Abstract

Uniqueness properties of phase problems in three or more dimensions are investigated. It is shown that an N-dimensional image is overdetermined by its continuous Fourier amplitude. A sampling of the Fourier amplitude, with a density approximately 22−N times the Nyquist density, is derived that is sufficient to uniquely determine an N-dimensional image. Both continuous and discrete images are considered. Practical implications for phase retrieval in multidimensional imaging, particularly in crystallography where the amplitude data are undersampled, are described. Simulations of phase retrieval for two- and three-dimensional images illustrate the practical implications of the theoretical results.

© 1996 Optical Society of America

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References

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  1. R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).
  2. N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer, Dordrecht, The Netherlands, 1989).
  3. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  4. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.
  5. R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
    [CrossRef]
  6. G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).
  7. J. Drenth, Principles of Protein X-Ray Crystallography (Springer-Verlag, New York, 1994).
    [CrossRef]
  8. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  9. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  10. R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Computer Vision Graphics Image Process. 25, 205–217 (1984).
    [CrossRef]
  11. R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
    [CrossRef]
  12. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
    [CrossRef]
  13. D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [CrossRef]
  14. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  15. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  16. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), Chap. 8, pp. 277–320.
  17. B. C. McCallum, J. M. Rodenburg, “Simultaneous reconstruction of object and aperture functions from multiple far-field intensity measurements,” J. Opt. Soc. Am. A 10, 231–239 (1993).
    [CrossRef]
  18. R. G. Paxman, J. R. Fienup, J. H. Seldin, J. C. Marron, “Phase retrieval with an opacity constraint,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 109–111.
  19. N. K. Bose, ed., feature on multidimensional signal processing, Proc. IEEE78(4) (1990).
    [CrossRef]
  20. R. P. Millane, “Redundancy in multidimensional deconvolution and phase retrieval,” In Digital Image Synthesis and Inverse Optics. A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 227–236 (1990).
    [CrossRef]
  21. R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
    [CrossRef]
  22. P. J. Bones, C. R. Parker, B. L. Satherley, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995).
    [CrossRef]
  23. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  24. R. P. Millane, “Phase problems for periodic images: effects of support and symmetry,” J. Opt. Soc. Am. A 10, 1037–1045 (1993).
    [CrossRef]
  25. R. P. Millane, “Properties and implications of phase problems for multidimensional images,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 115–117.
  26. R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
    [CrossRef]
  27. R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Computing in Crystallography 4, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.
  28. H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
    [CrossRef] [PubMed]
  29. H. V. Deighton, M. S. Scivier, H. M. Berenyi, M. A. Fiddy, “Practical phase retrieval based on theoretical models for multi-dimensional band-limited signals,” in Inverse Optics II, A. J. Devancy, R. H. T. Bates, eds., Proc. SPIE558, 65–72 (1985).
    [CrossRef]

1995 (1)

1993 (2)

1990 (2)

1987 (2)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

1985 (1)

1984 (2)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Computer Vision Graphics Image Process. 25, 205–217 (1984).
[CrossRef]

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

1979 (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1978 (1)

1963 (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Barakat, R.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Computer Vision Graphics Image Process. 25, 205–217 (1984).
[CrossRef]

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
[CrossRef]

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

Berenyi, H. M.

H. V. Deighton, M. S. Scivier, H. M. Berenyi, M. A. Fiddy, “Practical phase retrieval based on theoretical models for multi-dimensional band-limited signals,” in Inverse Optics II, A. J. Devancy, R. H. T. Bates, eds., Proc. SPIE558, 65–72 (1985).
[CrossRef]

Bones, P. J.

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.

Deighton, H. V.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
[CrossRef] [PubMed]

H. V. Deighton, M. S. Scivier, H. M. Berenyi, M. A. Fiddy, “Practical phase retrieval based on theoretical models for multi-dimensional band-limited signals,” in Inverse Optics II, A. J. Devancy, R. H. T. Bates, eds., Proc. SPIE558, 65–72 (1985).
[CrossRef]

Drenth, J.

J. Drenth, Principles of Protein X-Ray Crystallography (Springer-Verlag, New York, 1994).
[CrossRef]

Fiddy, M. A.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
[CrossRef] [PubMed]

H. V. Deighton, M. S. Scivier, H. M. Berenyi, M. A. Fiddy, “Practical phase retrieval based on theoretical models for multi-dimensional band-limited signals,” in Inverse Optics II, A. J. Devancy, R. H. T. Bates, eds., Proc. SPIE558, 65–72 (1985).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

R. G. Paxman, J. R. Fienup, J. H. Seldin, J. C. Marron, “Phase retrieval with an opacity constraint,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 109–111.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Hayes, M. H.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer, Dordrecht, The Netherlands, 1989).

Izraelevitz, D.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Jensen, L. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

Lane, R. G.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), Chap. 8, pp. 277–320.

Lim, J. S.

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Marron, J. C.

R. G. Paxman, J. R. Fienup, J. H. Seldin, J. C. Marron, “Phase retrieval with an opacity constraint,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 109–111.

McCallum, B. C.

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

Millane, R. P.

R. P. Millane, “Phase problems for periodic images: effects of support and symmetry,” J. Opt. Soc. Am. A 10, 1037–1045 (1993).
[CrossRef]

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

R. P. Millane, “Redundancy in multidimensional deconvolution and phase retrieval,” In Digital Image Synthesis and Inverse Optics. A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 227–236 (1990).
[CrossRef]

R. P. Millane, “Properties and implications of phase problems for multidimensional images,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 115–117.

R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
[CrossRef]

R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Computing in Crystallography 4, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.

Mnyama, D.

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
[CrossRef]

Newsam, G.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Parker, C. R.

Paxman, R. G.

R. G. Paxman, J. R. Fienup, J. H. Seldin, J. C. Marron, “Phase retrieval with an opacity constraint,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 109–111.

Quek, B. K.

Rodenburg, J. M.

Satherley, B. L.

Scivier, M. S.

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
[CrossRef] [PubMed]

H. V. Deighton, M. S. Scivier, H. M. Berenyi, M. A. Fiddy, “Practical phase retrieval based on theoretical models for multi-dimensional band-limited signals,” in Inverse Optics II, A. J. Devancy, R. H. T. Bates, eds., Proc. SPIE558, 65–72 (1985).
[CrossRef]

Seldin, J. H.

R. G. Paxman, J. R. Fienup, J. H. Seldin, J. C. Marron, “Phase retrieval with an opacity constraint,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 109–111.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Stark, H.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), Chap. 8, pp. 277–320.

Stout, G. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Watson, R. W.

Appl. Opt. (1)

Computer Vision Graphics Image Process. (1)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Computer Vision Graphics Image Process. 25, 205–217 (1984).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (3)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987).
[CrossRef]

D. Izraelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

J. Math. Phys. (1)

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Opt. Commun. (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Opt. Lett. (2)

Other (14)

H. V. Deighton, M. S. Scivier, H. M. Berenyi, M. A. Fiddy, “Practical phase retrieval based on theoretical models for multi-dimensional band-limited signals,” in Inverse Optics II, A. J. Devancy, R. H. T. Bates, eds., Proc. SPIE558, 65–72 (1985).
[CrossRef]

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), Chap. 8, pp. 277–320.

R. P. Millane, “Properties and implications of phase problems for multidimensional images,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 115–117.

R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
[CrossRef]

R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Computing in Crystallography 4, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.

R. G. Paxman, J. R. Fienup, J. H. Seldin, J. C. Marron, “Phase retrieval with an opacity constraint,” in Signal Recovery and Synthesis, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 109–111.

N. K. Bose, ed., feature on multidimensional signal processing, Proc. IEEE78(4) (1990).
[CrossRef]

R. P. Millane, “Redundancy in multidimensional deconvolution and phase retrieval,” In Digital Image Synthesis and Inverse Optics. A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 227–236 (1990).
[CrossRef]

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer, Dordrecht, The Netherlands, 1989).

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.

R. H. T. Bates, D. Mnyama, “The status of practical Fourier phase retrieval,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, Orlando, Fla., 1986), Vol. 67, pp. 1–64.
[CrossRef]

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

J. Drenth, Principles of Protein X-Ray Crystallography (Springer-Verlag, New York, 1994).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Error versus iteration number for reconstructions of a three-dimensional image, for cases (a) 1, (b) 2, (c) 3, (d) 4, as described in the text.

Fig. 2
Fig. 2

Error versus iteration number for reconstructions of a two-dimensional image, for cases (a) 1, (b) 3, and (c) 4, as described in the text.

Fig. 3
Fig. 3

(a) Original and (b)–(e) reconstructed three-dimensional images for cases (b) 1, (c) 2, (d) 3, and (e) 4, as described in the text. Each row in the figure shows the four planes of the three-dimensional image.

Fig. 4
Fig. 4

(a) Original and (b)–(d) reconstructed two-dimensional images for cases (b) 1, (c) 3, and (d) 4, as described in the text.

Tables (1)

Tables Icon

Table 1 Amplitude Data Used for Reconstruction of M × N and M × N × P Images

Equations (33)

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S [ f ( x , y ) ] = ( - a / 2 , a / 2 ) × ( - b / 2 , b / 2 ) .
F ( u , v ) = - f ( x , y ) exp [ i 2 π ( u x + v y ) ] d x d y .
F ( u , v ) = F ( u , v ) exp { i P [ F ( u , v ) ] } ,
f ( x , y ) = - F ( u , v ) exp [ - i 2 π ( u x + v y ) ] d u d v .
F ( u , v ) 2 A ( x , y ) = - f ( x , y ) f * ( x + x , y + y ) × d x d y ,
S [ A ( x , y ) ] = ( - a , a ) × ( - b , b )
f ^ ( x , y ) = f ( s x - α , s y - β ) exp ( i ϕ )
F ^ ( u , v ) = F ( u , v ) exp [ i ( ϕ + α u + β v ) ]             or             F * ( u , v ) exp [ i ( ϕ + α u + β v ) ] ,
F ^ ( u , v ) = F ( u , v ) exp ( i { k P [ F ( u , v ) ] + ϕ + α u + β v } ) ,
S [ f ( x , y , z ) ] = ( - a / 2 , a / 2 ) × ( - b / 2 , b / 2 ) × ( - c / 2 , c / 2 ) ,
f p ( x , y ) = - f ( x , y , z ) exp ( i 2 π p z / c ) d z .
S [ f p ( x , y ) ] = ( - a / 2 , a / 2 ) × ( - b / 2 , b / 2 ) .
f ( x , y , z ) = ( 1 / c ) p = - f p ( x , y ) exp ( - i 2 π p z / c ) .
f ^ p ( x , y ) = f p ( s p x - α p , s p y - β p ) exp ( i ϕ p )
F ^ p ( u , v ) = F ( u , v , p / c ) exp ( i { k p P [ F ( u , v , p / c ) ] + ϕ p + α p u + β p v } ) ,
F ( 0 , v , w ) exp ( i { k P [ F ( 0 , v , w ) ] + ϕ + β v + γ w } ) .
F ˜ ( u , 0 , w ) = F ( u , 0 , w ) exp ( i { k P [ F ( u , 0 , w ) ] + ϕ + α u + γ w } )
F ˜ ( 0 , 0 , w ) = F ( 0 , 0 , w ) exp ( i { k P [ F ( 0 , 0 , w ) ] + ϕ + γ w } ) ,
F ˜ p ( 0 , v ) = F ( 0 , v , p / c ) exp ( i { k p P [ F ( 0 , v , p / c ) ] + ϕ p + β p v } ) ,
F ˜ p ( u , 0 ) = F ( u , 0 , p / c ) exp ( i { k p P [ F ( u , 0 , p / c ) ] + ϕ p + α p u } ) ,
T 3 = { ( u , v , p / c ) , ( 0 , v , w ) , ( u , 0 , w ) ; u , v , w R , p I }
S 3 = { ( m / 2 a , n / 2 b , p / c ) , ( 0 , n / 2 b , p / 2 c ) , ( m / 2 a , 0 , p / 2 c ) ; m , n , p I } .
T 3 = { ( u , v , p / c ) , ( 0 , v , w ) ; u , v , w R , p I } ,
T 3 = { ( u , v , p / c ) ; u , v R , p I } ,
{ ( u , v , p / c , q / d ) , ( 0 , v , w , q / d ) , ( u , 0 , w , q / d ) } ,
T 4 = { ( u , v , p / c , q / d ) , ( 0 , v , w , q / d ) , ( u , 0 , w , q / d ) , ( 0 , 0 , w , t ) , ( 0 , v , 0 , t ) , ( u , 0 , 0 , t ) ; u , v , w , t R , p , q I }
T 4 = { ( u , v , p / c , q / d ) , ( 0 , v , w , q / d ) , ( 0 , 0 , w , t ) ; u , v , w , t R , p , q I } .
T 4 = { ( u , v , p / c , q / d ) ; u , v , R , p , q I } ,
F [ q , r ] = m = 0 2 M - 1 n = 0 2 N - 1 f [ m , n ] exp [ i 2 π ( m q / 2 M + n r / 2 N ) ] .
f [ m , n ] = 1 4 M N q = 0 2 M - 1 r = 0 2 N - 1 F [ q , r ] exp [ - i 2 π ( m q / 2 M + n r / 2 N ) ] .
f [ m , n , p ] = 1 8 M N P q = 0 M - 1 r = 0 N - 1 s = 0 P - 1 F [ 2 q , 2 r , 2 s ] × exp [ - i 2 π ( m q / M + n r / N + p s / P ) ]
{ [ q , r , 2 s ] , [ 0 , r , s ] , [ q , 0 , s ] ; q = 0 , 1 , , 2 M - 1 , r = 0 , 1 , , 2 N - 1 , s = 0 , 1 , , P - 1 , s = 0 , 1 , , 2 P - 1 }
I p ( R ) = 0 2 π F ( R , ψ , p / c ) 2 d ψ ,

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