Abstract

The iterative Fourier-transform algorithm has been demonstrated to be a practical method for reconstructing an object from the modulus of its Fourier transform (i.e., solving the problem of recovering phase from a single intensity measurement). In some circumstances the algorithm may stagnate. New methods are described that allow the algorithm to overcome three different modes of stagnation: those characterized by (1) twin images, (2) stripes, and (3) truncation of the image by the support constraint. Curious properties of Fourier transforms of images are also described: the zero reversal for the striped images and the relationship between the zero lines of the real and imaginary parts of the Fourier transform. A detailed description of the reconstruction method is given to aid those employing the iterative transform algorithm.

© 1986 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, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  4. G. H. Stout, L. H. Jenson, X-Ray Structure Determination (Macmillan, London, 1968).
  5. C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
    [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 61, 247–262 (1982); K. L. Garden, R. H. T. Bates, “II: One-dimensional considerations,” Optik 62, 131–142 (1982); W. R. Fright, R. H. T. Bates, “III: Computational examples for two dimension,” Optik 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. 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. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  14. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  15. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  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. G. B. Feldkamp, J. R. Fienup, “Noise properties of images reconstructed from Fourier modulus,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 84–93 (1980).
    [CrossRef]
  18. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]
  19. D. C. Youla, “Generalized image restoration by method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  20. 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]
  21. J. R. Fienup, “Phase retrieval from a single intensity distribution,” in Optics in Modern Science and Technology, proceedings of the International Commission for Optics-13, August 20–24, 1984, Sapporo, Japan (Optics-13 Conference Committee, Sapporo, Japan, 1984).
  22. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multi-dimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  23. J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 95–102 (1980).
    [CrossRef]
  24. R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
    [CrossRef]
  25. R. J. Sault, “Two procedures for phase estimation from visibility magnitudes,” Aust. J. Phys. 37, 209–229 (1984).
    [CrossRef]
  26. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
    [CrossRef]
  27. P. vanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
    [CrossRef]
  28. J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Indirect Imaging, Proceedings of URSI/IAU Symposium (Cambridge U., Cambridge, 1984), pp. 99–109.
  29. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978), pp. 271–272.
  30. J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).

1985 (2)

1984 (4)

R. J. Sault, “Two procedures for phase estimation from visibility magnitudes,” Aust. J. Phys. 37, 209–229 (1984).
[CrossRef]

P. vanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
[CrossRef]

J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).

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]

1983 (3)

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. 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]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

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]

1982 (3)

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

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

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]

1979 (1)

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

1978 (3)

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

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

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 (3)

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]

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

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 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. 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, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978), pp. 271–272.

Brames, B. J.

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. 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]

Crimmins, T. R.

Currie, D. G.

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

Dainty, J. C.

Deighton, H. V.

Feldkamp, G. B.

G. B. Feldkamp, J. R. Fienup, “Noise properties of images reconstructed from Fourier modulus,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 84–93 (1980).
[CrossRef]

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 95–102 (1980).
[CrossRef]

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).

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

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]

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

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

J. R. Fienup, “Phase retrieval from a single intensity distribution,” in Optics in Modern Science and Technology, proceedings of the International Commission for Optics-13, August 20–24, 1984, Sapporo, Japan (Optics-13 Conference Committee, Sapporo, Japan, 1984).

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 95–102 (1980).
[CrossRef]

G. B. Feldkamp, J. R. Fienup, “Noise properties of images reconstructed from Fourier modulus,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 84–93 (1980).
[CrossRef]

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

J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Indirect Imaging, Proceedings of URSI/IAU Symposium (Cambridge U., Cambridge, 1984), pp. 99–109.

Frieden, B. R.

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

Fright, W. R.

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 35, 237–246 (1972).

Greenaway, A. H.

P. vanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
[CrossRef]

Holsztynski, W.

Huiser, A. M. J.

P. vanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
[CrossRef]

Jenson, L. H.

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

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Levi, A.

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. 15, 427–430 (1974).

Sault, R. J.

R. J. Sault, “Two procedures for phase estimation from visibility magnitudes,” Aust. J. Phys. 37, 209–229 (1984).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 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).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

vanToorn, P.

P. vanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
[CrossRef]

Wackerman, C. C.

J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).

Warner, P. J.

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

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).

Astrophys. J. Lett. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Aust. J. Phys. (1)

R. J. Sault, “Two procedures for phase estimation from visibility magnitudes,” Aust. J. Phys. 37, 209–229 (1984).
[CrossRef]

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. (3)

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 (2)

P. vanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
[CrossRef]

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

Opt. Commun. (2)

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

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. 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. Eng. (1)

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Opt. Lett. (3)

Optik (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

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

Other (8)

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

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

G. B. Feldkamp, J. R. Fienup, “Noise properties of images reconstructed from Fourier modulus,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 84–93 (1980).
[CrossRef]

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

J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Indirect Imaging, Proceedings of URSI/IAU Symposium (Cambridge U., Cambridge, 1984), pp. 99–109.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978), pp. 271–272.

J. R. Fienup, “Phase retrieval from a single intensity distribution,” in Optics in Modern Science and Technology, proceedings of the International Commission for Optics-13, August 20–24, 1984, Sapporo, Japan (Optics-13 Conference Committee, Sapporo, Japan, 1984).

J. R. Fienup, G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.243, 95–102 (1980).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Block diagram of the iterative transform algorithm.

Fig. 2
Fig. 2

Simultaneous twin-images problem. (A) Object f(x). (B) Twin image f*(−x). (C) Output image from the iterative transform algorithm that has stagnated with features of both.

Fig. 3
Fig. 3

Reduced-area support constraint method for overcoming the problem of simultaneous twin images. (A) Stagnated output image. (B) Mask defining the temporary reduced-area support constraint. (C) Output image after 10 iterations using temporary support. (D)–(F) Output image after further iterations using the correct support.

Fig. 4
Fig. 4

Voting method for eliminating stripes in the output image. (A) The object. (B)–(D) Output images from the iterative transform algorithm, each with different stripes. (E) Output of voting method. (F) Output image after further iterations.

Fig. 5
Fig. 5

Patching method for eliminating stripes in the output image. (A) The object. (B), (C) Output images from the iterative transform algorithm, each with different stripes. (D) Output of patching method.

Fig. 6
Fig. 6

Same as Fig. 5 but overexposed to emphasize the stripes. and full 128 × 128 arrays are shown.

Fig. 7
Fig. 7

Details of the patching method. (A) Mask used to isolate the stripes of the output images. (B) Stripes isolated from the first image. (C) Modulus of the Fourier transform of the isolated stripes from the first image. (D) Fourier mask obtained by thresholding and smoothing (C). (E)–(G) Same as (B)-(D) but for the second image.

Fig. 8
Fig. 8

Fourier phases. (A) Fourier phase of the object. (B) Up-samples phase from the area in (A) outlined by the square. (C) Fourier phase of the striped output image. (D) Upsampled phase from the area in (C) outlined by the square. The (u, υ) zeros of the complex Fourier transforms are reversed in the areas enclosed in squares in (B) and (D).

Fig. 9
Fig. 9

Locations of the zeros of the real part (dark lines) and the imaginary part (white lines) of the Fourier transform of the object. The object was translated to be causal, and the area of its Fourier transform shown here is that shown in Fig. 8(B).

Equations (22)

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

F ( u , υ ) = | F ( u , υ ) | exp [ i ψ ( u , υ ) ] = F [ f ( x , y ) ] = f ( x , y ) exp [ i 2 π ( u x + υ y ) ] d x d y
F ( u ) = x = 0 N 1 f ( x ) exp ( i 2 π u x / N )
f ( x ) = N 2 u = 0 N 1 F ( u ) exp ( i 2 π u x / N ) ,
G k ( u ) = | G k ( u ) | exp [ i ϕ k ( u ) ] = F [ g k ( x ) ] ,
G k ( u ) = | F ( u ) | exp [ i ϕ k ( u ) ] ,
g k ( x ) = F 1 [ G k ( u ) ] ,
g k + 1 ( x ) = { g k ( x ) x γ 0 , x γ ,
g k + 1 ( x ) = | f ( x ) | exp [ i θ k + 1 ( x ) ] = | f ( x ) | exp [ i θ k ( x ) ] ,
E F 2 = N 2 u [ | G ( u ) | | F ( u ) | ] 2 ,
E 0 2 = x γ | g k ( x ) | 2 ,
g k + 1 ( x ) = { g k ( x ) x γ g k ( x ) β g k ( x ) x γ ,
F I ( u , υ ) = 1 π P F R ( u , υ ) υ υ d υ
= 1 π P F R ( u , υ ) u u d u ,
G k ( u ) = G k ( u ) | F ( u ) | / [ | G k ( u ) | + δ ] ,
E | F ˆ | = [ u [ | F ˆ ( u ) | | F ( u ) | ] 2 u [ | F ( u ) | ] 2 ] 1 / 2 .
E 0 = [ x γ | g k ( x ) | 2 x | g k ( x ) | 2 ] 1 / 2
F ( u , υ ) = F R ( u , υ ) + i F I ( u , υ )
= f ( x , y ) exp [ i 2 π ( u x + υ y ) ] d x d y = [ f ( x , y ) exp ( i 2 π u x ) d x ] exp ( i 2 π υ y ) d y = f ˜ ( u , y ) exp ( i 2 π υ y ) d y ,
F I ( u , υ ) = 1 π P F R ( u , υ ) υ υ d υ
F R ( u , υ ) = 1 π P F I ( u , υ ) υ υ d υ ,
F I ( u , υ ) = 1 π P F R ( u , υ ) u u d u
F R ( u , υ ) = 1 π P F I ( u , υ ) u u d u .

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