Abstract

A method to solve the phase-retrieval problem from two intensities observed at the Fourier transform of an object function in one dimension is proposed. This method involves the solution of the linear equations consisting of the data of two intensities, obtained with and without an exponential filter at the object plane, and unknown coefficients in the Fourier series expansion of phase. There is no need to treat the nonlinear equation for zero location in the complex plane. The usefulness of the method is shown in computer simulation studies of the reconstruction of the one-dimensional phase object from the observable moduli at the Fourier-transform plane of the object.

© 1987 Optical Society of America

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References

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  1. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
    [CrossRef]
  2. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  3. G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
    [CrossRef]
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  5. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982); “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef] [PubMed]
  6. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I. Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982); R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,”J. Opt. Soc. Am. 73, 358–365 (1983).
    [CrossRef]
  7. 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); H. H. Arsenault, K. Chalasinska-Macukow, “A solution to the phase-retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  8. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  9. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  10. 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]
  11. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  12. B. J. Hoenders, “On the solution of the phase retrieval problem,”J. Math. Phys. 16, 1719–1725 (1975).
    [CrossRef]
  13. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
    [CrossRef]
  14. N. Nakajima, T. Asakura, “Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information,”J. Phys. D 19, 319–331 (1986); “A new approach to two-dimensional phase retrieval,” Opt. Acta 32, 647–658 (1985).
    [CrossRef]
  15. 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]
  16. J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  17. J. W. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
    [CrossRef] [PubMed]
  18. N. Nakajima, T. Asakura, “Study of zero location by means of an exponential filter in the phase retrieval problem,” Optik (Stuttgart) 60, 289–305 (1982).
  19. J. W. Wood, T. J. Hall, M. A. Fiddy, “A comparison study of some computational methods for locating the zeros of entire functions,” Opt. Acta 30, 511–527 (1983).
    [CrossRef]

1986 (1)

N. Nakajima, T. Asakura, “Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information,”J. Phys. D 19, 319–331 (1986); “A new approach to two-dimensional phase retrieval,” Opt. Acta 32, 647–658 (1985).
[CrossRef]

1985 (2)

1983 (2)

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]

J. W. Wood, T. J. Hall, M. A. Fiddy, “A comparison study of some computational methods for locating the zeros of entire functions,” Opt. Acta 30, 511–527 (1983).
[CrossRef]

1982 (4)

N. Nakajima, T. Asakura, “Study of zero location by means of an exponential filter in the phase retrieval problem,” Optik (Stuttgart) 60, 289–305 (1982).

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982); “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef] [PubMed]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I. Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982); R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,”J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

1981 (2)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

J. W. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
[CrossRef] [PubMed]

1979 (1)

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

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

1975 (1)

B. J. Hoenders, “On the solution of the phase retrieval problem,”J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

1972 (1)

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

1963 (1)

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

Arsenault, H. H.

Asakura, T.

N. Nakajima, T. Asakura, “Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information,”J. Phys. D 19, 319–331 (1986); “A new approach to two-dimensional phase retrieval,” Opt. Acta 32, 647–658 (1985).
[CrossRef]

N. Nakajima, T. Asakura, “Study of zero location by means of an exponential filter in the phase retrieval problem,” Optik (Stuttgart) 60, 289–305 (1982).

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I. Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982); R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,”J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

Bruck, Y. M.

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

Burge, R. E.

J. W. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
[CrossRef] [PubMed]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Chalasinska-Macukow, K.

Deighton, H. V.

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
[CrossRef]

Fiddy, M. A.

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]

J. W. Wood, T. J. Hall, M. A. Fiddy, “A comparison study of some computational methods for locating the zeros of entire functions,” Opt. Acta 30, 511–527 (1983).
[CrossRef]

J. W. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
[CrossRef] [PubMed]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Fienup, J. R.

Gerchberg, R. W.

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

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Hall, T. J.

J. W. Wood, T. J. Hall, M. A. Fiddy, “A comparison study of some computational methods for locating the zeros of entire functions,” Opt. Acta 30, 511–527 (1983).
[CrossRef]

Hayes, M. H.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

Hoenders, B. J.

B. J. Hoenders, “On the solution of the phase retrieval problem,”J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

Huang, T. S.

Nakajima, N.

N. Nakajima, T. Asakura, “Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information,”J. Phys. D 19, 319–331 (1986); “A new approach to two-dimensional phase retrieval,” Opt. Acta 32, 647–658 (1985).
[CrossRef]

N. Nakajima, T. Asakura, “Study of zero location by means of an exponential filter in the phase retrieval problem,” Optik (Stuttgart) 60, 289–305 (1982).

Nieto-Vesperinas, M.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Sanz, J. L. C.

Saxton, W. O.

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

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

Scivier, M. S.

Sodin, L. G.

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

Walker, J. G.

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Walther, A.

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

Wood, J. W.

J. W. Wood, T. J. Hall, M. A. Fiddy, “A comparison study of some computational methods for locating the zeros of entire functions,” Opt. Acta 30, 511–527 (1983).
[CrossRef]

J. W. Wood, M. A. Fiddy, R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. 6, 514–516 (1981).
[CrossRef] [PubMed]

Appl. Opt. (1)

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

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

J. Math. Phys. (1)

B. J. Hoenders, “On the solution of the phase retrieval problem,”J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

N. Nakajima, T. Asakura, “Two-dimensional phase retrieval using the logarithmic Hilbert transform and the estimation technique of zero information,”J. Phys. D 19, 319–331 (1986); “A new approach to two-dimensional phase retrieval,” Opt. Acta 32, 647–658 (1985).
[CrossRef]

Opt. Acta (3)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

J. W. Wood, T. J. Hall, M. A. Fiddy, “A comparison study of some computational methods for locating the zeros of entire functions,” Opt. Acta 30, 511–527 (1983).
[CrossRef]

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

Opt. Commun. (1)

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

Opt. Lett. (2)

Optik (Stuttgart) (3)

N. Nakajima, T. Asakura, “Study of zero location by means of an exponential filter in the phase retrieval problem,” Optik (Stuttgart) 60, 289–305 (1982).

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

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I. Underlying theory,” Optik (Stuttgart) 61, 247–262 (1982); R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,”J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
[CrossRef]

Other (3)

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
[CrossRef]

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

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

The original phase object function of Eq. (12): (a) the modulus and (b) the phase of the object function.

Fig. 2
Fig. 2

The moduli for phase retrieval at the Fourier-transform plane of the object in Fig. 1: (a) the modulus |F(x)| observed at the Fourier-transform plane, (b) the modulus | F ^ (x)| of the Fourier transform of the phase object multiplied by an exponential filter exp[−2π(0.08u)], and (c) the modulus |M(xi0.08)| evaluated from the modulus in (a) by using Eqs. (6) and (7).

Fig. 3
Fig. 3

Reconstructed phase object function from the data of the moduli in Fig. 2: (a) the modulus and (b) the phase of the reconstructed object.

Fig. 4
Fig. 4

The noisy moduli for phase retrieval at the Fourier-transform plane of the object in Fig. 1: (a) the noisy modulus |Fn(x)| obtained at the Fourier-transform plane, (b) the noisy modulus | F ^ n (x)| of the Fourier transform of the object multiplied by an exponential filter exp [−2π(0.08u)], and (c) the noisy modulus |Mn(xi0.08)| produced from the noisy modulus in (a) by using Eqs. (6) and (7).

Fig. 5
Fig. 5

Reconstructed phase object function from the data of the noisy moduli in Fig. 4: (a) the modulus and (b) the phase of the reconstructed object.

Equations (13)

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F ( x ) = a b f ( u ) exp ( - 2 π i x u ) d u ,
F ( x ) = M ( x ) exp [ i ϕ ( x ) ] ,
F ^ ( x ) = a b f ( u ) exp ( - 2 π y c u ) exp ( - 2 π i x u ) d u .
F ^ ( x ) = F ( x - i y c ) .
F ( x - i y c ) = M ( x - i y c ) exp [ i ϕ ( x - i y c ) ] ,
[ M ( x - i y c ) ] 2 = F ( x - i y c ) F * ( x + i y c ) = - ( b - a ) b - a exp ( - 2 π y c τ ) a b f ( u + τ ) f * ( u ) d u × exp ( - 2 π i x τ ) d τ ,
a b f ( u + τ ) f * ( u ) d u = - F ( x ) 2 exp ( 2 π i x τ ) d x
F ^ ( x ) = M ( x - i y c ) exp [ - ϕ I ( x ) ] ,
ϕ ( x ) n = 1 N ( a n cos n π l x + b n sin n π l x ) ,
ϕ I ( x ) n = 1 N [ a n sinh ( n π l y c ) sin n π l x - b n sinh ( n π l y c ) cos n π l x ] .
ln F ^ ( x ) M ( x - i y c ) n = 1 N [ - a n sinh ( n π l y c ) sin n π l x + b n sinh ( n π l y c ) cos n π l x ] .
f ( u ) = { exp { i 2 [ cos ( 2 π u ) + sin ( 0.1 π u ) ] } - 0.625 u 0.625 0 otherwise
SNR = m = 1 M F ^ ( x m ) 2 m = 1 M n ( x m ) 2

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