Abstract

Finite plotter resolution introduces quantization into the representation of digital holograms. By utilizing the statistical properties of the Fourier transform of random phase images, optimum quantization schemes are derived and tabulated for the representation of these transforms. These schemes are applied to Lohmann and Lee type holograms where it is found that measured quantization errors agree with theoretical predictions. Methods for predicting optimum quantization schemes and associated mean squared errors are simplified to table-lookup procedures.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Gabel, Appl. Opt. 14, 2252 (1975).
    [CrossRef] [PubMed]
  2. W. A. Pearlman, Ph.D. Thesis, Stanford U., Stanford Calif., 1974 (University Microfilms, Ann Arbor, Mich.).
  3. R. S. Powers, J. W. Goodman, Appl. Opt. 14, 1690 (1975).
    [CrossRef] [PubMed]
  4. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  5. W. H. Lee, Appl. Opt. 9, 639 (1970).
    [CrossRef] [PubMed]
  6. R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
    [CrossRef] [PubMed]
  7. N. C. Gallagher, B. Liu, Optik 42, 65 (1975).
  8. L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1960).
    [CrossRef]
  9. J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
    [CrossRef]
  10. P. E. Fleischer, IEEE Int. Conv. Rec., Part I, 104 (1964).
  11. W. J. Dallas, Appl. Opt. 13, 2274 (1974).
    [CrossRef] [PubMed]
  12. W. H. Lee, Appl, Opt. 13, 1677 (1974).
    [CrossRef]
  13. J. P. Allebach, N. C. Gallagher, B. Liu, Appl. Opt. 15, 2183 (1976).
    [CrossRef] [PubMed]

1976

1975

1974

1970

1967

1964

P. E. Fleischer, IEEE Int. Conv. Rec., Part I, 104 (1964).

1960

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1960).
[CrossRef]

J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
[CrossRef]

Allebach, J. P.

Dallas, W. J.

Fleischer, P. E.

P. E. Fleischer, IEEE Int. Conv. Rec., Part I, 104 (1964).

Gabel, R. A.

Gallagher, N. C.

Goodman, J. W.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1960).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1960).
[CrossRef]

Lee, W. H.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1960).
[CrossRef]

Liu, B.

Lohmann, A. W.

Max, J.

J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
[CrossRef]

Paris, D. P.

Pearlman, W. A.

W. A. Pearlman, Ph.D. Thesis, Stanford U., Stanford Calif., 1974 (University Microfilms, Ann Arbor, Mich.).

Powers, R. S.

Appl, Opt.

W. H. Lee, Appl, Opt. 13, 1677 (1974).
[CrossRef]

Appl. Opt.

IBM J. Res. Dev.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1960).
[CrossRef]

IEEE Int. Conv. Rec.

P. E. Fleischer, IEEE Int. Conv. Rec., Part I, 104 (1964).

IRE Trans. Inf. Theory

J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
[CrossRef]

Optik

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

Other

W. A. Pearlman, Ph.D. Thesis, Stanford U., Stanford Calif., 1974 (University Microfilms, Ann Arbor, Mich.).

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 (2)

Fig. 1
Fig. 1

Reconstructed image of NASA from Lohmann hologram.

Fig. 2
Fig. 2

Reconstructed image of NASA from Lee hologram.

Tables (2)

Tables Icon

Table I Parameters for the Optimum Quantizer

Tables Icon

Table II Parameters for the Optimum Equally Spaced Level Quantizer

Equations (36)

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

f ( a , θ ) = g ( a ) h ( θ ) ,
E = n = 1 N 1 m = 1 N 2 e n e n + 1 γ m γ m + 1 | ae i θ x n e i α m | 2 f ( a , θ ) da d θ ,
E e n = 0 n = 2 , . . . , N 1 ,
E x n = 0 n = 1 , . . . , N 1 ,
E α m = 0 m = 1 , . . . , N 2 ,
E γ m = 0 m = 2 , . . . , N 2 ,
e n = x n + x n 1 2 π / N 2 sin π / N 2 ,
e n e n + 1 ( x n a sin π / N 2 π / N 2 ) g ( a ) da = 0 ,
γ m = ( α m + α m 1 ) / 2 ,
α m = ( γ m + γ m + 1 ) / 2 .
( x n e n + 1 ) exp ( e n + 1 2 / 2 σ 2 ) + ( e n x n ) exp ( e n 2 / 2 σ 2 ) + σ ( π 2 ) 1 / 2 ] erf ( e n + 1 σ 2 ) erf ( e n σ 2 ) ] = 0 ,
g ( a ) = a σ 2 exp ( a 2 / 2 σ 2 ) , a 0 .
erf ( x ) = 2 π 0 x exp ( t 2 ) dt .
E = n = 1 N 1 e n e n + 1 ( a x n ) 2 g ( a ) da + 2 ( 1 sin π / N 2 π / N 2 ) × n = 1 N 1 x n e n e n + 1 ag ( a ) da .
n = 1 N 1 x n e n e n + 1 ag ( a ) da 0 a 2 g ( a ) da = 2 σ 2 ;
E = n = 1 N 1 e n e n + 1 ( a x n ) 2 g ( a ) da + 4 σ 2 ( 1 sin π / N 2 π / N 2 ) .
E p = 4 σ 2 ( 1 sin π / N 2 π / N 2 ) ,
E M = n = 1 N 1 e n e n + 1 ( a x n ) 2 g ( a ) da = n = 1 N 1 { exp ( e n 2 / 2 σ 2 ) [ ( e n x n ) 2 + 2 σ 2 ] exp ( e n + 1 2 / 2 σ 2 ) × [ ( e n + 1 x n ) 2 + 2 σ 2 ] x n σ ( 2 π ) 1 / 2 [ erf ( e n + 1 / σ 2 ) erf ( e n / σ 2 ) ] } .
e n = ( n 1 ) Δ
x n = ( Δ / 2 ) ( 2 n 1 ) , n = 1 , 2 , . . . , N 1 ,
δ 2 + n = 1 N 1 1 { ( 2 π ) 1 / 2 erf ( n δ 2 ) + 2 n δ exp [ ( n δ ) 2 / 2 ] } ( π 2 ) 1 / 2 ( 2 N 1 ) = 0 ,
E M = n = 1 N 1 e n e n + 1 ( a x n ) 2 g ( a ) da = σ 2 ( δ 2 4 + 2 ) + δ ( π 2 ) 1 / 2 [ 2 n = 1 N 1 1 erf ( n δ 2 ) ( 2 N 1 1 ) ] } .
E / 2 σ 2 = ϵ [ | ( z 1 z ̂ 1 ) + i ( z 2 z ̂ 2 ) | 2 ] / 2 σ 2 ,
E / 2 σ 2 = 1 σ 2 ϵ [ ( z 1 z ̂ 1 ) 2 ] = 1 σ 2 ϵ [ ( z 2 z ̂ 2 ) 2 ] .
N 1 N 2 N 1 N 2 400 .
A exp ( i ψ ) = ( R P R n ) + i ( I P I n ) ,
e n = ( Δ / 2 ) + ( n 2 ) Δ x n = ( n 1 ) Δ , n = 2 , . . . , N 1 ,
Δ = 1.67 × 10 3
E = 1.96 × 10 6 .
2 σ 2 = 1 N 2 p , q N = 0 A pq 2 ,
Δ = ( 0.256 ) ( 6.12 × 10 13 ) = 1.57 × 10 3 .
E = E M + E P = 1.94 × 10 6 .
Δ = 1.41 × 10 3
E = 3.76 × 10 7 .
Δ = ( 0.2315 ) σ = ( 0.2315 ) ( 6.12 × 10 3 ) = 1.42 × 10 3
E = ( 5.355 × 10 3 ) ( 2 σ 2 ) = ( 5.355 × 10 3 ) ( 7.49 × 10 5 ) = 4.01 × 10 7 .

Metrics