Abstract

Both a new iterative grid-search technique and the iterative Fourier-transform algorithm are used to illuminate the relationships among the ambiguous images nearest a given object, error metric minima, and stagnation points of phase-retrieval algorithms. Analytic expressions for the subspace of ambiguous solutions to the phase-retrieval problem are derived for 2 × 2 and 3 × 2 objects. Monte Carlo digital experiments using a reduced-gradient search of these subspaces are used to estimate the probability that the worst-case nearest ambiguous image to a given object has a Fourier modulus error of less than a prescribed amount. Probability distributions for nearest ambiguities are estimated for different object-domain constraints.

© 1990 Optical Society of America

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References

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  1. E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence,” Proc. Phys. Soc. (London) 80, 1269–1272 (1962).
    [CrossRef]
  2. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  3. E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
    [CrossRef]
  4. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  5. 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).
  6. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  7. G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).
  8. W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 94–98 (1980).
    [CrossRef]
  9. M. Nieto-Vesperinas, “Dispersion relations in two dimensions: application to the phase problem,” Optik (Stuttgart) 56, 377–384 (1980).
  10. I. Manolitsakis, “Two-dimensional scattered fields: a description in terms of the zeros of entire functions,” J. Math. Phys. 23, 2291–2298 (1982).
    [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. J. L. C. Sanz, T. S. Huang, “Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude,” J. Opt. Soc. Am. 73, 1446–1450 (1983).
    [CrossRef]
  13. I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985).
    [CrossRef]
  14. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  15. L. Carlitz, “The distribution of irreducible polynomials in several indeterminates,” Ill. J. Math. 7, 371–375 (1963).
  16. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  17. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef]
  18. 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]
  19. B. J. Brames, “Unique phase retrieval with explicit support information,” Opt. Lett. 11, 61–63 (1986).
    [CrossRef]
  20. J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  21. T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am A 4, 124–134 (1987).
    [CrossRef]
  22. 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]
  23. J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  24. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  25. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  26. P. VanToorn, A. H. Greenaway, A. M. J. Huiser, “Phaseless object reconstruction,” Opt. Acta 7, 767–774 (1984).
    [CrossRef]
  27. J. R. Fienup, “Experimental evidence of the uniqueness of phase retrieval from intensity data,” in Indirect Imaging, J. A. Roberts, ed., Proceedings of International Union of Radio Science/International Astronomical Union Symposium, August 30–September 2, 1983, Sydney, Australia (Cambridge U. Press, Cambridge, 1984), pp. 99–109.
  28. 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]
  29. R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
    [CrossRef]
  30. A. M. J. Huiser, P. VanToorn, “Ambiguity of the phase-reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
    [CrossRef]
  31. E. N. Leith, J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [CrossRef]
  32. 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]
  33. M. Nieto-Vesperinas, R. Navarro, F. J. Fuentes, “Performance of a simulated-annealing algorithm for phase retrieval,” J. Opt. Soc. Am. A 5, 30–38 (1988).
    [CrossRef]
  34. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).
  35. J. H. Seldin, J. R. Fienup, “Numerical investigation of phase retrieval uniqueness,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 120–123.

1988

1987

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am A 4, 124–134 (1987).
[CrossRef]

1986

1985

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]

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

1984

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

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

1983

1982

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

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

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

1980

A. M. J. Huiser, P. VanToorn, “Ambiguity of the phase-reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
[CrossRef]

M. Nieto-Vesperinas, “Dispersion relations in two dimensions: application to the phase problem,” Optik (Stuttgart) 56, 377–384 (1980).

1979

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

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

1978

1974

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).

1964

E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

1963

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

L. Carlitz, “The distribution of irreducible polynomials in several indeterminates,” Ill. J. Math. 7, 371–375 (1963).

1962

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence,” Proc. Phys. Soc. (London) 80, 1269–1272 (1962).
[CrossRef]

E. N. Leith, J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
[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.

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).

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]

Carlitz, L.

L. Carlitz, “The distribution of irreducible polynomials in several indeterminates,” Ill. J. Math. 7, 371–375 (1963).

Clinthorne, J. T.

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

Crimmins, T. R.

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am A 4, 124–134 (1987).
[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]

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]

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

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 Indirect Imaging, J. A. Roberts, ed., Proceedings of International Union of Radio Science/International Astronomical Union Symposium, August 30–September 2, 1983, Sydney, Australia (Cambridge U. Press, Cambridge, 1984), pp. 99–109.

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

J. H. Seldin, J. R. Fienup, “Numerical investigation of phase retrieval uniqueness,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 120–123.

Fuentes, F. J.

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Greenaway, A. H.

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

Hayes, M. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Hofstetter, E. M.

E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Huang, T. S.

Huiser, A. M. J.

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

A. M. J. Huiser, P. VanToorn, “Ambiguity of the phase-reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
[CrossRef]

Jensen, L. H.

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

Lawton, W.

W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 94–98 (1980).
[CrossRef]

Leith, E. N.

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]

McClellan, J. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

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).

Navarro, R.

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]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, R. Navarro, F. J. Fuentes, “Performance of a simulated-annealing algorithm for phase retrieval,” J. Opt. Soc. Am. A 5, 30–38 (1988).
[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]

M. Nieto-Vesperinas, “Dispersion relations in two dimensions: application to the phase problem,” Optik (Stuttgart) 56, 377–384 (1980).

Paxman, R. G.

R. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

Sanz, J. L. C.

Saxton, W. O.

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

Seldin, J. H.

J. H. Seldin, J. R. Fienup, “Numerical investigation of phase retrieval uniqueness,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 120–123.

Sodin, L. G.

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

Stefanescu, I. S.

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

Stout, G. H.

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

Upatnieks, J.

VanToorn, P.

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

A. M. J. Huiser, P. VanToorn, “Ambiguity of the phase-reconstruction problem,” Opt. Lett. 5, 499–501 (1980).
[CrossRef]

Wackerman, C. C.

Walther, A.

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

Wolf, E.

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence,” Proc. Phys. Soc. (London) 80, 1269–1272 (1962).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Appl. Opt.

Astron. Astrophys. Suppl.

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. Inf. Theory

E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Ill. J. Math.

L. Carlitz, “The distribution of irreducible polynomials in several indeterminates,” Ill. J. Math. 7, 371–375 (1963).

J. Math. Phys.

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

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

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

J. Opt. Soc. Am A

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am A 4, 124–134 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

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

Opt. Commun.

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.

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

Opt. Lett.

Optik (Stuttgart)

M. Nieto-Vesperinas, “Dispersion relations in two dimensions: application to the phase problem,” Optik (Stuttgart) 56, 377–384 (1980).

Proc. IEEE

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Proc. Phys. Soc. (London)

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence,” Proc. Phys. Soc. (London) 80, 1269–1272 (1962).
[CrossRef]

Other

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

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

W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” in 1980 International Optical Computing Conference, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 94–98 (1980).
[CrossRef]

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

J. H. Seldin, J. R. Fienup, “Numerical investigation of phase retrieval uniqueness,” in Digest of Topical Meeting on Signal Recovery and Synthesis III (Optical Society of America, Washington, D.C., 1989), pp. 120–123.

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

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. G. Paxman, J. R. Fienup, J. T. Clinthorne, “Effect of tapered illumination and Fourier intensity errors on phase retrieval,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 184–189 (1987).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Object-space to Fourier-modulus-space mappings of a unique object f and a pair of ambiguous images (ga, gac), with error metrics δ and .

Fig. 2
Fig. 2

Fourier-modulus error versus object-domain error δ for a five-step grid search with step size Δs = 1. The minimum value of (excluding g = f) is boxed.

Fig. 3
Fig. 3

Fourier-modulus error versus object-domain error δ for a five-step grid search with Δs = 1/3 about the minimum of the grid search of Fig. 2. All points satisfying < 0.125 are shown here.

Fig. 4
Fig. 4

(g, f) and δ(g, f) versus t for g = f + t(gminf), the line joining f and gmin.

Fig. 5
Fig. 5

(g, f) and δ(g, f) versus t for g = f + t(gstagf), the line joining f and gstag.

Fig. 6
Fig. 6

Object-space to Fourier-modulus-space mappings of an object f, two stagnated images gmin and gstag, and the nearest ambiguous image to f with respect to the Fourier-modulus error (ga, gac).

Fig. 7
Fig. 7

Flow chart for determining the ambiguity of the 3 × 2 realvalued image of Eq. (33). Multiple conditions in a box must all be satisfied for “YES,” except where “or” is specified.

Fig. 8
Fig. 8

Gradient-projection constrained-minimization algorithm. The search direction is determined by projecting the negative gradient of the objective function onto the tangent plane to the constraint surface.

Fig. 9
Fig. 9

Fourier-modulus error versus object-domain error δ for worst-case nearest ambiguities to 2 × 2 objects. (a) No nonnegativity constraint, 4752 objects; (b) nonnegativity constraint, 4486 objects.

Fig. 10
Fig. 10

Monte Carlo estimates of the probability that the worst-case nearest ambiguity to 2 × 2 objects with and without a nonnegativity constraint has a Fourier-modulus error less than and an object-domain error greater than K∊(K = 4 and K = 10).

Fig. 11
Fig. 11

Fourier-modulus error versus object-domain error δ for worst-case nearest ambiguities to 3 × 2 objects. (a) No nonnegativity constraint, 4112 objects; (b) nonnegativity constraint, 4601 objects.

Fig. 12
Fig. 12

Monte Carlo estimates of the probability that the worst-case nearest ambiguity to 3 × 2 objects with and without a nonnegativity constraint has a Fourier-modulus error less than and an object-domain error greater than K∊(K = 4 and K = 10).

Fig. 13
Fig. 13

Fourier-modulus error versus object-domain error δ for worst-case nearest ambiguities to 3 × 2, nonnegative, L-shaped objects, (a) L robustness > 10%, 3190 objects; (b) L robustness > 25%, 2714 objects.

Fig. 14
Fig. 14

Monte Carlo estimates of the probability that the worst-case nearest ambiguity to 3 × 2, nonnegative, L-shaped objects with L robustness greater than R% (R = 10 and R = 25) has a Fourier-modulus error less than and an object-domain error greater than K∊(K = 4 and K = 10).

Equations (78)

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

F ( u , υ ) = | F ( u , υ ) | exp [ i ψ ( u , υ ) ] = [ f ( x , y ) ] = f ( x , y ) exp [ 2 π ( u x + υ y ) ] d x d y .
r ( x , y ) = 1 | F ( u , υ ) | 2 .
F ( u , υ ) = | F ( u , υ ) | exp [ i ψ ( u , υ ) ] = DFT [ f ( x , y ) ] = x = 0 M 1 y = 0 N 1 f ( x , y ) exp [ j 2 π ( u x 2 M + u y 2 N ) ] ,
f 1 ( x , y ) = g ( x , y ) * h ( x , y )
f 2 ( x , y ) = g ( x , y ) * h * ( x , y )
F 1 ( u , υ ) = G ( u , υ ) H ( u , υ )
F 2 ( u , υ ) = G ( u , υ ) H * ( u , υ )
| F 1 ( u , υ ) | = | F 2 ( u , υ ) | = | G ( u , υ ) | | H ( u , υ ) | ,
( g , f ) [ u , υ [ α f | G ( u , υ ) | | F ( u , υ ) | ] 2 u , υ | F ( u , υ ) | 2 ] 1 / 2 ,
α f = [ u , υ | F ( u , υ ) | 2 u , υ | G ( u , υ ) | 2 ] 1 / 2
δ ( g , f ) [ x , y [ α 0 g ( x , y ) f ( x , y ) ] 2 x , y f 2 ( x , y ) ] 1 / 2 ,
α 0 = α f sign [ x , y f ( x , y ) g ( x , y ) ]
g inc = [ s 1 s 2 s 3 s 4 s 5 s 6 ] ,
f = [ 1 2 1 2 1 2 ] .
g 0 = [ 1 1 2 1 1 2 ] ,
g 1 = [ 2 3 2 3 2 1 2 3 7 3 ] ,
g 2 = [ 2 3 7 9 17 9 10 9 2 3 22 9 ] ,
g inc = [ s 1 s 2 s 3 s 4 s 5 s 6 ]
g min = [ 0.623 0.749 1.871 1.149 0.659 2.530 ] ,
g = f + t ( g min f ) .
g a = [ 0.594 1.624 1.211 2.330 1.415 1.730 ] ,
g ac = [ 0.363 0.618 1.987 1.422 0.600 2.837 ] ,
g stag = [ 0.694 1.778 1.010 2.235 1.355 1.856 ] ,
f = [ 0.476 3.244 1.379 1.659 2.939 1.102 ] ,
g a = [ 0.867 3.521 1.278 1.679 2.651 0.796 ] ,
g ac = [ 0.350 2.146 3.171 0.677 2.475 1.974 ] ,
g s 1 = [ 0.353 2.143 3.172 0.684 2.470 1.976 ]
g s 2 = [ 0.266 1.876 2.971 0.746 2.711 2.222 ]
[ a b c d ] = [ e f ] * [ g h ] = [ e g e h f g f h ] ,
a = e g ,
b = e h ,
c = f g ,
d = f h .
a d = b c .
a d = b c ,
| b | | a | | c | .
[ a b c d e f ] = [ g h ] * [ i j k l ] = [ g i h i + g j h j g k h k + g l h l ] ,
( a f c d ) 2 ( ae bd ) ( bf be ) = 0 .
b = 1 2 [ e ( c f + a d ) ± ( e 2 4 d f ) 1 / 2 ( c f a d ) ] .
E ( g , f ) = u , υ [ | G ( u , υ ) | | F ( u , υ ) | ] 2 ,
2 × 2 Images ( L = 4 ) h ( x ¯ ) = a d b c = 0 ,
3 × 2 Images ( L = 6 ) h ( x ¯ ) = ( a f c d ) 2 ( a e b d ) ( b f c e ) = 0 .
f = [ 1.48155 0 0 2.01553 3.97050 0.16831 ] ,
g a = [ 1.48170 6.29 E ­ 4 2.78 E ­ 3 2.01419 3.97109 0.16907 ] = [ 1 0.04354 ] * [ 1.48170 0.06388 2.01419 3.88340 ] ,
g ac = [ 0.045354 1 ] * [ 1.48170 0.06388 2.01419 3.88340 ] = [ 0.06451 1.47892 0.06388 0.08769 2.18326 3.88340 ] .
R 100 = min { a , f } / [ ( a 2 + d 2 + e 2 + f 2 ) / 4 ] 1 / 2 .
[ a b c d e f g h i ] ,
( a h b g ) 2 ( a e b d ) ( d h e g ) = 0
( a h b g ) ( a f c d ) ( a e b d ) ( a i c g ) = 0 .
δ ( g , f ) = [ x , y [ α 0 g ( x , y ) f ( x , y ) ] 2 / x , y f 2 ( x , y ) ] 1 / 2 = [ u , υ | ± α f G ( u , υ ) F ( u , υ ) | 2 / u , υ | F ( u , υ ) | 2 ] 1 / 2 .
| ± α f G ( u , υ ) F ( u , υ ) | 2 [ α f | G ( u , υ ) | | F ( u , υ ) | ] 2 .
δ ( g , f ) [ u , υ [ α f | G ( u , υ ) | | F ( u , υ ) | ] 2 / u , υ | F ( u , υ ) | 2 ] 1 / 2 ( g , f ) .
a = g i ,
b = h i + g j ,
c = h j ,
d = g k ,
e = h k + g l ,
f = h l .
a f = ghil ,
c d = ghjk .
( a f c d ) 2 = g 2 h 2 ( i l j k ) 2 .
( b f c e ) = h 2 ( i l j k ) ,
( a e b d ) = g 2 ( i l j k ) .
( a e b d ) ( b f c e ) = g 2 h 2 ( i l j k ) 2 .
( a f c d ) 2 ( a e b d ) ( b f c e ) = 0 .
b 2 ( d f ) b ( a e f + c d e ) + a c e 2 + ( a f c d ) 2 = 0 .
b = [ e ( a f + c d ) ± ( e 2 4 d f ) 1 / 2 ( a f c d ) ] 2 d f = 1 2 [ e ( c f + a d ) ± ( e 2 4 d f ) 1 / 2 ( c f a d ) ] .
h ( x ¯ ) = ( h x 1 h x 2 h x L ) t .
b ¯ 1 = [ ( h x 1 ) 1 ( h x 2 ) 1 0 0 0 ] t , b ¯ 2 = [ ( h x 1 ) 1 ( h x 3 ) 0 1 0 0 ] t , b ¯ L 1 = [ ( h x 1 ) 1 ( h x L ) 0 0 0 1 ] t .
Z [ b ¯ 1 b ¯ 2 b ¯ L 1 ] ,
p ¯ = Z ( Z t Z ) 1 Z t ω ¯ .
p ¯ = Z ( Z t Z ) 1 Z t E ( x ¯ ) .
p ¯ k = Z k ( Z k t Z k ) 1 Z k t E ( x ¯ k ) .
x ¯ k + 1 = x ¯ k + γ k p ¯ k + q ¯ k ,
h ( x ¯ k + 1 ) = 0 ,
E g ( x , y ) = 2 M N [ g ( x , y ) g ( x , y ) ] ,
DFT { g ( x , y ) } = | F ( u , υ ) | | G ( u , υ ) | G ( u , υ ) .
E ( x ¯ k + 1 ) p ¯ k + 1 < ζ

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