Abstract

It is shown that a priori knowledge of the edges of an object is not sufficient to ensure that it can be uniquely reconstructed from the modulus of its Fourier transform (or from its autocorrelation function). Furthermore, even in those cases for which the ultimate solution is unique, in intermediate steps in the solution by the recursive Hayes–Quatieri algorithm there can be ambiguities. An extension of the recursive algorithm that finds the solution (or solutions) is suggested, and it is shown that the recursive method can be applied to complex-valued objects.

© 1986 Optical Society of America

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References

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  1. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  2. C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
    [CrossRef]
  3. J. W. Goodman, “Analogy between holography and interferometric image formation,”J. Opt. Soc. Am. 60, 506–509 (1970).
    [CrossRef]
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978); R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).
  5. W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik 44, 45–59 (1975).
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982); “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978); “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef] [PubMed]
  7. P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. Ser. 15, 427–430 (1974).
  8. H. A. Arsenault, K. Chalasinka-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  9. G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).
  10. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,”J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  11. J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  12. M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,”J. Opt. Soc. Am. 73, 1427–1433 (1983).
    [CrossRef]
  13. M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.
  14. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  15. J. R. Fienup, “Image reconstruction for stellar interferometry,” in Current Trends in Optics, F. T. Arecchi, F. R. Aussenegg, eds. (Taylor and Francis, London, 1981), pp. 95–102.
  16. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]
  17. A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10–12 (1977); T. R. Crimmins, J. R. Fienup, “Ambiguity of phase retrieval for functions with disconnected supported,”J. Opt. Soc. Am. 71, 1026–1028 (1981).
    [CrossRef] [PubMed]
  18. T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for functions with sufficiently disconnected support,”J. Opt. Soc. Am. 73, 218–221 (1983).
    [CrossRef]
  19. J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Indirect Imaging, J. A. Roberts, ed., Conference Proceedings of URSI/IAU Symposium, Syndey, Australia, August 30–September 2, 1983 (Cambridge U. Press, Cambridge, 1984), pp. 99–109.

1983 (5)

1982 (2)

1979 (1)

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

1977 (1)

1975 (1)

W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik 44, 45–59 (1975).

1974 (1)

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. Ser. 15, 427–430 (1974).

1973 (1)

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978); R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

1970 (1)

Arsenault, H. A.

H. A. Arsenault, K. Chalasinka-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Bates, R. H. T.

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. Ser. 15, 427–430 (1974).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Brames, B. J.

Bruck, Yu. M.

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

Chalasinka-Macukow, K.

H. A. Arsenault, K. Chalasinka-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Crimmins, T. R.

Dainty, J. C.

Dallas, W. J.

W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik 44, 45–59 (1975).

Fiddy, M. A.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978); R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

Goodman, J. W.

Greenaway, A. H.

Hayes, M. H.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,”J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

Holsztynski, W.

Jensen, L. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).

Liu, C. Y. C.

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Lohmann, A. W.

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Napier, P. J.

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. Ser. 15, 427–430 (1974).

Quatieri, T. F.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,”J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978); R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

Sodin, L. G.

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

Stout, G. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).

Appl. Opt. (1)

Astron. Astrophys. Suppl. Ser. (1)

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. Ser. 15, 427–430 (1974).

J. Opt. Soc. Am. (5)

Opt. Commun. (3)

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

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

H. A. Arsenault, K. Chalasinka-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Opt. Lett. (2)

Optik (1)

W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik 44, 45–59 (1975).

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978); R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

Other (5)

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).

J. R. Fienup, “Image reconstruction for stellar interferometry,” in Current Trends in Optics, F. T. Arecchi, F. R. Aussenegg, eds. (Taylor and Francis, London, 1981), pp. 95–102.

M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Indirect Imaging, J. A. Roberts, ed., Conference Proceedings of URSI/IAU Symposium, Syndey, Australia, August 30–September 2, 1983 (Cambridge U. Press, Cambridge, 1984), pp. 99–109.

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Figures (4)

Fig. 1
Fig. 1

Example 1. Two different objects, (a) and (b), have the same boundary values and also have the same Fourier modulus (not shown) and the same autocorrelation (c).

Fig. 2
Fig. 2

Functions (a) and (b), which generate the object shown in Fig. 1(a) by cross correlation and in Fig. 1(b) by convolution. In (c) is the general form of the objects that have the autocorrelation shown in Fig. 1(c).

Fig. 3
Fig. 3

Example 2. An object (a), which is uniquely related to its autocorrelation function (b). For this example Eqs. (5) do not have a unique solution.

Fig. 4
Fig. 4

Example 3. A complex-valued object (a), which is uniquely related to its autocorrelation function (b). The right half of (b) (not shown) is the Hermitian conjugate of the left half (shown). For this example Eqs. (5) have a unique solution.

Equations (21)

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F ( p , q ) = F ( p , q ) exp [ i ψ ( p , q ) ] = F [ f ( m , n ) ] = m = 0 P - 1 n = 0 Q - 1 f ( m , n ) exp [ - i 2 π ( m p / P + n q / Q ) ] ,
r ( m , n ) = j = 0 M - 1 k = 0 N - 1 f ( j , k ) f * ( j - m , k - n )
= j = 0 M - 1 k = 0 N - 1 f * ( j , k ) f ( j + m , k + n )
= F - 1 [ F ( p , q ) 2 ] ,
r ( m , N - 2 ) = j = 0 M - 1 k = 0 N - 1 f * ( j , k ) f ( j + m , k + N - 2 ) = j = 0 M - 1 f * ( j , 0 ) f ( j + m , N - 2 ) + j = 0 M - 1 f * ( j , 1 ) f ( j + m , N - 1 ) = j = 0 M - 1 α * ( j ) f ( j + m , N - 2 ) + j = 0 M - 1 f * ( j , 1 ) β ( j + m )
27 = 12 + a + 2 f ,
52 = 12 + 2 a + b + 2 e + 3 f ,
77 = 12 + 2 a + 2 b + c + 2 d + 3 e + 3 f ,
88 = 13 + 2 a + 2 b + 2 c + 3 d + 3 e + 3 f ,
73 = 13 + a + 2 b + 2 c + 3 d + 3 e + f ,
48 = 13 + b + 2 c + 3 d + e ,
23 = 13 + c + d .
c = ( 15 - a + 2 b ) / 4 ,
d = ( 25 + a - 2 b ) / 4 ,
e = ( 35 - a - 2 b ) / 4 ,
f = ( 15 - a ) / 2.
52 = 12 + 2 a + 2 c + d + 3 f
100 = 12 + 4 b + 3 c + 2 d + 4 e + a f .
a = 15 - 2 b ,
b 2 - 10 b + 24 = 0
b = 4 or 6.

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