Abstract

The quality of reconstructed images from in-line holograms can be seriously degraded by the linear superposition of twin images having the same information but different foci. Starting from the reconstructed field at the real image plane, we make use of the uncontaminated information contained in the out-of-focus wave (virtual image) outside the in-focus wave (real image) support, together with a finite-support constraint, to form an iterative procedure for twin-image elimination. This algorithm can reconstruct complex objects, provided that they are not recorded in very near-field conditions. For real objects additional constraints can be imposed, extending the algorithm application to very near-field conditions. The algorithm’s convergence properties are studied in both cases, and some examples are shown.

© 1993 Optical Society of America

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  1. M. R. Howells, M. A. Iarocci, J. Kirz, “Experiments in x-ray holographic microscopy using synchrotron radiation,” J. Opt. Soc. Am. A 3, 2171–2178 (1986).
    [CrossRef]
  2. C. Jacobsen, M. R. Howells, J. Kirz, S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A 7, 1847–1860 (1990).
    [CrossRef]
  3. D. Joyeux, S. Lowenthal, F. Polack, A. Bernstein, “X-ray microscopy with holography at L.U.R.E.,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 246–252.
    [CrossRef]
  4. J. B. De Velis, G. B. Parrent, B. J. Thompson, “Image reconstruction with Fraunhofer holograms,” J. Opt. Soc. Am. 56, 423–427 (1966).
    [CrossRef]
  5. G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied particle analysis—a reassessment,” Opt. Acta 23, 685–700 (1976).
    [CrossRef]
  6. G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
    [CrossRef]
  7. L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
    [CrossRef]
  8. K. A. Nugent, “Twin-image elimination in Gabor holography,” Opt. Commun. 78, 293–299 (1990).
    [CrossRef]
  9. G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
    [CrossRef]
  10. K. H. S. Marie, J. C. Benett, A. P. Anderson, “Digital processing technique for suppressing the interfering outputs in the image from an in-line hologram,” Electron. Lett. 15, 241–243 (1979).
    [CrossRef]
  11. A. Lannes, “Correction des effets de l’image jumelle et du term quadratique en holographie de Gabor,” Opt. Commun. 20, 356–358 (1977).
    [CrossRef]
  12. G. Koren, D. Joyeux, F. Polack, “Twin-image elimination in in-line holography of finite-support complex objects,” Opt. Lett. 16, 1979–1981 (1991).
    [CrossRef] [PubMed]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–55.
  14. R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
    [CrossRef]
  15. V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
    [CrossRef]
  16. W. Rudin, Principles of Mathematical Analysis, 3rd ed. (McGraw-Hill, New York, 1976), pp. 220–221.
  17. Theorem (contraction mapping): If ℳ is a complete metric space and if Tis a contraction of ℳ into ℳ, then there exists one and only one u∈ ℳ such that T(u) = u[note: all Euclidean spaces are complete (see Ref. 16, pp. 53, 54)].

1991 (2)

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

G. Koren, D. Joyeux, F. Polack, “Twin-image elimination in in-line holography of finite-support complex objects,” Opt. Lett. 16, 1979–1981 (1991).
[CrossRef] [PubMed]

1990 (2)

1987 (2)

1986 (1)

1981 (2)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

1979 (1)

K. H. S. Marie, J. C. Benett, A. P. Anderson, “Digital processing technique for suppressing the interfering outputs in the image from an in-line hologram,” Electron. Lett. 15, 241–243 (1979).
[CrossRef]

1977 (1)

A. Lannes, “Correction des effets de l’image jumelle et du term quadratique en holographie de Gabor,” Opt. Commun. 20, 356–358 (1977).
[CrossRef]

1976 (1)

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

1966 (1)

Anderson, A. P.

K. H. S. Marie, J. C. Benett, A. P. Anderson, “Digital processing technique for suppressing the interfering outputs in the image from an in-line hologram,” Electron. Lett. 15, 241–243 (1979).
[CrossRef]

Benett, J. C.

K. H. S. Marie, J. C. Benett, A. P. Anderson, “Digital processing technique for suppressing the interfering outputs in the image from an in-line hologram,” Electron. Lett. 15, 241–243 (1979).
[CrossRef]

Bernstein, A.

D. Joyeux, S. Lowenthal, F. Polack, A. Bernstein, “X-ray microscopy with holography at L.U.R.E.,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 246–252.
[CrossRef]

De Velis, J. B.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–55.

Hayes, M. H.

V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Howells, M. R.

Hua, L. F.

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

Iarocci, M. A.

Jacobsen, C.

Joyeux, D.

G. Koren, D. Joyeux, F. Polack, “Twin-image elimination in in-line holography of finite-support complex objects,” Opt. Lett. 16, 1979–1981 (1991).
[CrossRef] [PubMed]

D. Joyeux, S. Lowenthal, F. Polack, A. Bernstein, “X-ray microscopy with holography at L.U.R.E.,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 246–252.
[CrossRef]

Kirz, J.

Koren, G.

Lannes, A.

A. Lannes, “Correction des effets de l’image jumelle et du term quadratique en holographie de Gabor,” Opt. Commun. 20, 356–358 (1977).
[CrossRef]

Liu, G.

Lowenthal, S.

D. Joyeux, S. Lowenthal, F. Polack, A. Bernstein, “X-ray microscopy with holography at L.U.R.E.,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 246–252.
[CrossRef]

Marie, K. H. S.

K. H. S. Marie, J. C. Benett, A. P. Anderson, “Digital processing technique for suppressing the interfering outputs in the image from an in-line hologram,” Electron. Lett. 15, 241–243 (1979).
[CrossRef]

McClellan, J. H.

V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Nugent, K. A.

K. A. Nugent, “Twin-image elimination in Gabor holography,” Opt. Commun. 78, 293–299 (1990).
[CrossRef]

Onural, L.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Parrent, G. B.

Patel, S.

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

Polack, F.

G. Koren, D. Joyeux, F. Polack, “Twin-image elimination in in-line holography of finite-support complex objects,” Opt. Lett. 16, 1979–1981 (1991).
[CrossRef] [PubMed]

D. Joyeux, S. Lowenthal, F. Polack, A. Bernstein, “X-ray microscopy with holography at L.U.R.E.,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 246–252.
[CrossRef]

Quartieri, T. F.

V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Rothman, S.

Rudin, W.

W. Rudin, Principles of Mathematical Analysis, 3rd ed. (McGraw-Hill, New York, 1976), pp. 220–221.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Scott, P. D.

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
[CrossRef]

Shaw, D. T.

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

Thompson, B. J.

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

J. B. De Velis, G. B. Parrent, B. J. Thompson, “Image reconstruction with Fraunhofer holograms,” J. Opt. Soc. Am. 56, 423–427 (1966).
[CrossRef]

Tom, V T.

V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

Tyler, G. A.

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

Xie, G. W.

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

Electron. Lett. (1)

K. H. S. Marie, J. C. Benett, A. P. Anderson, “Digital processing technique for suppressing the interfering outputs in the image from an in-line hologram,” Electron. Lett. 15, 241–243 (1979).
[CrossRef]

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

V T. Tom, T. F. Quartieri, M. H. Hayes, J. H. McClellan, “Convergence of iterative nonexpansive signal reconstruction algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052–1058 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

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

Opt. Commun. (2)

K. A. Nugent, “Twin-image elimination in Gabor holography,” Opt. Commun. 78, 293–299 (1990).
[CrossRef]

A. Lannes, “Correction des effets de l’image jumelle et du term quadratique en holographie de Gabor,” Opt. Commun. 20, 356–358 (1977).
[CrossRef]

Opt. Eng. (2)

G. W. Xie, L. F. Hua, S. Patel, P. D. Scott, D. T. Shaw, “Using dynamic holography for iron fibers with sub-micron diameter and high velocity,” Opt. Eng. 30, 1306–1314 (1991).
[CrossRef]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Other (4)

D. Joyeux, S. Lowenthal, F. Polack, A. Bernstein, “X-ray microscopy with holography at L.U.R.E.,” in X-ray Microscopy II, D. Sayre, M. Howells, J. Kirz, H. Rarback, eds., Vol. 56 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1988), pp. 246–252.
[CrossRef]

W. Rudin, Principles of Mathematical Analysis, 3rd ed. (McGraw-Hill, New York, 1976), pp. 220–221.

Theorem (contraction mapping): If ℳ is a complete metric space and if Tis a contraction of ℳ into ℳ, then there exists one and only one u∈ ℳ such that T(u) = u[note: all Euclidean spaces are complete (see Ref. 16, pp. 53, 54)].

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–55.

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Figures (8)

Fig. 1
Fig. 1

Block diagram of the twin-image elimination algorithm.

Fig. 2
Fig. 2

Results of twin-image elimination algorithm applied to an in-line hologram of a complex object, Nf = 100. (a) Original object amplitude, (b) original object phase, (c) naturally reconstructed amplitude, (d) naturally reconstructed phase, (e) reconstructed object amplitude after ten iterations, (f) reconstructed object phase. (g), (h), (i), (j), (k), (1) are vertical sections of (a), (b), (c), (d), (e), (f), respectively.

Fig. 3
Fig. 3

Same as Fig. 2 but Nf = 10.

Fig. 4
Fig. 4

Rms error estimation, i.e., rms difference between the true object and its estimation as a function of the number of iterations (complex objects). Recording conditions: Nf = 100 (dotted curve) and Nf = 10 (dashed curve).

Fig. 5
Fig. 5

Block diagram of the modified twin-image elimination algorithm for real (i.e., zero phase) objects.

Fig. 6
Fig. 6

Results of application of the modified twin-image elimination algorithm to an in-line hologram of a real object, Nf = 100. (a) Original object amplitude, (b) original object phase, (c) naturally reconstructed amplitude, (d) naturally reconstructed phase, (e) reconstructed object amplitude after ten iterations, (f) reconstructed object phase. (g), (h), (i), (j), (k), (l) are vertical sections of (a), (b), (c), (d), (e), (f), respectively.

Fig. 7
Fig. 7

Same as Fig. 6, but Nf = 10.

Fig. 8
Fig. 8

Rms error between the true object and its estimation, as a function of the number of iterations, for the modified algorithm applied to a real object. Recording conditions: Nf = 100 (dotted curve) and Nf = 10 (dashed curve).

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

U h ( x ) = H z + [ 1 a ( x ) ] ,
t h ( x ) = α + β I h ( x ) .
U r ( x ) = H z + [ t h ( x ) ] .
U r ( x ) = D a * ( x ) H 2 z + [ a ( x ) ] + H z + { | H z + [ a ( x ) ] | 2 } .
G ( x ) = | grad | U r ( x ) | 2 | .
Û o ( 1 ) ( x ) = { U r ( x ) x Σ Ū r ( x ) x Σ ,
Û i ( 1 ) ( x ) = H 2 z [ Û o ( 1 ) ( x ) ] .
Û i ( 2 ) ( x ) = C Σ D [ Û i ( 1 ) ( x ) ] ,
C Σ D [ Û ( x ) ] = { D D Û ( x ) γ x Σ Û ( x ) x Σ .
Û o ( 2 ) ( x ) = H 2 z + [ Û i ( 2 ) ( x ) ] .
Û o ( 2 ) ( x ) = S Σ [ Û o ( 2 ) ( x ) ] ,
S Σ [ Û o ( x ) ] = { Û o ( x ) x Σ U r ( x ) x Σ .
Û o ( k + 1 ) ( x ) = S Σ H 2 z + C Σ D H 2 z [ Û o ( k ) ( x ) ] ,
Û i ( k + 1 ) ( x ) = C Σ D H 2 z S Σ H 2 z + [ Û i ( k ) ( x ) ] .
d [ T ( u ) , T ( υ ) ] c d ( u , υ ) for all u , υ ,
d [ U 1 ( x ) , U 2 ( x ) ] = { x | U 1 ( x ) U 2 ( x ) | 2 d x } 1 / 2 .
d ( U 1 , U 2 ) = U 1 U 2 .
C Σ D [ Û i ( x ) ] C Σ D [ Û i ( x ) ] = { x | C Σ D [ Û i ( x ) ] C Σ D [ Û i ( x ) ] | 2 d x } 1 / 2 = { x Σ | Û i ( x ) Û i ( x ) | 2 d x + x Σ | D D Û i ( x ) γ ( D D Û i γ ) | 2 d x } 1 / 2 = { x Σ | Û i ( x ) Û i ( x ) | 2 + 1 γ x Σ | Û i ( x ) Û i ( x ) | 2 d x } 1 / 2 .
C Σ D [ Û i ( x ) ] C Σ D [ Û i ( x ) ] Û i ( x ) Û i ( x ) ,
S Σ [ Û o ( x ) ] S Σ [ Û o ( x ) ] Û o ( x ) Û o ( x ) ,
Û o ( x ) Û o ( x ) = H 2 z + [ Û i ( x ) ] H 2 z + [ Û i ( x ) ] = { x | h 2 z + * [ Û i ( x ) Û i ( x ) ] | 2 d x } 1 / 2 .
x | h 2 z ( x ) * Û i ( x ) | 2 d x = x | Û i ( x ) | 2 d x ;
Û o ( x ) Û o ( x ) = Û i ( x ) Û i ( x ) .
O = C Σ D H 2 z S Σ H 2 z + .
Û i ( k + 1 ) ( x ) U i ( x ) = O [ Û i ( k ) ( x ) ] O [ U i ( x ) ] ,
Û i ( k + 1 ) ( x ) U i ( x ) Û ( k ) i ( x ) U i ( x ) .
k + 1 k .
Δ k = { x | [ Û i ( k ) ( x ) Û i ( k 1 ) ( x ) ] | 2 d x } 1 / 2 .
k ( rms ) = { x | [ Û i ( k ) ( x ) U i ( x ) ] | 2 d x } 1 / 2 .
Im { H 2 z + [ a ( x ) ] } = Im [ U r ( x ) ] ,
Z [ Û i ( x ) ] = | Û i ( x ) |
P Σ [ Û o ( k ) ( x ) ] = { Re [ Û o ( k ) ( x ) ] + j Im [ U r ( x ) ] x Σ Û o ( k ) ( x ) x Σ
Û o ( 1 ) ( x ) = { | Û o ( 1 ) ( x ) | exp [ j φ r ( x ) ] x Σ U r ( x ) x Σ ,
| Û o ( 1 ) ( x ) | = D 2 + | U r ( x ) | 2
Û i ( k + 1 ) ( x ) = Z C Σ D H 2 z P Σ S Σ H 2 z + [ Û i ( k ) ( x ) ] .
Z [ Û i ( x ) ] = | Û i ( x ) | , Z [ Û i ( x ) ] = | Û i ( x ) | .
Z [ Û i ( x ) ] Z [ Û i ( x ) ] = | Û i ( x ) | | Û i ( x ) | .
| Û i ( x ) | | Û i ( x ) | Û i ( x ) Û i ( x ) ,
P Σ [ Û i ( x ) ] P Σ [ Û i ( x ) ] = Re [ Û i ( x ) ] Re [ Û i ( x ) ] Û i ( x ) Û i ( x ) ,

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