Abstract

The iterative Fourier transform algorithm, although it has been demonstrated to be a practical phase retrieval algorithm, suffers from certain stagnation problems. Specifically, there exists a stripe stagnation problem, in which stagnated reconstructed images exhibit stripelike features throughout the image, which is particularly difficult to overcome. Previous solutions to this problem used multiple reconstructions and did not address the cause. In this paper a new procedure that uses only a single image is developed that estimates the locations of real-plane zeros from either the measured Fourier modulus data or a stagnated reconstruction and uses this information in the iterative Fourier transform algorithm to force the complex-valued Fourier data to have real-plane zeros at the correct locations. It is shown that this procedure overcomes the stripe stagnation.

© 1991 Optical Society of America

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References

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  1. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  3. G. H. Stout, L. H. Jenson, X-Ray Structure Determination (Macmillan, London, 1968).
  4. H. Stark, ed., Image Recovery: Theory and Application (Academic, San Diego, Calif., 1987).
  5. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer Academic, Dordrecht, The Netherlands, 1989).
    [CrossRef]
  6. P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).
  7. B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).
  8. J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).
  9. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “II: One-dimensional considerations,” Optik (Stuttgart) 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219–230 (1982).
  10. H. H. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukow, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase-retrieval problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
    [CrossRef]
  11. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  12. 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]
  13. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  14. 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).
  15. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  16. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  17. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal ProcessingW. T. Rhodes, J. R. Fienup, B. E. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]
  18. D. C. Youla, “Generalized image restoration by method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  19. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  20. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” presented at the Topical Meeting on Signal Recovery and Synthesis III, North Falmouth, Mass., June 1989.

1986 (1)

1985 (2)

1984 (1)

1983 (1)

H. H. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukow, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase-retrieval problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

1982 (2)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “II: One-dimensional considerations,” Optik (Stuttgart) 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219–230 (1982).

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

1978 (3)

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

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

D. C. Youla, “Generalized image restoration by method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

1976 (1)

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

1974 (2)

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

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[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).

Arsenault, H. H.

H. H. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukow, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase-retrieval problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

Baldwin, J. E.

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “II: One-dimensional considerations,” Optik (Stuttgart) 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219–230 (1982).

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

Chalasinska-Macukow, K.

H. H. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukow, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase-retrieval problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

Currie, D. G.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

Deighton, H. V.

Fiddy, M. A.

Fienup, J. R.

Frieden, B. R.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

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).

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer Academic, Dordrecht, The Netherlands, 1989).
[CrossRef]

Jenson, L. H.

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

Levi, A.

Napier, P. J.

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

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).

Scivier, M. S.

Stark, H.

Stout, G. H.

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

Wackerman, C. C.

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” presented at the Topical Meeting on Signal Recovery and Synthesis III, North Falmouth, Mass., June 1989.

Warner, P. J.

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

Yagle, A. E.

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” presented at the Topical Meeting on Signal Recovery and Synthesis III, North Falmouth, Mass., June 1989.

Youla, D. C.

D. C. Youla, “Generalized image restoration by method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. Suppl. (1)

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

IEEE Trans. Circuits Syst. (1)

D. C. Youla, “Generalized image restoration by method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

J. Opt. Soc. Am. A (3)

Mon. Not. R. Astron. Soc. (1)

J. E. Baldwin, P. J. Warner, “Phaseless aperture synthesis,” Mon. Not. R. Astron. Soc. 182, 411–422 (1978).

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Commun. (1)

H. H. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukow, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase-retrieval problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (2)

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).

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “II: One-dimensional considerations,” Optik (Stuttgart) 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219–230 (1982).

Other (6)

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

H. Stark, ed., Image Recovery: Theory and Application (Academic, San Diego, Calif., 1987).

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer Academic, Dordrecht, The Netherlands, 1989).
[CrossRef]

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal ProcessingW. T. Rhodes, J. R. Fienup, B. E. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” presented at the Topical Meeting on Signal Recovery and Synthesis III, North Falmouth, Mass., June 1989.

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Figures (6)

Fig. 1
Fig. 1

Flow diagram for the iterative Fourier transform algorithm.

Fig. 2
Fig. 2

Examples of stripe stagnation. The upper image is the original object; the bottom four images show examples of stripe stagnation reconstructions. Error metrics indicate the degree of disagreement between the reconstructions and the constraints.

Fig. 3
Fig. 3

Example of the causes of stripe stagnation. (A) Fourier phases of original object. (B) Upsampled phases from within the white square in (A). (C) Fourier phases from stagnated reconstruction. (D) Upsampled phases from (C). The small squares in (B) and (D) indicate shifted real-plane zeros.

Fig. 4
Fig. 4

Superposition of the peaks of W(u, υ) (bright dots) with the zero curves of the original object; the dark curves indicate real-part zero lines, and the bright curves indicate imaginary-part zero lines. Note that peaks of W(u, υ) lie at the intersection points, i.e., the location of zeros.

Fig. 5
Fig. 5

Example of a Fourier phase mask constructed from estimates of the phase values around a real-plane zero (small boxes) and phase values taken from the stagnated reconstruction. The gray locations around the small boxes indicate regions where no Fourier phase is specified.

Fig. 6
Fig. 6

Example of creating real-plane zeros with the use of Fourier phase masks in the iterative algorithm. Part (a) is a small portion of Fourier phases for the original object, and part (b) displays the same phases for a stagnated reconstruction; the small white squares indicate the location of real-plane zeros not present in the reconstruction. Part (c) presents the same phases after correction for the real-plane zero errors with the algorithm developed in the text.

Equations (18)

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F ( u , υ ) = | F ( u , υ ) | exp [ i P ( u , υ ) ] = f ( x , y ) exp [ i 2 π ( u x + υ y ) ] d x d y
F ( m , n ) = j = 0 N 1 k = 0 N 1 f ( j , k ) exp [ i 2 π ( m j + n k ) N ]
f ( j , k ) = 1 N 2 m = 0 N 1 n = 0 N 1 F ( m , n ) exp [ i 2 π ( m j + n k ) N ] ,
E F 2 = 1 N 2 m = 0 N 1 n = 0 N 1 [ | G ( m , n ) | | F ( m , n ) | ] 2 ,
E 0 2 = ( j , k ) V | g ( j , k ) | 2 j = 0 N 1 k = 0 N 1 | g ( j , k ) | 2 ,
E ( m , n ) = ABS [ | F ( m , n ) | | G ( m , n ) | ]
W ( m , n ) = | F ( m , n ) | m m | F ( m , n ) | n n | F ( m , n ) | 2
F r ( m , n ) = a r m + b r n ,
F i ( m , n ) = a i m + b i n ,
[ F r F i ] = [ a r b r a i b i ] [ m n ] .
[ F r F i ] = [ cos ( a ) sin ( a ) sin ( a ) ± cos ( a ) ] [ s 1 0 0 s 2 ] × [ cos ( b ) sin ( b ) sin ( b ) ± cos ( b ) ] [ m n ] ,
F ( m , n ) = e i a ( s 1 m ± i s 2 n ) ,
| F ( m , n ) | 2 = ( a r 2 + a i 2 ) m 2 + ( b r 2 + b i 2 ) n 2 + 2 ( a r b r + a i b i ) m n .
C 1 = a r 2 + a i 2 ,
C 2 = b r 2 + b i 2 ,
C 3 = 2 ( a r b r + a i b i ) .
λ = { ( C 1 + C 2 ) ± [ ( C 1 C 2 ) 2 + C 3 ] 1 / 2 } / 2 ,
A t A = [ C 1 C 3 / 2 C 3 / 2 C 2 ] ,

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