Abstract

A simple method for reconstructing a two-dimensional image from power spectral data is proposed. By locating the points at which the Fourier intensity data are zero, we use a product of point-zero factors to approximate the complex spectrum. A simple iterative method then successfully refines this phase estimate.

© 1994 Optical Society of America

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References

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  1. R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
    [CrossRef]
  2. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  3. M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 13.
  4. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one dimension,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  5. 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]
  6. J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
    [CrossRef]
  7. 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]
  8. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  9. 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]
  10. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
    [CrossRef]

1991 (1)

1990 (2)

1985 (1)

1984 (1)

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[CrossRef]

1983 (3)

1982 (2)

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

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

Baranova, N. B.

Bates, R. H. T.

Brames, B. J.

Dainty, J. C.

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[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]

Fiddy, M. A.

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]

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[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]

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 13.

Fienup, J. R.

Hayes, M. H.

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

Huang, T. S.

Mamaev, A. V.

McClellan, J. H.

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

Parker, C. R.

Pilipetsky, N. F.

Quek, B. K.

Sanz, J. L. C.

Scivier, M. S.

Seldin, J. H.

Shkunov, V. V.

Wackerman, C. C.

Yagle, A. E.

Zel’dovich, B. Ya.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

J. C. Dainty, M. A. Fiddy, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

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

Other (1)

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 13.

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Figures (9)

Fig. 1
Fig. 1

Object satisfying Eisenstein’s criterion.

Fig. 2
Fig. 2

Fourier intensity data from Fig. 1.

Fig. 3
Fig. 3

Fourier phase from Fig. 1.

Fig. 4
Fig. 4

Point zeros from intensity data (Fig. 2).

Fig. 5
Fig. 5

Phase from the polynomial representation [Eq. (7)].

Fig. 6
Fig. 6

Object recovered from the polynomial representation model.

Fig. 7
Fig. 7

Object recovered from the Fourier intensity with the phase from the polynomial representation model.

Fig. 8
Fig. 8

Object recovered after 50 iterations with initial phase from the model.

Fig. 9
Fig. 9

Object recovered after 50 iterations with zero initial phase.

Equations (8)

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F ( u , υ ) = s ( p , q ) f ( p , q ) exp [ i ( u p + υ q ) ] d p d q ,
F ( u , υ ) = j = 1 N { F j ( u , υ ) exp [ γ j ( u , υ ) ] } L j N ;
F ( α , β ) = m n f ( m , n ) α m β n .
F ( x 1 , x 2 ) = ( f 0 , 1 + f 1 , 1 x 1 + . . . + f M 1 , 1 x 1 M 1 ) x 2 + ( f 0 , 2 + f 1 , 2 x 1 + . . . + f M 1 , 2 x 1 M 1 ) x 2 2 + . . . + ( f 0 , M + f 1 , M x 1 + . . . + f M 1 , M x 1 M 1 ) x 2 M ,
F ( u , υ ) = ( A 1 u + i A 2 υ ) ,
arg [ F ( u , υ ) ] = arctan ( A 2 υ / A 1 u ) .
F ( u , υ ) = ( u u 0 ) + i A 2 ( υ υ 0 )
F ( u , υ ) = n = 1 N ( U n + i A n V n ) ,

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