Abstract

The use of the discrete Fourier transform in digital holography introduces aliasing error in the reconstructed image. Spectrum shaping to reduce dynamic range may also result in a serious increase in aliasing error. The effect of aliasing in digital holography is analyzed. It is proposed that the bandwidth be constrained when shaping the image spectrum. Experimental results show the approach to be quite effective.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G-AE Subcommittee on Measurement Concepts, IEEE Trans. Audio Electroacoust. AU-15, 45 (1967).
  2. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [CrossRef]
  3. C. B. Burckhardt, Appl. Opt. 9, 695 (1970).
    [CrossRef] [PubMed]
  4. Y. Takeda, Y. Oshida, Y. Miyamura, Appl. Opt. 11, 818 (1972).
    [CrossRef] [PubMed]
  5. R. H. Katyl, Appl. Opt. 11, 198 (1972).
    [CrossRef] [PubMed]
  6. W. J. Dallas, Appl. Opt. 12, 1179 (1973).
    [CrossRef]
  7. D. C. Chu, J. Opt. Soc. Am. 64, 1395A (1974).
  8. D. R. Rothschild, Ph.D. Thesis, University of Michigan, Ann Arbor (1966).
  9. P. M. Hirsch et al., U.S. Patent3,619,022 (9November1971).
  10. N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  11. J. P. Allebach, B. Liu, to appear in Appl. Opt.
  12. D. C. Chu, Ph.D. Thesis, Stanford University, 93 (1974).
  13. J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).
  14. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 345.
  15. D. C. Chu, J. R. Fienup, J. W. Goodman, Appl. Opt. 12, 1386 (1973).
    [CrossRef] [PubMed]
  16. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  17. J. Upatnieks, Appl. Opt. 6, 1905 (1967).
    [CrossRef] [PubMed]
  18. E. N. Leith, J. Upatnieks, Appl. Opt. 7, 2085 (1968).
    [CrossRef] [PubMed]
  19. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
    [CrossRef] [PubMed]
  20. N. C. Gallagher, B. Liu, Optik 42, 65 (1975).
  21. R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
    [CrossRef] [PubMed]
  22. B. Liu, N. C. Gallagher, J. Opt. Soc. Am. 64, 1227 (1974).
    [CrossRef]

1975 (1)

N. C. Gallagher, B. Liu, Optik 42, 65 (1975).

1974 (3)

D. C. Chu, J. Opt. Soc. Am. 64, 1395A (1974).

J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).

B. Liu, N. C. Gallagher, J. Opt. Soc. Am. 64, 1227 (1974).
[CrossRef]

1973 (3)

1972 (2)

1970 (2)

1968 (2)

1967 (3)

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
[CrossRef] [PubMed]

J. Upatnieks, Appl. Opt. 6, 1905 (1967).
[CrossRef] [PubMed]

G-AE Subcommittee on Measurement Concepts, IEEE Trans. Audio Electroacoust. AU-15, 45 (1967).

1964 (1)

Allebach, J. P.

J. P. Allebach, B. Liu, to appear in Appl. Opt.

Burckhardt, C. B.

C. B. Burckhardt, Appl. Opt. 9, 695 (1970).
[CrossRef] [PubMed]

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 345.

Chu, D. C.

D. C. Chu, J. Opt. Soc. Am. 64, 1395A (1974).

D. C. Chu, J. R. Fienup, J. W. Goodman, Appl. Opt. 12, 1386 (1973).
[CrossRef] [PubMed]

D. C. Chu, Ph.D. Thesis, Stanford University, 93 (1974).

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 345.

Dallas, W. J.

Fienup, J. R.

Gabel, R. A.

Gallagher, N. C.

Gerritsen, H. J.

Goodman, J. W.

Hannan, W. J.

Hirsch, P. M.

P. M. Hirsch et al., U.S. Patent3,619,022 (9November1971).

Katyl, R. H.

Leith, E. N.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 345.

Liu, B.

Lohmann, A. W.

Miyamura, Y.

Oshida, Y.

Paris, D. P.

Ramberg, E. G.

Rothschild, D. R.

D. R. Rothschild, Ph.D. Thesis, University of Michigan, Ann Arbor (1966).

Takeda, Y.

Upatnieks, J.

Appl. Opt. (11)

IEEE Trans. Audio Electroacoust. (1)

G-AE Subcommittee on Measurement Concepts, IEEE Trans. Audio Electroacoust. AU-15, 45 (1967).

J. Opt. Soc. Am. (4)

D. C. Chu, J. Opt. Soc. Am. 64, 1395A (1974).

J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[CrossRef]

B. Liu, N. C. Gallagher, J. Opt. Soc. Am. 64, 1227 (1974).
[CrossRef]

Optik (1)

N. C. Gallagher, B. Liu, Optik 42, 65 (1975).

Other (5)

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 345.

D. R. Rothschild, Ph.D. Thesis, University of Michigan, Ann Arbor (1966).

P. M. Hirsch et al., U.S. Patent3,619,022 (9November1971).

J. P. Allebach, B. Liu, to appear in Appl. Opt.

D. C. Chu, Ph.D. Thesis, Stanford University, 93 (1974).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Original image used for experimental work.

Fig. 2
Fig. 2

Second letter E of reconstructed image when the phase of the sampled image is zero.

Fig. 3
Fig. 3

Second letter E of reconstructed image when the phase of the sampled image is randomly distributed on [−π,π).

Fig. 4
Fig. 4

Second letter E of reconstructed image when the phase of the sampled image is chosen via iterative method in which spectrum is shaped to occupy the full spectral window.

Fig. 5
Fig. 5

Second letter E of reconstructed image when the phase of the sampled image is chosen via iterative method in which spectrum is shaped to occupy half of the spectral window.

Fig. 6
Fig. 6

Image reconstructed from Lohmann hologram when the phase of the sampled image is zero.

Fig. 7
Fig. 7

Image reconstructed from Lohmann hologram when the phase of the sampled image is randomly distributed on [−π,−π).

Fig. 8
Fig. 8

Image reconstructed from Lohmann hologram when the phase of the sampled image is chosen via iterative method in which spectrum is shaped to occupy the full spectral window.

Fig. 9
Fig. 9

Image reconstructed from Lohmann hologram when the phase of the sampled image is chosen via iterative method in which spectrum is shaped to occupy half of the spectral window.

Fig. 10
Fig. 10

Image reconstructed when spectrum is repeated on a 4 × 4 grid. The phase of the sampled image was chosen via iterative method in which spectrum is shaped to occupy the full spectral window

Fig. 11
Fig. 11

Image reconstructed when the spectrum is repeated on a 4 × 4 grid. The phase of the sampled image was chosen via iterative method in which spectrum is shaped to occupy half of the spectral window.

Tables (1)

Tables Icon

Table I Statistics of the Reconstructed Image for Four Choices of Image Phase

Equations (22)

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

a ( x , y ) = 0 , x , y > X / 2.
a ( x , y ) = 1 X 2 m , n = - A m n exp [ 2 π i ( m x + n y ) / X ] ,
A m n = - X / 2 X / 2 a ( x , y ) exp [ - 2 π i ( m x + n y ) / X ] d x d y .
H ( u , v ) = m , N / 2 - 1 n = - N / 2 A m n f ( u - m B / N B / N ) f [ v - n B / N B / N ] ,
a r ( x , y ) = 1 X 2 m , N / 2 - 1 n = - N / 2 A m n exp [ 2 π i ( m x + n y ) / X ] ,
H ( u ) = n = - N / 2 N / 2 - 1 A n f ( u - n B / N ) B / N
a r ( x ) = 1 X n = - N / 2 N / 2 - 1 A n exp ( 2 π i n x / X ) .
A n = - X / 2 X / 2 a ( x ) exp ( - 2 π i n x / X ) d x ,
A K * = X N n = - N / 2 N / 2 - 1 a ( n X / N ) exp ( - 2 π i n k / N ) , - N / 2 k < N / 2 .
A K * = n = - A k + n N .
H ( u ) = n = - N / 2 N / 2 - 1 A n * f ( u - n B / N B / N ) ,
a r ( x ) = 1 X n = - N / 2 N / 2 - 1 A n * exp ( 2 π i n x / X ) .
a r ( x ) = n = - N 2 N 2 - 1 a ( n X / N ) 1 N m = - N 2 N 2 - 1 exp [ 2 π i m ( x - n X / N ) / X ] .
g * ( x ) = k = - g ( x + k X ) ,
g * ( n ) = k = - g ( n + k N ) ,
1 N m = - N 2 N 2 - 1 exp ( 2 π i m x / X ) = k = - sinc ( x - k X X / N ) - i N sin ( π N x / X ) .
a r ( x ) = n = - N 2 N 2 - 1 a ( n X / N ) sinc * ( x - n X / N ) - i N sin ( π N x / X ) n = - N 2 N 2 - 1 ( - 1 ) n a ( n X / N ) .
a r ( n X / N ) = a ( n X / N ) , - N / 2 n < N / 2.
Φ = - X / 2 X / 2 [ a r ( x ) - b ( x ) ] 2 d x .
Φ = 1 X n I A n 2 + 1 X n I k = - A n - k N 2 ,
1 X n I A n 2 Φ 2 X n I A n 2 ,
W 2 = 1 B n = - N 2 N 2 - 1 W n A n * 2

Metrics