Abstract

Previously it was shown that one can reconstruct an object from the modulus of its Fourier transform (solve the phase-retrieval problem) by using the iterative Fourier-transform algorithm if one has a nonnegativity constraint and a loose support constraint on the object. In this paper it is shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint. Sufficiently strong support constraints include certain special shapes and separated supports. Reconstruction results are shown, including the effect of tapered edges on the object’s support.

© 1987 Optical Society of America

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References

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  1. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  2. J. L. C. Sanz, T. S. Huang, F. Cukierman, “Stability of unique Fourier-transform phase reconstruction,”J. Opt. Soc. Am. 73, 1442–1445 (1983).
    [CrossRef]
  3. I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
    [CrossRef]
  4. I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
    [CrossRef]
  5. 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]
  6. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  7. 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]
  8. M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
    [CrossRef]
  9. J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  10. B. J. Brames, “Unique phase retrieval with explicit support information,” Opt. Lett. 11, 61–63 (1986).
    [CrossRef] [PubMed]
  11. T. R. Crimmins, “Phase retrieval for discrete functions with support constraints: summary,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986); “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987).
  12. 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]
  13. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  14. J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  15. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  16. J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986); “Improved phase-retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).
    [CrossRef]
  17. 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).
  18. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  19. J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Proceedings of URSI/IAU Symposium on Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, Cambridge, 1984), pp. 99–109.
  20. J. R. Fienup, “Phase retrieval: algorithm improvements, uniqueness, and complex objects,” Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 40–43.
  21. 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]
  22. R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
    [CrossRef]
  23. D. C. Youla, “Generalized image restoration by method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  24. 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]
  25. J. L. C. Sanz, T. S. Huang, T.-F. Wu, “A note on iterative Fourier transform phase reconstruction from magnitude,”IEEE Trans. Acoust, Speech Signal Process. ASSP-32, 1251–1254 (1984).
    [CrossRef]
  26. E. N. Leith, J. Upatnieks, “Reconstructed wavefronts and communication theory,”J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [CrossRef]
  27. J. W. Goodman, “Analogy between holography and interferometric image formation,”J. Opt. Soc. Am. 60, 506–509 (1970).
    [CrossRef]
  28. J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.
  29. J. R. Fienup, “Phase retrieval from a single intensity distribution,” in ICO-13, Optics in Modern Science and Technology (ICO-13 Organizing Committee, Sapporo, Japan, 1984), pp. 606–609.
  30. J. R. Fienup, “Phase retrieval using a support constraint,” presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va., October 28–30, 1985.

1986 (2)

1985 (2)

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[CrossRef]

1984 (2)

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]

J. L. C. Sanz, T. S. Huang, T.-F. Wu, “A note on iterative Fourier transform phase reconstruction from magnitude,”IEEE Trans. Acoust, Speech Signal Process. ASSP-32, 1251–1254 (1984).
[CrossRef]

1983 (4)

1982 (3)

1979 (2)

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

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

1978 (2)

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]

1974 (1)

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 35, 237–246 (1972); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

1970 (1)

1963 (1)

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

1962 (1)

Bates, R. H. T.

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

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]

Cederquist, J. N.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.

Crimmins, T. R.

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]

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]

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints: summary,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986); “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987).

Cukierman, F.

Dainty, J. C.

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[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]

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]

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986); “Improved phase-retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).
[CrossRef]

J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
[CrossRef]

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]

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

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]

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

J. R. Fienup, “Phase retrieval: algorithm improvements, uniqueness, and complex objects,” Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 40–43.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.

J. R. Fienup, “Phase retrieval from a single intensity distribution,” in ICO-13, Optics in Modern Science and Technology (ICO-13 Organizing Committee, Sapporo, Japan, 1984), pp. 606–609.

J. R. Fienup, “Phase retrieval using a support constraint,” presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va., October 28–30, 1985.

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

Goodman, J. W.

Holsztynski, W.

Huang, T. S.

J. L. C. Sanz, T. S. Huang, T.-F. Wu, “A note on iterative Fourier transform phase reconstruction from magnitude,”IEEE Trans. Acoust, Speech Signal Process. ASSP-32, 1251–1254 (1984).
[CrossRef]

J. L. C. Sanz, T. S. Huang, F. Cukierman, “Stability of unique Fourier-transform phase reconstruction,”J. Opt. Soc. Am. 73, 1442–1445 (1983).
[CrossRef]

Kryskowski, D.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.

Leith, E. N.

Levi, A.

Manolitsakis, I.

I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[CrossRef]

Robinson, S. R.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.

Sanz, J. L. C.

J. L. C. Sanz, T. S. Huang, T.-F. Wu, “A note on iterative Fourier transform phase reconstruction from magnitude,”IEEE Trans. Acoust, Speech Signal Process. ASSP-32, 1251–1254 (1984).
[CrossRef]

J. L. C. Sanz, T. S. Huang, F. Cukierman, “Stability of unique Fourier-transform phase reconstruction,”J. Opt. Soc. Am. 73, 1442–1445 (1983).
[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).

Sodin, L. G.

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

Stark, H.

Stefanescu, I. S.

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

Tan, D. G. H.

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Upatnieks, J.

Wackerman, C. C.

J. R. Fienup, C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986); “Improved phase-retrieval algorithm,” J. Opt. Soc. Am. A 1, 1320 (A) (1984).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.

Walther, A.

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

Wu, T.-F.

J. L. C. Sanz, T. S. Huang, T.-F. Wu, “A note on iterative Fourier transform phase reconstruction from magnitude,”IEEE Trans. Acoust, Speech Signal Process. ASSP-32, 1251–1254 (1984).
[CrossRef]

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)

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

J. L. C. Sanz, T. S. Huang, T.-F. Wu, “A note on iterative Fourier transform phase reconstruction from magnitude,”IEEE Trans. Acoust, Speech Signal Process. ASSP-32, 1251–1254 (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. Math. Phys. (2)

I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,”J. Math. Phys. 23, 2291–2298 (1982).
[CrossRef]

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

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

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

Opt. Commun. (2)

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

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (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 (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); W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Other (8)

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramer-Rao lower bound on Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), postdeadline paper.

J. R. Fienup, “Phase retrieval from a single intensity distribution,” in ICO-13, Optics in Modern Science and Technology (ICO-13 Organizing Committee, Sapporo, Japan, 1984), pp. 606–609.

J. R. Fienup, “Phase retrieval using a support constraint,” presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va., October 28–30, 1985.

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints: summary,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986); “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987).

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

J. R. Fienup, “Phase retrieval: algorithm improvements, uniqueness, and complex objects,” Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 40–43.

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]

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Examples of reconstructing complex-valued objects from the moduli of their Fourier transforms by using a support constraint. (A), (D), (G), and (J): moduli of the complex-valued objects, each having a different support; (B), (E), (H), and (K): the moduli of the images reconstructed by using the iterative Fourier-transform algorithm from the corresponding Fourier moduli, shown in (C), (F), (I), and (L), respectively.

Fig. 2
Fig. 2

Error E0 as a function of iteration number for reconstruction examples. Curve 1 corresponds to the case of Figs. 1(A), 1(B), and 1(C); curve 2 corresponds to Figs. 1(D), 1(E), and 1(F); curve 3 corresponds to Figs. 1(G), 1(H), and 1(I); curve 4 corresponds to Figs. 1(J), 1(K), and 1(L); curve 5 corresponds to the case of the object being just the larger ellipse in Fig. 1(D).

Fig. 3
Fig. 3

Examples of reconstructing objects from the moduli of their Fourier transforms by using a triangular support constraint. (A) Object with sharp edges and bright corners, (B) reconstructed image; (C) object with sharp edges and zeroed corners, (D) reconstructed image; (E) object with tapered edges and zeroed corners, (F) partially reconstructed image.

Equations (12)

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

F ( u , v ) = F ( u , v ) exp [ i ψ ( u , v ) ] = [ f ( x , y ) ] ,
g k + 1 ( x , y ) = { g k ( x , y ) , ( x , y ) S g k ( x , y ) - β g k ( x , y ) , ( x , y ) S ,
g k + 1 ( x , y ) = { g k ( x , y ) , ( x , y ) S 0 , ( x , y ) S .
E 0 k = [ ( x , y ) S g k ( x , y ) 2 ( x , y ) g k ( x , y ) 2 ] 1 / 2 .
E F k = { ( u , v ) [ G k ( u , v ) - F ( u , v ) 2 ] ( u , v ) F ( u , v ) 2 } 1 / 2 .
E F ( k + 1 ) E 0 k E F k E 0 ( k - 1 ) .
G k ( u , v ) = G k ( u , v ) [ F ( u , v ) / G k ( u , v ) ] .
G k ( u , v ) = G k ( u , v ) [ F ( u , v ) / ( G k ( u , v ) + δ ) ] ,
f ( x , y ) = A δ ( x - x 0 , y ) + f 1 ( x , y )
F ( u , v ) = A exp ( - i 2 π u x 0 ) + F 1 ( u , v ) ,
F ( u , v ) 2 = A exp ( - i 2 π u x 0 ) + F 1 ( u , v ) 2 = A 2 + F 1 ( u , v ) 2 + 2 A F 1 ( u , v ) cos [ 2 π u x 0 + ψ 1 ( u , v ) ] .
f ( x , y ) = f 0 ( x , y ) + f 1 ( x , y )

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