Abstract

A phase-retrieval system with a Gaussian filter of known width is proposed. Using this system, one can retrieve the Fourier phase of a complex-valued object function from intensity measurements of two Fourier transforms of the object modulated by the Gaussian filter and the object modulated by the laterally shifted Gaussian filter with known displacement. The shift of the Gaussian filter has the same effect as the modulation of the object with an exponential filter, which is used in the previous phase-retrieval method based on the properties of analytic functions. In addition, the same situation as for the exponential filtering with variable inclination can be easily produced by changing the displacement quantity of the Gaussian filter.

© 1998 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.
  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.
  4. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.
  5. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  6. M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 499–529.
  7. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer Academic, Dordrecht, The Netherlands, 1989).
  8. 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).
  9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  10. J. G. Walker, “Computer simulation of a method for object reconstruction from stellar speckle interferometry data,” Appl. Opt. 21, 3132–3137 (1982).
    [CrossRef] [PubMed]
  11. N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.
  12. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  13. N. Nakajima, “Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion,” J. Opt. Soc. Am. A 5, 257–262 (1988).
    [CrossRef]
  14. N. Nakajima, “Two-dimensional phase retrieval by exponential filtering,” Appl. Opt. 28, 1489–1493 (1989).
    [CrossRef] [PubMed]

1989

1988

1987

1982

1972

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).

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

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.

Fiddy, M. A.

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.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 499–529.

Fienup, J. R.

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

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

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).

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer Academic, Dordrecht, The Netherlands, 1989).

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Nakajima, N.

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.

Ross, G.

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.

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).

Stark, H.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Walker, J. G.

Appl. Opt.

J. Opt. Soc. Am. A

Optik (Stuttgart)

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).

Other

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.

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.

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.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 499–529.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer Academic, Dordrecht, The Netherlands, 1989).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the geometry of the phase-retrieval system. The object function is reconstructed from two intensity measurements of the Fourier transform of the object modulated by a Gaussian filter (dashed rectangle) and by its laterally shifted Gaussian filter (solid rectangle).

Fig. 2
Fig. 2

Original object function: (a) modulus (solid curve), (b) phase of the object function. The functions of the Gaussian filter and its shifted Gaussian filter are represented by the dashed and the dotted–dashed curves, respectively, in (a).

Fig. 3
Fig. 3

Moduli obtained in the Fourier plane: (a), (b) moduli of the Fourier transforms of the object modulated by the Gaussian filter and by the shifted Gaussian filter, respectively.

Fig. 4
Fig. 4

Object reconstructed from the intensities shown in Fig. 3: (a) modulus, (b) phase.

Fig. 5
Fig. 5

Same as in Fig. 3, except that noise was added to the moduli in Fig. 3.

Fig. 6
Fig. 6

Same as in Fig. 4 but for reconstruction from the noisy moduli in Fig. 5.

Equations (15)

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g(u-u0)=A exp-(u-u0)2W2,
H1(x)=-f(u)g(u-u0)exp(-2πiux)du,
H2(x)=-f(u)g(u-u0-c)exp(-2πiux)du,
H1(x)=-F(x-x)G(x)exp(-2πiu0x)dx,
H2(x)=-F(x-x)G(x)exp[-2πi(u0+c)x]dx,
G(x)exp[-2πi(u0+c)x]=AπW exp-(u0+c)2W2×exp-π2W2x+i u0+cπW22=exp-(u0+c)2W2G[x+i(u0+c)/πW2].
H2(x)=exp-c2+2u0cW2H1x+i cπW2.
H1(x)=M(x)exp[iϕ(x)].
|H1(x+ic/πW2)|=|M(x+ic/πW2)|×exp[-Im ϕ(x+ic/πW2)],
ln |H2(x)||M(x+ic/πW2)|+c2W2=-Im ϕ(x+ic/πW2)-2u0cW2.
M(x+ic/πW2)=--M(x)exp(2πiux)dx×exp(2cu/W2)exp(-2πixu)du.
D(x)=-Im ψ(x+ic/πW2),
ψ(x)n=1Nan cos nπlx+bn sin nπlx,
Im ψ(x+ic/πW2)n=1N-an sin nπlx+bn cos nπlx×sinhncW2l.
D(x)n=1Nan sin nπlx-bn cos nπlxsinhncW2l.

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