Abstract

Fourier phase-retrieval algorithms are modified and applied to in-line holography, where phase is lost during the hologram recording process. Retrieval of phase permits separation of real-object distributions from the twin-image interference that accompanies conventional optical reconstruction. The rate of convergence is enhanced by the availability of a good initial guess based on the digital equivalent of conventional optical reconstruction.

© 1987 Optical Society of America

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References

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  1. G. Liu, “Digital signal processing methods in optical holography: phase retrieval, reconstruction and noise effects,” Ph.D. dissertation (State University of New York at Buffalo, Amherst, N.Y., 1985).
  2. G. Liu, P. D. Scott, D. T. Shaw, “Modeling and sampling strategies for digital in-line holography,” J. Opt. Soc. Am. A 1, 1220 (A) (1984).
  3. G. Liu, P. D. Scott, “Phase retrieval for in line holograms,” in Proceedings of the Nineteenth Annual Conference on Information Sciences and Systems (Johns Hopkins U. Press, Baltimore, Md., 1985), pp. 237–241.
  4. L. Onural, P. D. Scott, “A digital filtering system for decoding in-line holograms,” in Proceedings of the 1985 IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 708–711.
  5. L. Onural, “Digital decoding of in-line holograms,” Ph.D. dissertation (State University of New York at Buffalo, Amherst, New York, 1985).
  6. G. A. Tyler, B. T. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
    [CrossRef]
  7. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  8. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  9. J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithms,”J. Opt. Soc. Am. 1, 1320 (A) (1984).
  10. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  11. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics. I: Test measurements,”J. Phys. D. 6, 2200–2216 (1973).
    [CrossRef]
  12. R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity measurements,” in 1980 International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–41 (1980).
    [CrossRef]
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2768 (1982).
    [CrossRef] [PubMed]
  14. R. Rolleston, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
    [CrossRef] [PubMed]

1986 (1)

1984 (2)

G. Liu, P. D. Scott, D. T. Shaw, “Modeling and sampling strategies for digital in-line holography,” J. Opt. Soc. Am. A 1, 1220 (A) (1984).

J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithms,”J. Opt. Soc. Am. 1, 1320 (A) (1984).

1982 (1)

1978 (1)

1976 (1)

G. A. Tyler, B. T. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

1973 (2)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics. I: Test measurements,”J. Phys. D. 6, 2200–2216 (1973).
[CrossRef]

1972 (1)

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

Boucher, R. H.

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity measurements,” in 1980 International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–41 (1980).
[CrossRef]

Fienup, J. R.

George, N.

Gerchberg, R. W.

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

Liu, G.

G. Liu, P. D. Scott, D. T. Shaw, “Modeling and sampling strategies for digital in-line holography,” J. Opt. Soc. Am. A 1, 1220 (A) (1984).

G. Liu, P. D. Scott, “Phase retrieval for in line holograms,” in Proceedings of the Nineteenth Annual Conference on Information Sciences and Systems (Johns Hopkins U. Press, Baltimore, Md., 1985), pp. 237–241.

G. Liu, “Digital signal processing methods in optical holography: phase retrieval, reconstruction and noise effects,” Ph.D. dissertation (State University of New York at Buffalo, Amherst, N.Y., 1985).

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics. I: Test measurements,”J. Phys. D. 6, 2200–2216 (1973).
[CrossRef]

Onural, L.

L. Onural, P. D. Scott, “A digital filtering system for decoding in-line holograms,” in Proceedings of the 1985 IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 708–711.

L. Onural, “Digital decoding of in-line holograms,” Ph.D. dissertation (State University of New York at Buffalo, Amherst, New York, 1985).

Rolleston, R.

Saxton, W. O.

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

Scott, P. D.

G. Liu, P. D. Scott, D. T. Shaw, “Modeling and sampling strategies for digital in-line holography,” J. Opt. Soc. Am. A 1, 1220 (A) (1984).

G. Liu, P. D. Scott, “Phase retrieval for in line holograms,” in Proceedings of the Nineteenth Annual Conference on Information Sciences and Systems (Johns Hopkins U. Press, Baltimore, Md., 1985), pp. 237–241.

L. Onural, P. D. Scott, “A digital filtering system for decoding in-line holograms,” in Proceedings of the 1985 IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 708–711.

Shaw, D. T.

G. Liu, P. D. Scott, D. T. Shaw, “Modeling and sampling strategies for digital in-line holography,” J. Opt. Soc. Am. A 1, 1220 (A) (1984).

Thompson, B. T.

G. A. Tyler, B. T. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Tyler, G. A.

G. A. Tyler, B. T. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Wackerman, C. C.

J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithms,”J. Opt. Soc. Am. 1, 1320 (A) (1984).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. R. Fienup, C. C. Wackerman, “Improved phase retrieval algorithms,”J. Opt. Soc. Am. 1, 1320 (A) (1984).

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

G. Liu, P. D. Scott, D. T. Shaw, “Modeling and sampling strategies for digital in-line holography,” J. Opt. Soc. Am. A 1, 1220 (A) (1984).

J. Phys. D (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,”J. Phys. D 6, L6–L9 (1973).
[CrossRef]

J. Phys. D. (1)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics. I: Test measurements,”J. Phys. D. 6, 2200–2216 (1973).
[CrossRef]

Opt. Acta (1)

G. A. Tyler, B. T. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Opt. Lett. (1)

Optik (1)

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

Other (5)

G. Liu, “Digital signal processing methods in optical holography: phase retrieval, reconstruction and noise effects,” Ph.D. dissertation (State University of New York at Buffalo, Amherst, N.Y., 1985).

G. Liu, P. D. Scott, “Phase retrieval for in line holograms,” in Proceedings of the Nineteenth Annual Conference on Information Sciences and Systems (Johns Hopkins U. Press, Baltimore, Md., 1985), pp. 237–241.

L. Onural, P. D. Scott, “A digital filtering system for decoding in-line holograms,” in Proceedings of the 1985 IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 708–711.

L. Onural, “Digital decoding of in-line holograms,” Ph.D. dissertation (State University of New York at Buffalo, Amherst, New York, 1985).

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity measurements,” in 1980 International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–41 (1980).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) In-line Fresnel hologram formation procedure. (b) Hologram-reconstruction procedure.

Fig. 2
Fig. 2

The hologram error-reduction loop block diagram. For modified input–output approach, take ψ(n, m) as the input (see text).

Fig. 3
Fig. 3

(a) Simulated one-dimensional object, (b) corresponding hologram, (c) reconstructed image. (d) Results of twin-image elimination after (d) 10 iterations, (e) 50 iterations, (f) 300 iterations, (g) 1500 iterations.

Fig. 4
Fig. 4

(a), (b), (c), (d), (e), (f) Corresponding profiles of Figs, 3(a), 3(c), 3(d), 3(f), and 3(g).

Fig. 5
Fig. 5

(a), (b), (c), (d) Rectangular object’s reconstructed image and estimated objects after 3, 10, and 400 iterations, respectively.

Fig. 6
Fig. 6

(a), (b), (c), (d) Simulation results of the reconstructed image and results after 3, 10, and 400 iterations, respectively. The object is a simulated aerosol.

Fig. 7
Fig. 7

(a) The real digitized optical hologram’s reconstruction. (b) Result after 5 iterations; (c), (d) another example.

Fig. 8
Fig. 8

The curve shows the relationship between the number of iterations and the squared error.

Equations (19)

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ψ ( x , y ) = B j λ z exp ( j κ z ) - - [ 1 - a ( ξ , η ) ] × exp { j κ [ ( x - ξ ) 2 + ( y - η ) 2 ] / 2 z } d ξ d η ,
ψ ( x , y ) = B exp ( j κ z ) [ 1 - a ( x , y ) ] * * h z ( x , y ) ,
h z ( x , y ) = 1 j λ z exp [ ( j π λ z ) ( x 2 + y 2 ) ] .
ψ ( x , y ) = [ 1 - a ( x , y ) ] * * h z ( x , y ) .
i x ( x , y ) = ψ ( x , y ) ψ * ( x , y ) = [ 1 - a ( x , y ) ] * * h x ( x , y ) 2 .
R z ( x , y ) = [ 1 + I z ( x , y ) ] * * h z ( x , y ) ,
I R ( x , y ) = R z ( x , y ) R z * ( x , y ) ,
h z ( x , y ) = 1 j λ z exp [ j π j λ z ( x 2 + y 2 ) ] FT H z ( μ , ν ) = exp [ - j λ z 4 π ( μ 2 + ν 2 ) ] ,
L y ( 2 z λ f ¯ y ) / [ 1 - ( λ f ¯ y ) 2 ] 1 / 2 ,
( λ f _ y ) [ 1 - ( λ f _ y ) 2 ] 1 / 2 = 2 ( λ f ¯ y ) M [ 1 - ( λ f ¯ y ) 2 ] 1 / 2 .
1 / f y = M 2 f ¯ y [ 1 - ( λ f ¯ y ) 1 - ( λ f _ y ) 2 ] 1 / 2 .
a ( x , y ) exp [ j θ ( x , y ) ] FT A ( μ , ν ) exp [ j θ ( μ , ν ) ] ,
R z ( x , y ) = 2 - 2 a ( x , y ) + I 2 z ( x , y ) .
h z ( x , y ) * * h z * ( x , y ) = δ ( x , y ) .
ψ ( x , y ) * * h z * ( x , y ) = 1 - a ( x , y )
ψ k + 1 ( x , y ) = ψ k ( x , y ) - γ ( x , y ) ψ k ( x , y ) ,
γ ( x , y ) = β [ ψ k ( x , y ) - ψ ( x , y ) ] .
β = 1 / ψ ( x , y ) max ,
= 10 log 10 [ n = 1 64 m = 1 64 a ( n , m ) - a ¯ k ( n , m ) 2 ] ,

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