Abstract

A simple recursive algorithm is proposed for reconstructing certain classes of two-dimensional objects from their autocorrelation functions (or equivalently from the modulus of their Fourier transforms—the phase-retrieval problem). The solution is shown to be unique in some cases. The objects contain reference points not satisfying the holography condition but satisfying weaker conditions. Included are objects described by Fiddy et al. [ Opt. Lett. 8, 96 ( 1983)] satisfying Eisenstein’s thorem.

© 1983 Optical Society of America

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References

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  1. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  2. C. Y. C. Liu and A. W. Lohmann, "High resolution image formation through the turbulent atmosphere," Opt. Commun. 8, 372–377 (1973).
    [CrossRef]
  3. J. W. Goodman, "Analogy between holography and interferometric image formation," J. Opt. Soc. Am. 60, 506–509 (1970).
    [CrossRef]
  4. R. W. Gerchberg and 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); R. H. Boucher, "Convergence of algorithms for phase retrieval from two intensity distributions," Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).
  5. W. J. Dallas, "Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography," Optik 44, 45–59 (1975).
  6. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  7. J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27–29 (1978); J. R. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. 18, 529–534 (1979).
    [CrossRef] [PubMed]
  8. P. J. Napier and R. H. T. Bates, "Inferring phase information from modulus information in two-dimensional aperture synthesis," Astron. Astrophys. Suppl. 15, 427–430 (1974).
  9. G. H. Stout and L. H. Jensen, X-Ray Structure Determination (Macmillan, London, 1968).
  10. J. R. Fienup, T. R. Crimmins, and W. Holsztynski, "Reconstruction of the support of an object from the support of its autocorrelation," J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  11. Yu. M. Bruck and L. G. Sodin, "On the ambiguity of the image reconstruction problem," Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  12. A. Walther, "The question of phase retrieval in optics," Opt. Acta 10,41–49 (1963); E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
    [CrossRef]
  13. M. A. Fiddy, B. J. Brames and J. C. Dainty, "Enforcing irreducibility for phase retrieval in two dimensions," Opt. Lett. 8,96–98 (1983).
    [CrossRef] [PubMed]
  14. M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," IEEE Trans. Acoust. Speech Signal Process. ASSP-82, 1545 (1982).
  15. J. R. Fienup, "Image reconstruction for stellar interferometry," in Current Trends in Optics, F. T. Arecchi and F. R. Aussenegg, eds. (Taylor and Francis, London, 1981), pp. 95–102

1983

1982

1979

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

1978

1975

W. J. Dallas, "Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography," Optik 44, 45–59 (1975).

1974

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

1973

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

1970

1963

A. Walther, "The question of phase retrieval in optics," Opt. Acta 10,41–49 (1963); E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Bates, R. H. T.

P. J. Napier and 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).

Brames, B. J.

Bruck, Yu. M.

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

Crimmins, T. R.

Dainty, J. C.

Dallas, W. J.

W. J. Dallas, "Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography," Optik 44, 45–59 (1975).

Fiddy, M. A.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and 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); R. H. Boucher, "Convergence of algorithms for phase retrieval from two intensity distributions," Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

Goodman, J. W.

Hayes, M. H.

M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," IEEE Trans. Acoust. Speech Signal Process. ASSP-82, 1545 (1982).

Holsztynski, W.

Jensen, L. H.

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

Liu, C. Y. C.

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

Napier, P. J.

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

Quatieri, T. F.

M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," IEEE Trans. Acoust. Speech Signal Process. ASSP-82, 1545 (1982).

Saxton, W. O.

R. W. Gerchberg and 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); R. H. Boucher, "Convergence of algorithms for phase retrieval from two intensity distributions," Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

Sodin, L. G.

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

Stout, G. H.

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

Walther, A.

A. Walther, "The question of phase retrieval in optics," Opt. Acta 10,41–49 (1963); E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Appl. Opt.

Astron. Astrophys. Suppl.

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

IEEE Trans. Acoust. Speech Signal Process.

M. H. Hayes and T. F. Quatieri, "The importance of boundary conditions in the phase retrieval problem," IEEE Trans. Acoust. Speech Signal Process. ASSP-82, 1545 (1982).

J. Opt. Soc. Am.

Opt. Acta

A. Walther, "The question of phase retrieval in optics," Opt. Acta 10,41–49 (1963); E. M. Hofstetter, "Construction of time-limited functions with specified autocorrelation functions," IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[CrossRef]

Opt. Commun.

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

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

Opt. Lett.

Optik

W. J. Dallas, "Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography," Optik 44, 45–59 (1975).

Other

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

R. W. Gerchberg and 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); R. H. Boucher, "Convergence of algorithms for phase retrieval from two intensity distributions," Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130–141 (1980).

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

J. R. Fienup, "Image reconstruction for stellar interferometry," in Current Trends in Optics, F. T. Arecchi and F. R. Aussenegg, eds. (Taylor and Francis, London, 1981), pp. 95–102

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Figures (6)

Fig. 1
Fig. 1

Fiddy–Brames–Dainty13 object. (a) FBD object support having two reference points, A and B; (b) object support assumed; (c) autocorrelation support. The object is uniquely reconstructed from its autocorrelation function.

Fig. 2
Fig. 2

Locator set containing all possible solutions, used to show that the support solution is unique.

Fig. 3
Fig. 3

Alternative case. (a) Object support; (b) autocorrelation support; (c) locator set.

Fig. 4
Fig. 4

Triangular-shaped object. (a) Object support; (b) autocorrelation support. The object is uniquely (among triangular-shaped solutions) reconstructed from its autocorrelation function.

Fig. 5
Fig. 5

Specific triangular-shaped object. (a) The object; (b) a second nontriangular-shaped solution; (c) the common autocorrelation function; (d) the function used to synthesize objects shown in (a) and (b).

Fig. 6
Fig. 6

Another case related to FBD objects. (a) Object support; (b) alternative support reconstruction; (c) autocorrelation support. The object is reconstructed from its autocorrelation function, with two solutions.

Equations (25)

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F ( p , q ) = F ( p , q ) exp [ i ψ ( p , q ) ] = F [ f ( m , n ) ] = m = 0 M - 1 n = 0 N - 1 f ( m , n ) exp [ - i 2 π ( m p / M + n q / N ) ] ,
r f ( m , n ) = j = 0 M - 1 k = 0 N - 1 f ( j , k ) f * ( j - m , k - n )
= F - 1 [ F ( p , q ) 2 ] ,
f ( m , n ) = A δ ( m - m 0 , n - n 0 ) + g ( m , n ) ,
r f ( m , n ) = A 2 δ ( m , n ) + r g ( m , n ) + A g * ( m 0 - m , n 0 - n ) + A * g ( m + m 0 , n + n 0 ) ,
f ( m , n ) = A δ ( m , n ) + g ( m , n ) ,
r f ( J , n ) = A * g ( J , n ) ,             n = 1 , , K ,
r f ( m , K ) = A * g ( m , K ) ,             m = 1 , , J .
r f ( - J + 1 , K - 1 ) = g ( 1 , K ) g * ( J , 1 ) = B * g ( 1 , K ) .
r f ( J , 1 ) = A * g ( J , 1 ) = A * B ,
r f ( 1 , K ) = A * g ( 1 , K ) .
A 2 = r f ( J , 1 ) r f * ( 1 , K ) r f * ( - J + 1 , K - 1 ) .
r f ( - J + 1 , K - 2 ) = g ( 1 , K ) g * ( J , 2 ) + g ( 1 , K - 1 ) g * ( J , 1 ) .
g ( 1 , K - 1 ) = [ r f ( - J + 1 , K - 2 ) - g ( 1 , K ) g * ( J , 2 ) ] / B * ,
r f ( - J + 1 , K - 3 ) = g ( 1 , K ) g * ( J , 3 ) + g ( 1 , K - 1 ) g * ( J , 2 ) + g ( 1 , K - 2 ) g * ( J , 1 ) .
r f ( J - 1 , n ) = g ( J - 1 , n ) A * + k = n + 1 K g ( J , k ) g * ( 1 , k - n ) ,
r ( 0 , K ) = f ( 0 , K ) f * ( 0 , 0 ) = C A * ,
r ( J , - K ) = f ( J , 0 ) f * ( 0 , K ) = B C * ,
r ( J , 0 ) = f ( J , 0 ) f * ( 0 , 0 ) = B A * .
A 2 = r * ( 0 , K ) r ( J , 0 ) r ( J , - K ) .
B = r ( J , 0 ) / A * ,
C = r ( 0 , K ) / A * .
f ( 0 , n ) = r ( - J , n ) / B * ,
f ( m , 0 ) = r ( m , - K ) / C * ,
f ( m , K - m ) = r ( m , K - m ) / A * .