–217 (1984).
[CrossRef]

1983 (1)

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262–1275 (1983).
[CrossRef]

1982 (2)

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

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

1981 (2)

G. T. Gullberg, T. F. Budinger, “The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography,” IEEE Trans. Biomed. Eng. 8, 142–157 (1981).
[CrossRef]

R. J. Jaszczak, R. E. Coleman, C. B. Lim, F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci. 28, 69–80 (1981).
[CrossRef]

1980 (1)

O. J. Tretiak, C. E. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39, 341–354 (1980).
[CrossRef]

1979 (1)

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

1975 (1)

K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms using Born and Rytov’s approximation,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 383 (1975).
[CrossRef]

1970 (1)

1969 (1)

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

Anastasio, M.

M. Anastasio, M. Kupinski, X. Pan, “Noise properties ofreconstructed images in ultrasound diffraction tomogra- phy,” in IEEE 1997 Nuclear Science Symposium and Medical Imaging Conference Record (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), vol. 2, pp. 1561–1565.

André, M. P.

H. S. Janée, J. P. Jones, M. P. André, “Analysis of scatter fields in diffraction tomography experiments,” Acoust. Imaging 22, 21–26 (1996).
[CrossRef]

M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

Barrett, H. H.

H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, N.Y., 1984), vol. XXI, pp. 219–286.

Berggren, M. J.

D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
[CrossRef] [PubMed]

Boas, M. L.

M. L. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983).

Bolomey, J.

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Borup, D. T.

D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
[CrossRef] [PubMed]

Budinger, T. F.

G. T. Gullberg, T. F. Budinger, “The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography,” IEEE Trans. Biomed. Eng. 8, 142–157 (1981).
[CrossRef]

T. F. Budinger, G. T. Gullberg, R. H. Huesman, “Emission computed tomography,” in Topics in Applied Physics: Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 147–246.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1969).

Coleman, R. E.

R. J. Jaszczak, R. E. Coleman, C. B. Lim, F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci. 28, 69–80 (1981).
[CrossRef]

Dean, S. R.

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Devaney, A. J.

G. A. Tsihrintzis, A. J. Devaney, “Application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography,” IEEE Trans. Med. Imaging 12, 545–554 (1993).
[CrossRef] [PubMed]

G. A. Tsihrintzis, A. J. Devaney, “Stochastic diffraction tomography: theory and computer simulation,” Signal Process. 30, 49–64 (1993).
[CrossRef]

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[CrossRef] [PubMed]

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

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

Dickens, T. A.

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

Duchêne, B.

B. Duchêne, D. Lesselier, W. Tabbara, “Experimental investigation of a diffraction tomography technique in fluid ultrasonics,” IEEE Trans. Biomed. Eng. 35, 437–444 (1988).

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.-J.

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in Proceedings of NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Bristol, Philadelphia, 1992).

Greenleaf, J. F.

J. F. Greenleaf, “3D and tomographic ultrasound,” Medical CT and Ultrasound: Current Technology and Applications, L. W. Goldman, J. B. Fowlkes, ed. (American Association of Physicists in Medicine, College Park, Md., 1995), pp. 267–284.

Gullberg, G. T.

G. T. Gullberg, T. F. Budinger, “The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography,” IEEE Trans. Biomed. Eng. 8, 142–157 (1981).
[CrossRef]

T. F. Budinger, G. T. Gullberg, R. H. Huesman, “Emission computed tomography,” in Topics in Applied Physics: Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 147–246.

Huesman, R. H.

T. F. Budinger, G. T. Gullberg, R. H. Huesman, “Emission computed tomography,” in Topics in Applied Physics: Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 147–246.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Medium Vol. 2 (Academic, New York, 1978).

Iverson, R. D.

R. K. Mueller, M. Kaveh, R. D. Iverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Holography, vol. 8, A. F. Metherell, ed. (Plenum, New York, 1980), pp. 615–628.

Iwata, K.

K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms using Born and Rytov’s approximation,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 383 (1975).
[CrossRef]

K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms,” J. Opt. Soc. Am. 60, 133–135 (1970).
[CrossRef]

Izadnegahdar, A.

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Janée, H. S.

H. S. Janée, J. P. Jones, M. P. André, “Analysis of scatter fields in diffraction tomography experiments,” Acoust. Imaging 22, 21–26 (1996).
[CrossRef]

M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

Jaszczak, R. J.

R. J. Jaszczak, R. E. Coleman, C. B. Lim, F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci. 28, 69–80 (1981).
[CrossRef]

Jofre, L.

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Johansen, I.

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in Proceedings of NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Bristol, Philadelphia, 1992).

Johnson, S. A.

D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
[CrossRef] [PubMed]

Jones, J. P.

H. S. Janée, J. P. Jones, M. P. André, “Analysis of scatter fields in diffraction tomography experiments,” Acoust. Imaging 22, 21–26 (1996).
[CrossRef]

M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

Kak, A. C.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262–1275 (1983).
[CrossRef]

M. Slaney, A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A. J. Devaney, ed. (SPIE, Bellingham, Wash., 1983), pp. 2–19.

Kaveh, M.

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

R. K. Mueller, M. Kaveh, R. D. Iverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Holography, vol. 8, A. F. Metherell, ed. (Plenum, New York, 1980), pp. 615–628.

Kim, W. W.

D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
[CrossRef] [PubMed]

Kupinski, M.

M. Anastasio, M. Kupinski, X. Pan, “Noise properties ofreconstructed images in ultrasound diffraction tomogra- phy,” in IEEE 1997 Nuclear Science Symposium and Medical Imaging Conference Record (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), vol. 2, pp. 1561–1565.

Ladas, K. T.

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

Lee, H.

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. 31, 195–217 (1984).
[CrossRef]

Lesselier, D.

B. Duchêne, D. Lesselier, W. Tabbara, “Experimental investigation of a diffraction tomography technique in fluid ultrasonics,” IEEE Trans. Biomed. Eng. 35, 437–444 (1988).

Lim, C. B.

R. J. Jaszczak, R. E. Coleman, C. B. Lim, F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci. 28, 69–80 (1981).
[CrossRef]

Lu, Z.

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

Martin, P. J.

M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

Metz, C. E.

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

X. Pan, C. E. Metz, “Analysis of noise properties of a class of exact methods of inverting the 2D exponential Radon transform,” IEEE Trans. Med. Imaging 14, 659–668 (1995).

O. J. Tretiak, C. E. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39, 341–354 (1980).
[CrossRef]

Mueller, R. K.

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

R. K. Mueller, M. Kaveh, R. D. Iverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Holography, vol. 8, A. F. Metherell, ed. (Plenum, New York, 1980), pp. 615–628.

Nagata, R.

K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms using Born and Rytov’s approximation,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 383 (1975).
[CrossRef]

K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms,” J. Opt. Soc. Am. 60, 133–135 (1970).
[CrossRef]

Natterer, F.

F. Natterer, F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

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

Otto, G. P.

M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

Pan, S. X.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262–1275 (1983).
[CrossRef]

Pan, X.

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

X. Pan, C. E. Metz, “Analysis of noise properties of a class of exact methods of inverting the 2D exponential Radon transform,” IEEE Trans. Med. Imaging 14, 659–668 (1995).

M. Anastasio, M. Kupinski, X. Pan, “Noise properties ofreconstructed images in ultrasound diffraction tomogra- phy,” in IEEE 1997 Nuclear Science Symposium and Medical Imaging Conference Record (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), vol. 2, pp. 1561–1565.

Papoulis, A.

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Peronnet, G.

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Pichot, C.

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
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D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[CrossRef]

Rouseff, D.

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

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C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. 31, 195–217 (1984).
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M. Slaney, A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A. J. Devaney, ed. (SPIE, Bellingham, Wash., 1983), pp. 2–19.

Solaimani, M.

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

Sponheim, N.

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in Proceedings of NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Bristol, Philadelphia, 1992).

Stamnes, J. J.

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in Proceedings of NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Bristol, Philadelphia, 1992).

Tabbara, W.

B. Duchêne, D. Lesselier, W. Tabbara, “Experimental investigation of a diffraction tomography technique in fluid ultrasonics,” IEEE Trans. Biomed. Eng. 35, 437–444 (1988).

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R. S. Wu, M. N. Toksöz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysics 52, 11–25 (1987).
[CrossRef]

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O. J. Tretiak, C. E. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39, 341–354 (1980).
[CrossRef]

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G. A. Tsihrintzis, A. J. Devaney, “Stochastic diffraction tomography: theory and computer simulation,” Signal Process. 30, 49–64 (1993).
[CrossRef]

G. A. Tsihrintzis, A. J. Devaney, “Application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography,” IEEE Trans. Med. Imaging 12, 545–554 (1993).
[CrossRef] [PubMed]

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C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. 31, 195–217 (1984).
[CrossRef]

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

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[CrossRef]

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T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

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]

Wu, R. S.

R. S. Wu, M. N. Toksöz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysics 52, 11–25 (1987).
[CrossRef]

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F. Natterer, F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
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Z. Lu, Y. Zhang, “Acoustical tomography based on the second-order Born transform perturbation approximation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 296–302 (1996).
[CrossRef]

Acoust. Imaging (2)

H. S. Janée, J. P. Jones, M. P. André, “Analysis of scatter fields in diffraction tomography experiments,” Acoust. Imaging 22, 21–26 (1996).
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M. P. André, H. S. Janée, G. P. Otto, P. J. Martin, J. P. Jones, “Reduction of phase aberration in a diffraction tomography system for breast imaging,” Acoust. Imaging 22, 151–157 (1996).
[CrossRef]

Geophysics (1)

R. S. Wu, M. N. Toksöz, “Diffraction tomography and multisource holography applied to seismic imaging,” Geophysics 52, 11–25 (1987).
[CrossRef]

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

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262–1275 (1983).
[CrossRef]

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G. T. Gullberg, T. F. Budinger, “The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography,” IEEE Trans. Biomed. Eng. 8, 142–157 (1981).
[CrossRef]

B. Duchêne, D. Lesselier, W. Tabbara, “Experimental investigation of a diffraction tomography technique in fluid ultrasonics,” IEEE Trans. Biomed. Eng. 35, 437–444 (1988).

IEEE Trans. Geosci. Remote Sens. (1)

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

IEEE Trans. Med. Imaging (3)

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

X. Pan, C. E. Metz, “Analysis of noise properties of a class of exact methods of inverting the 2D exponential Radon transform,” IEEE Trans. Med. Imaging 14, 659–668 (1995).

G. A. Tsihrintzis, A. J. Devaney, “Application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography,” IEEE Trans. Med. Imaging 12, 545–554 (1993).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theor. Tech. (1)

J. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theor. Tech. MTT-30, 1998–2000 (1982).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

R. J. Jaszczak, R. E. Coleman, C. B. Lim, F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci. 28, 69–80 (1981).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

C. F. Schueler, H. Lee, G. Wade, “Fundamentals of digital ultrasonic imaging,” IEEE Trans. Sonics Ultrason. 31, 195–217 (1984).
[CrossRef]

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

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

Inverse Probl. (2)

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

F. Natterer, F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[CrossRef]

J. Acoust. Soc. Am. (2)

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

T. A. Dickens, G. A. Winbow, “Spatial resolution of diffraction tomography,” J. Acoust. Soc. Am. 101, 77–86 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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K. Iwata, R. Nagata, “Calculation of three-dimensional refractive index distribution from interferograms using Born and Rytov’s approximation,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 383 (1975).
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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]

Proc. IEEE (1)

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

SIAM J. Appl. Math. (1)

O. J. Tretiak, C. E. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39, 341–354 (1980).
[CrossRef]

Signal Process. (1)

G. A. Tsihrintzis, A. J. Devaney, “Stochastic diffraction tomography: theory and computer simulation,” Signal Process. 30, 49–64 (1993).
[CrossRef]

Ultrason. Imaging (3)

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

D. T. Borup, S. A. Johnson, W. W. Kim, M. J. Berggren, “Nonperturbative diffraction tomography via Gauss–Newton iteration applied to the scattering integral equation,” Ultrason. Imaging 14, 69–85 (1992).
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A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[CrossRef] [PubMed]

Other (13)

H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, N.Y., 1984), vol. XXI, pp. 219–286.

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

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1969).

R. K. Mueller, M. Kaveh, R. D. Iverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Holography, vol. 8, A. F. Metherell, ed. (Plenum, New York, 1980), pp. 615–628.

M. Slaney, A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A. J. Devaney, ed. (SPIE, Bellingham, Wash., 1983), pp. 2–19.

L.-J. Gelius, I. Johansen, N. Sponheim, J. J. Stamnes, “Diffraction tomography applications in medicine and seismics,” in Proceedings of NATO Advanced Workshop on Inverse Problems in Scattering and Imaging (Bristol, Philadelphia, 1992).

J. F. Greenleaf, “3D and tomographic ultrasound,” Medical CT and Ultrasound: Current Technology and Applications, L. W. Goldman, J. B. Fowlkes, ed. (American Association of Physicists in Medicine, College Park, Md., 1995), pp. 267–284.

A. Ishimaru, Wave Propagation and Scattering in Random Medium Vol. 2 (Academic, New York, 1978).

T. F. Budinger, G. T. Gullberg, R. H. Huesman, “Emission computed tomography,” in Topics in Applied Physics: Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 147–246.

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

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

M. L. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983).

M. Anastasio, M. Kupinski, X. Pan, “Noise properties ofreconstructed images in ultrasound diffraction tomogra- phy,” in IEEE 1997 Nuclear Science Symposium and Medical Imaging Conference Record (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), vol. 2, pp. 1561–1565.

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Figures (5)

Fig. 1
Fig. 1

(a) 2D Radon transform p(ξ, ϕ0) is an integral of the 2D object function a(r) along a straight line, which is specified by two parameters, ξ (the distance of the line to the origin of the coordinate systems) and ϕ0 (the angle between the ξ and x axes). (b) According to the central-slice theorem, the 1D Fourier transform P(νa, ϕ0) of the Radon transform equals the 2D Fourier transform A(νx, νy) of the object function along a straight line with a slope of tan ϕ0 in the 2D Fourier space.

Fig. 2
Fig. 2

(a) In 2D diffraction tomography, one measures at η=l the scattered field from a scattering object that is illuminated by a plane wave propagating along the η axis. (b) In the Born (or the Rytov) approximation, the modified sinogram is related to the scattered field (or the unwrapped complex phase). According to the generalized central-slice theorem, the 1D Fourier transform M(νm, ϕ) of the modified sinogram equals the 2D Fourier transform A(νx, νy) of the object function along a semicircle AOB that has a radius of ν0 and a center at νη=-ν0.

Fig. 3
Fig. 3

Relationship between the spatial frequency νm of the modified sinogram and the corresponding spatial frequency νa of the ideal sinogram. For a given value of 0νa2ν0, there are two distinct values of νm as suggested in Eq. (19). The values of νm and νa are expressed in units of ν0, the frequency of the incident plane-wave radiation.

Fig. 4
Fig. 4

As the projection angle ϕ varies from 0 to 2π, the semicircle AOB rotates around the origin of the Fourier space of the object function, and two distinct coverages, (a) and (b), of the Fourier space are generated by the two segments OA and OB, respectively, of the semicircle AOB.

Fig. 5
Fig. 5

Complex plane for the combination coefficient ω. For uncorrelated or stationary data noise, the optimal combination coefficient ωop is a point on the vertical line that passes through the real axis at Re(ω)=12. The proposed ωk(n)(νm) is also a point on this vertical line for any value of n. Specifically, for n=0, ωk(n)(νm) is the crossing point between the vertical line and the real axis.

Equations (83)

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p(ξ, ϕ0)=--a(r)δ[ξ-r cos(ϕ0-θ)]dr,
P(νa, ϕ0)=-p(ξ, ϕ0)exp(-j2πνaξ)dξ=--a(r)exp[-j2πνar cos(ϕ0-θ)]dr,
2us+4π2ν02us=-4π2ν02a(r)(ui+us),
Us(νm, ϕ)=2π2ν02U0jνexp(j2πνl)--a(r)×exp{-j2π[νmξ+(ν-ν0)η]}drif|νm|ν00if|νm|>ν0,
ν=ν02-νm2.
2(uiψs)+4π2ν02(uiψs)=-4π2ν02a(r)ui.
Ψs(νm, ϕ)=2π2ν02U0jνexp[j2π(ν-ν0)l]×--a(r)×exp{-j2π[νmξ+(ν-ν0)η]}drif|νm|ν00if|νm|>ν0,
M(νm, ϕ)=--a(r)exp{-j2π[νmξ+(ν-ν0)η]}dr.
m(ξ, ϕ)=-M(νm, ϕ)exp(j2πνmξ)dνm,
Pk(νa)=02π-p(ξ, ϕ0)exp(-j2πνaξ)×exp(-jkϕ0)dξdϕ0,
Pk(νa)=(-j)kθ=02πr=0a(r)exp(-jkθ)Jk(2πνar)rdrdθ,
Pk(νa)=(-1)kPk(-νa)
Mk(νm)=12π02π-m(ξ, ϕ)exp(-j2πνmξ)×exp(-jkϕ)dξdϕ.
Mk(νm)=r=0θ=02πa(r)12πϕ=02π×exp{μr sin(ϕ-θ)-j[kϕ+2πνm×r cos(ϕ-θ)]}dϕrdrdθ,
Mk(νm)=(-j)kνm+νμ(νm2-νμ2)1/2k×r=0θ=02πa(r)exp(-jkθ)×Jk[2πr(νm2-νμ2)1/2]rdrdθifνm2νμ2,
νμ=μ2π=j[(ν02-νm2)1/2-ν0].
Pk(νa)=(νm2-νμ2)1/2νm+νμkMk(νm)if|νm|ν0,
νa2=νm2-νμ2.
νm1=-νm2=νm=νa(1-νa2/4ν02)1/2,
γ(νm)=(νm2-νμ2)1/2νm+νμ
γ(-νm)=-γ(νm)-1,γ(νm)*=γ(νm)-1.
|γ(νm)|2=γ(νm)γ(νm)*=γ(νm)γ(νm)-1=1,
γ(νm)=exp[jβ(νm)],
Pk(νa)=[γ(νm1)]kMk(νm1)=[γ(νm)]kMk(νm)
Pk(νa)=[γ(νm2)]kMk(νm2)=(-1)k[γ(νm)]-kMk(-νm)
M(νm, ϕ)=f(νm)S(νm, ϕ),
S(νm, ϕ)=Us(νm, ϕ)fortheBornapproximationΨs(νm, ϕ)fortheRytovapproximation,
f(νm)=jν2π2ν02U0exp(-j2πνl)fortheBornapproximationjν2π2ν02U0exp[-j2π(ν-ν0)l]fortheRytovapproximation.
covar{Mk(νm), Mk(νm)}=f(νm)f*(νm)4π2ϕ,ϕ=02πξ,ξ=-×covar{s(ξ, ϕ), s(ξ, ϕ)}×exp[-j(kϕ-kϕ)]×exp[-j2π(νmξ-νmξ)]dξdξdϕdϕ.
covar{Mk(νm), Mk(-νm)}=|f(νm)|24π2ϕ,ϕ=02πξ,ξ=-×covar{s(ξ, ϕ), s(ξ, ϕ)}×exp[-jk(ϕ-ϕ)]×exp[-j2πνm(ξ+ξ)]dξdξdϕdϕ,
τk+2(νm)=var{Mk(νm)}=covar{Mk(νm), Mk(νm)}=|f(νm)|24π2ϕ,ϕ=02πξ,ξ=-×covar{s(ξ, ϕ), s(ξ, ϕ)}×exp[-jk(ϕ-ϕ)]×exp[-j2πνm(ξ-ξ)]dξdξdϕdϕ,
τk-2(νm)=var{Mk(-νm)}=covar{Mk(-νm), Mk(-νm)}=|f(νm)|24π2ϕ,ϕ=02πξ,ξ=-×covar{s(ξ, ϕ), s(ξ, ϕ)}×exp[-jk(ϕ-ϕ)]×exp[j2πνm(ξ-ξ)]dξdξdϕdϕ.
covar{s(ξ, ϕ), s(ξ, ϕ)}=σ2(ξ, ϕ)δ(ξ-ξ)δ(ϕ-ϕ),
covar{Mk(νm), Mk(-νm)}=|f(νm)|24π2ϕ=02πξ=-σ2(ξ, ϕ)exp(-j4πνmξ)dξdϕ,
τk+2(νm)=τk-2(νm)=τ(νm)2=|f(νm)|24π2ϕ=02πξ=-σ2(ξ, ϕ)dξdϕ.
a(r)=2πk=-jkνa=0Pk(νa)exp(jkθ)×Jk(2πνar)νadνa.
θ=02πr=0var{a(r, θ)}rdrdθ=2πk=-νa=0var{Pk(νa)}νadνa.
Pk(νa)=ωγkMk(νm)+(1-ω)(-1)kγ-kMk(-νm),
var{Pk(νa)}=τk+(νm)τk-(νm)×{[t++t--2(-1)kρk(r)(νm)]×(R2+I2)-[2t--2(-1)kρk(r)(νm)]×R-(-1)kρk(i)(νm)I+t-},
t+=1/t-=τk+(νm)τk-(νm)
ρk(r)(νm)=Reγ2k covar{Mk(νm), Mk(-νm)}τk+(νm)τk-(νm)
ρk(i)(νm)=Imγ2k covar{Mk(νm), Mk(-νm)}τk+(νm)τk-(νm),
var{Pk(νa)}R=0andvar{Pk(νa)}I=0.
Rop=t--(-1)kρk(r)(νm)t++t--2(-1)kρk(r)(νm)
Iop=(-1)kρk(i)(νm)t++t--2(-1)kρk(r)(νm),
var{Pk(νa)}min=τk+(νm)τk-(νm)×1-|ρk(νm)|2t++t--2(-1)kρk(r)(νm),
2var{Pk(νa)}R22var{Pk(νa)}I22var{Pk(νa)}RI2
Rop=12andIop=12(-1)kρk(i)(νm)1-(-1)kρk(r)(νm)
var{Pk(νa)}min=τ2(νm)21-|ρk(νm)|21-(-1)kρk(r)(νm),
ω=ωk(n)(νm)γ-nkγ-nk+γnk,
[ωk(n)(νm)]*=ωk(n)(-νm)=1-ωk(n)(νm),
ωk(n)(νm)=12-j12tan[nkβ(νm)].
Pk(n)(νa)=ωk(n)(νm)γkMk(νm)+ωk(n)(-νm)(-1)k×γ-kMk(-νm)=γ-(n-1)kγ-nk+γnkMk(νm)+(-1)k γ(n-1)kγ-nk+γnkMk(-νm).
aFBPP(r)=12ϕ=02πνm=-ν0ν0 ν0ν|νm|M(νm, ϕ)×exp(j2πνmξ+2πνμη)dνmdϕ,
aFBPP(n)(r)=12ϕ=02πνm=-ν0ν0ν0ν|νm|M(n)(νm, ϕ)×exp(j2πνmξ+2πνμη)dνmdϕ,
M(n)(νm, ϕ)=k=-2ωk(n)(νm)Mk(νm)exp(jkϕ)
f(n)(νm, ϕ)k=-2ωk(n)(νm)exp(jkϕ).
a(n)(r)=2πk=-jkνa=0Pk(n)(νa)×exp(jkθ)Jk(2πνar)νadνa.
a(n)(r)=2πk=-jkνa=02ν0[ωk(n)(νm)γkMk(νm)+ωk(n)(-νm)(-1)kγ-kMk(-νm)]×exp(jkθ)Jk(2πνar)νadνa.
a(n)(r)=2πk=-jk exp(jkθ)×νm=0ν0 ν0ν[ωk(n)(νm)γkMk(νm)+ωk(n)(-νm)(-1)kγ-kMk(-νm)]×Jk[2π(νm2-νμ2)1/2r]νmdνm.
aFBPP(n)(r)=k=-ϕ=02πνm=-ν0ν0ν0ν|νm|ωk(n)(νm)Mk(νm)×exp(jkϕ+j2πνmξ+2πνμη)dνmdϕ=k=-ϕ=02πνm=0ν0ν0ννmωk(n)(νm)Mk(νm)×exp(jkϕ+j2πνmξ+2πνμη)dνmdϕ+k=-ϕ=02πνm=-ν00ν0ν(-νm)ωk(n)(νm)×Mk(νm)exp(jkϕ+j2πνmξ+2πνμη)dνmdϕ=2πk=-νm=0ν0ν0ννmωk(n)(νm)Mk(νm)×12πϕ=02π exp(jkϕ+j2πνmξ+2πνμη)dϕdνm+2πk=-νm=0ν0ν0ννmωk(n)(-νm)Mk(-νm)×12πϕ=02π exp(jkϕ-j2πνmξ+2πνμη)dϕdνm.
aFBPP(n)(r)=2πk=-jkνm=0ν0 ν0ν[ωk(n)(νm)γkMk(νm)+ωk(n)(-νm)(-1)kγ-kMk(-νm)]×exp(jkθ)Jk[2π(νm2-νμ2)1/2r]νmdνm.
a(n)(r)=aFBPP(n)(r).
P(νa, ϕ0)=--a(r)exp[-j2πνar cos(ϕ0-θ)]dr.
M(νm, ϕ)=--a(r)exp{-j2π[νmξ+(ν-ν0)η]}dr,
M(νm, ϕ)=--a(r)exp-j2π[νm2+(ν-ν0)2]1/2×νm[νm2+(ν-ν0)2]1/2ξ+ν-ν0[νm2+(ν-ν0)2]1/2ηdr.
cos ϕ=νm[νm2+(ν-ν0)2]1/2
sin ϕ=ν-ν0[νm2+(ν-ν0)2]1/2
M(νm, ϕ)=--a(r)×exp{-j2π[νm2+(ν-ν0)2]1/2r×[cos ϕ cos(ϕ-θ)-sin ϕ sin(ϕ-θ)]}dr=--a(r)×exp{-j2π[νm2+(ν-ν0)2]1/2r×cos(ϕ+ϕ-θ)}dr.
νa=[νm2+(ν-ν0)2]1/2andϕ0=ϕ+ϕ.
νm=νa(1-νa2/4ν02)1/2.
ϕ=ϕ0+arcsin(νa/2ν0)forνmϕ0-arcsin(νa/2ν0)-πfor-νm.
P(νa, ϕ0)=M(νm, ϕ),
P(νa, ϕ0)=M(-νm, ϕ),
Mk(νm)=12πϕ=02πM(νm, ϕ)exp(-jkϕ)dϕ.
Mk(νm)=exp[-jk arcsin(νa/2ν0)]×12πϕ0=02πP(νa, ϕ0)exp(-jkϕ0)dϕ0=exp[-jk arcsin(νa/2ν0)]Pk(νa).
exp[-j arcsin(νa/2ν0)]=cosarcsinνa2ν0-j sin[arcsin(νa/2ν0)]=1-νa24ν021/2-jνa2ν0.
[γ(νm)]-1=νm[νm2+(ν02-νm2-ν0)2]1/2-j ν0-(ν02-νm2)1/2[νm2+(ν02-νm2-ν0)2]1/2.
[γ(νm)]-1=1-νa24ν021/2-jνa2ν0.
Pk(νm)=[γ(νm)]kMk(νm),
Mk(-νm)=12πϕ=02πM(-νm, ϕ)exp(-jkϕ)dϕ.
Mk(-νm)=exp(-jkπ)expjk arcsinνa2ν0×12πϕ0=02πP(νa, ϕ0)exp(-jkϕ0)dϕ0=(-1)k exp[jk arcsin(νa/2ν0)]Pk(νa).
Pk(νm)=(-1)k[γ(νm)]-kMk(-νm),

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