Abstract

Two phase retrieval methods are used for the reconstruction of a 2-D object function from its. Fourier transformed moduli obtained with and without an exponential filter at the object plane. One of the methods is a phase retrieval method using only the Fourier series expansion, and the other is a phase retrieval method using the logarithmic Hilbert transform and the Fourier series expansion. These methods extend the range that successful reconstructions of 2-D objects can be obtained by making use of the properties of entire functions of the exponential type. The usefulness of these methods is shown in computer simulation studies of the reconstructions of 2-D real and phase objects from their Fourier moduli.

© 1989 Optical Society of America

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References

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  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 (1972).
  2. J. R. Fienup, “Phase Retrieval Algorithms: a Comparison,” Appl. Opt. 21, 2758 (1982).
    [Crossref] [PubMed]
  3. J. G. Walker, “Computer Simulation of a Method for Object Reconstruction from Stellar Speckle Interferometry Data,” Appl. Opt. 21, 3132 (1982).
    [Crossref] [PubMed]
  4. R. H. T. Bates, W. R. Fright, “Composite Two-Dimensional Phase-Restoration Procedure,” J. Opt. Soc. Am. 73, 358 (1983).
    [Crossref]
  5. A. Levi, H. Stark, “Image Restoration by the Method of Generalized Projections with Application to Restoration from Magnitude,” J. Opt. Soc. Am. A 1, 932 (1984).
    [Crossref]
  6. J. R. Fienup, C. C. Wackerman, “Phase-Retrieval Stagnation Problems and Solutions,” J. Opt. Soc. Am. A 3, 1897 (1986).
    [Crossref]
  7. A. Walther, “The question of Phase Retrieval in Optics,” Opt. Act 10, 41 (1963).
    [Crossref]
  8. B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys. 16, 1719 (1975).
    [Crossref]
  9. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The Phase Problem,” Proc. R. Soc. London Ser. A 350, 191 (1976).
    [Crossref]
  10. 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 (1986).
    [Crossref]
  11. N. Nakajima, “Phase Retrieval from Two Intensity Measurements Using the Fourier Series Expansion,” J. Opt. Soc. Am. A 4, 154 (1987).
    [Crossref]
  12. N. Nakajima, “Phase Retrieval Using the Logarithmic Hilbert Transform and the Fourier-Series Expansion,” J. Opt. Soc. Am. A 5, 257 (1988).
    [Crossref]

1988 (1)

1987 (1)

1986 (2)

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 (1986).
[Crossref]

J. R. Fienup, C. C. Wackerman, “Phase-Retrieval Stagnation Problems and Solutions,” J. Opt. Soc. Am. A 3, 1897 (1986).
[Crossref]

1984 (1)

1983 (1)

1982 (2)

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The Phase Problem,” Proc. R. Soc. London Ser. A 350, 191 (1976).
[Crossref]

1975 (1)

B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys. 16, 1719 (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 (1972).

1963 (1)

A. Walther, “The question of Phase Retrieval in Optics,” Opt. Act 10, 41 (1963).
[Crossref]

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 (1986).
[Crossref]

Bates, R. H. T.

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The Phase Problem,” Proc. R. Soc. London Ser. A 350, 191 (1976).
[Crossref]

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The Phase Problem,” Proc. R. Soc. London Ser. A 350, 191 (1976).
[Crossref]

Fienup, J. R.

Fright, W. 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 (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 (1976).
[Crossref]

Hoenders, B. J.

B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys. 16, 1719 (1975).
[Crossref]

Levi, A.

Nakajima, N.

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The Phase Problem,” Proc. R. Soc. London Ser. A 350, 191 (1976).
[Crossref]

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 (1972).

Stark, H.

Wackerman, C. C.

Walker, J. G.

Walther, A.

A. Walther, “The question of Phase Retrieval in Optics,” Opt. Act 10, 41 (1963).
[Crossref]

Appl. Opt. (2)

J. Math. Phys. (1)

B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys. 16, 1719 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

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

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 (1986).
[Crossref]

Opt. Act (1)

A. Walther, “The question of Phase Retrieval in Optics,” Opt. Act 10, 41 (1963).
[Crossref]

Optik (Stuttgart) (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 (1972).

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 (1976).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Reconstruction of a real object function by the phase retrieval method using only the Fourier series expansion: (a) the original real object function; (b) the reconstructed object function from Fourier moduli obtained with and without an exponential filter at the object plane; (c) the reconstructed object function from noisy moduli in the same way as in the noiseless case (b).

Fig. 2
Fig. 2

Reconstruction of a phase object function by the phase retrieval method using only the Fourier series expansion: (a) the modulus and (b) the phase of the original object function; (c) the modulus and (d) the phase of the reconstructed object function from Fourier moduli obtained with and without an exponential filter at the object plane; (e) the modulus and (f) the phase of the reconstructed object function from noisy moduli in the same way as in the noiseless case of (c) and (d).

Fig. 3
Fig. 3

Reconstruction of a Hermitian object function by the phase retrieval method using the logarithmic Hilbert transform and the Fourier series expansion: (a) the modulus and (b) the phase of the original Hermitian object function; (c) the modulus and (d) the phase of the reconstructed object function from Fourier moduli obtained with and without an exponential filter at the object plane; (e) the modulus and (f) the phase of the reconstructed object function from noisy moduli in the same way as in the noiseless case of (c) and (d).

Equations (17)

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F ( x 1 , x 2 ) = R f ( u 1 , u 2 ) exp [ 2 π i ( x 1 u 1 + x 2 u 2 ) ] d u 1 d u 2 ,
F ( x 1 , x 2 ) = | F ( x 1 , x 2 ) | exp [ i ϕ ( x 1 , x 2 ) ] ,
F ( x 1 i y c , x 2 ) = R exp ( 2 π y c u 1 ) f ( u 1 , u 2 ) × exp [ 2 π i ( x 1 u 1 + x 2 u 2 ) ] d u 1 d u 2 ,
F ( x 1 i y c , C ) = M ( x 1 i y c , C ) exp [ i ϕ ( x 1 i y c , C ) ] ,
M ( x 1 , x 2 ) = | F ( x 1 , x 2 ) | .
| F ( x 1 i y c , C ) | = | M ( x 1 i y c , C ) | exp [ ϕ I ( x 1 , C ) ] ,
ϕ ( x 1 i y c , C ) = ϕ R ( x 1 , C ) + i ϕ I ( x 1 , C ) .
ϕ ( x 1 , C ) n = 1 N [ a n ( C ) cos n π l x 1 + b n ( C ) sin n π l x 1 ] ,
ln | F ( x 1 i y c , C ) | | M ( x 1 i y c , C ) | n = 1 N [ a n ( C ) sin n π l x 1 + b n ( C ) cos n π l x 1 ] × sinh ( n π l y c ) .
ϕ ( x 1 , x 2 ) = ϕ ( x 1 , C ) + ϕ ( 0 , x 2 ) ,
ϕ ( x 1 , C ) = ϕ H ( x 1 , C ) + ϕ Z ( x 1 , C ) ,
ϕ H ( x 1 , C ) = x 1 π P ln | F ( x 1 , C ) | x 1 ( x 1 x 1 ) d x 1 ,
ϕ Z ( x 1 , C ) = 2 j = 1 N arg ( x 1 z j ) 2 j = 1 N arg ( z j ) ,
F ( x 1 , C ) = F H ( x 1 , C ) exp [ i ϕ Z ( x 1 , C ) ] ,
F H ( x 1 , C ) = | F ( x 1 , C ) | exp [ i ϕ H ( x 1 , C ) ] ,
ln | F ( x 1 i y c , C ) | | F H ( x 1 i y c , C ) | n = 1 N [ a n ( C ) sin n π l x 1 + b n ( C ) cos n π l x 1 ] × sinh ( n π l y c ) ,
ϕ Z ( x 1 , C ) n = 1 N [ a n ( C ) cos n π l x 1 + b n ( C ) sin n π l x 1 ] .

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