Abstract

A new approach to spectrum shaping, in which the maximum of the spectrum is minimized subject to a constraint on the bandwidth, is considered. The results can be applied to a variety of methods of digital holography. A lower bound for the maximum and a figure of merit are obtained. Practical methods for minimax shaping of the spectrum are described and compared via computer simulations. The results of these simulations indicate that the lower bound is relatively tight, and the figure of merit is a good one.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 230.
  2. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), Chap. 8.
  3. D. C. Chu, J. R. Fienup, J. W. Goodman, Appl. Opt. 12, 1386 (1973).
    [CrossRef] [PubMed]
  4. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  5. R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
    [CrossRef] [PubMed]
  6. G-AE Subcommittee on Measurement Concepts, IEEE Trans. Audio Electroacoust. AU-15, 45 (1967).
  7. J. P. Allebach, N. C. Gallagher, B. Liu, to appear in Appl. Opt.
  8. C. B. Burckhardt, Appl. Opt. 9, 695 (1970).
    [CrossRef] [PubMed]
  9. Y. Takeda, Y. Oshida, Y. Miyamura, Appl. Opt. 11, 218 (1972).
    [CrossRef]
  10. R. H. Katyl, Appl. Opt. 11, 198 (1972).
    [CrossRef] [PubMed]
  11. W. J. Dallas, Appl. Opt. 12, 1179 (1973).
    [CrossRef] [PubMed]
  12. D. C. Chu, J. Opt. Soc. Am. 64, 1395A (1974).
  13. D. R. Rothschild, Ph.D. Thesis, University of Michigan, Ann Arbor (1966).
  14. P. M. Hirsch et al., U.S. Patent3,619,022 (9November1971).
  15. N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  16. H. J. Caulfield, Appl. Opt. 9, 2587 (1970).
    [CrossRef] [PubMed]
  17. D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
    [CrossRef] [PubMed]
  18. D. C. Chu, Ph.D. Thesis, Stanford University, 93 (1974).
  19. J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).
  20. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 62.
  21. L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  22. A. W. Lohmann, “Comments about Phase-only Holograms,” in Acoustical Holography, A. F. Metherell, L. Larmore, Eds. (Plenum, New York, 1970), p. 203.
    [CrossRef]
  23. B. Liu, N. C. Gallagher, Appl. Opt. 13, 2470 (1974).
    [CrossRef] [PubMed]
  24. R. Fletcher, M. J. D. Powell, Comput. J. 6, 163 (1963).
    [CrossRef]
  25. S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).
  26. M. J. D. Powell, “A Method for Nonlinear Constraints in Minimization Problems,” in Optimization, R. Fletcher, Ed. (Academic, New York, 1969), p. 283.
  27. “System/360 scientific subroutine package (360A-CM-03X) version III, programmers manual,” IBM Data Processing Division, White Plains, N.Y., Document H20-0205-3 (1968).

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, Appl. Opt. 13, 2470 (1974).
[CrossRef] [PubMed]

1973 (3)

1972 (3)

1970 (3)

1969 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1967 (2)

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

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

1963 (1)

R. Fletcher, M. J. D. Powell, Comput. J. 6, 163 (1963).
[CrossRef]

Allebach, J. P.

J. P. Allebach, N. C. Gallagher, 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), Chap. 8.

Caulfield, H. J.

Chu, D. C.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), Chap. 8.

Dallas, W. J.

Fienup, J. R.

Fletcher, R.

R. Fletcher, M. J. D. Powell, Comput. J. 6, 163 (1963).
[CrossRef]

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 (1969).
[CrossRef]

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

Jacoby, S. L. S.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Katyl, R. H.

Kowalik, J. S.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), Chap. 8.

Liu, B.

Lohmann, A. W.

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

A. W. Lohmann, “Comments about Phase-only Holograms,” in Acoustical Holography, A. F. Metherell, L. Larmore, Eds. (Plenum, New York, 1970), p. 203.
[CrossRef]

Miyamura, Y.

Y. Takeda, Y. Oshida, Y. Miyamura, Appl. Opt. 11, 218 (1972).
[CrossRef]

Oshida, Y.

Y. Takeda, Y. Oshida, Y. Miyamura, Appl. Opt. 11, 218 (1972).
[CrossRef]

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 62.

Paris, D. P.

Pizzo, J. T.

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Powell, M. J. D.

R. Fletcher, M. J. D. Powell, Comput. J. 6, 163 (1963).
[CrossRef]

M. J. D. Powell, “A Method for Nonlinear Constraints in Minimization Problems,” in Optimization, R. Fletcher, Ed. (Academic, New York, 1969), p. 283.

Rothschild, D. R.

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

Takeda, Y.

Y. Takeda, Y. Oshida, Y. Miyamura, Appl. Opt. 11, 218 (1972).
[CrossRef]

Appl. Opt. (11)

Comput. J. (1)

R. Fletcher, M. J. D. Powell, Comput. J. 6, 163 (1963).
[CrossRef]

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

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

J. Opt. Soc. Am. (2)

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

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

Other (11)

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 62.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 230.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), Chap. 8.

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

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, N. C. Gallagher, B. Liu, to appear in Appl. Opt.

A. W. Lohmann, “Comments about Phase-only Holograms,” in Acoustical Holography, A. F. Metherell, L. Larmore, Eds. (Plenum, New York, 1970), p. 203.
[CrossRef]

S. L. S. Jacoby, J. S. Kowalik, J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

M. J. D. Powell, “A Method for Nonlinear Constraints in Minimization Problems,” in Optimization, R. Fletcher, Ed. (Academic, New York, 1969), p. 283.

“System/360 scientific subroutine package (360A-CM-03X) version III, programmers manual,” IBM Data Processing Division, White Plains, N.Y., Document H20-0205-3 (1968).

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 (20)

Fig. 1
Fig. 1

Optimum unrealizable solution.

Fig. 2
Fig. 2

Optimum unrealizable solution for discrete case.

Fig. 3
Fig. 3

Expected value of bandwidth { W ¯ B 2 } as a function of width of uniform distribution from which image phase is taken. Interval of distribution is [−cπ,cπ). Image magnitude is shown in Fig. 4.

Fig. 4
Fig. 4

Image magnitude used to evaluate spectrum shaping methods.

Fig. 5
Fig. 5

Successive approximation. j is the iteration number.

Fig. 6
Fig. 6

Magnitude of spectrum for image shown in Fig. 5 when image phase is zero.

Fig. 7
Fig. 7

Effect of successive approximation on bandwidth.

Fig. 8
Fig. 8

Effect of successive approximation on supremum.

Fig. 9
Fig. 9

Effect of successive approximation on bandwidth-supremum-squared product.

Fig. 10
Fig. 10

Effect of gradient search methods on bandwidth.

Fig. 11
Fig. 11

Effect of gradient search methods on supremum.

Fig. 12
Fig. 12

Effect of gradient search methods on bandwidth-supremum-squared product.

Fig. 13
Fig. 13

Magnitude of spectrum for image of Fig. 5 after being shaped by successive approximation method B2 and gradient search method C2.

Fig. 14
Fig. 14

Effect of bandwidth on efficiency for gradient search method C2.

Fig. 15
Fig. 15

Image reconstructed from kinoform when image phase is zero. Markers indicate sampling points.

Fig. 16
Fig. 16

Image reconstructed from kinoform when image phase is random.

Fig. 17
Fig. 17

Effect of bandwidth constrained spectrum shaping on image reconstructed from kinoform. Constraint bandwidth is 7.50.

Fig. 18
Fig. 18

Effect of bandwidth constrained spectrum shaping on image reconstructed from kinoform. Constraint bandwidth is 25.50.

Fig. 19
Fig. 19

Effect of bandwidth constrained spectrum shaping on image reconstructed from kinoform. Constraint bandwidth is 93.50.

Fig. 20
Fig. 20

Effect of bandwidth constrained spectrum shaping on two-dimensional image reconstructed from kinoform. Top row, left to right: original image, reconstructed image when phase is zero, and random. Bottom row, left to right: reconstructed image after spectrum shaping with constraint bandwidths 22.0, 51.0, and 187.0.

Tables (4)

Tables Icon

Table I Performance of Random Phasea

Tables Icon

Table II Effect of Constraint Bandwidth on Successive Approximation a

Tables Icon

Table III Error in the Reconstruction from a Kinoform of the Sampled Image of Fig. 4 for Different Choices of Image Phasea

Tables Icon

Table IV Error in the Reconstruction from a Kinoform of a Two-Dimensional Image for Different Choices of Image Phase a

Equations (32)

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

A ( u ) = a ( x ) exp ( i 2 π u x ) d x .
E = b 2 ( x ) d x ,
B 2 ( u ) d u = E .
W B 2 = W ( u ) B 2 ( u ) d u
W B 2 = ( 1 2 π ) 2 [ | d b ( x ) d x | 2 + b 2 ( x ) | d α ( x ) d x | 2 ] d x .
B ( u ) b ( x ) d x ,
α ( x ) 0 ,
B ̂ 2 ( u ) d u = E
W B ̂ 2 = W 0 2 .
P B = W ¯ B B sup 2 ,
A p = 1 N n = 0 N 1 a n exp ( i 2 π N P n ) .
W P = { ( P + 1 ) 2 P = 0 , 1 , . . . , N / 2 1 ( N P ) 2 P = N / 2 , . . . , N 1 .
W B 2 = p = 0 N 1 W p B p 2 .
W B 2 = 1 N n = 0 N 1 m = 0 N 1 w n m b n b m exp [ i ( α n α m ) ] ,
W ¯ B ̂ 2 = ( k + 1 ) ( k + 2 ) 12 ,
{ W B 2 } = 1 N Σ Σ m , n = 0 N 1 w n m b n b m { exp [ i ( α n α m ) ] } ,
{ exp [ i ( α n α m ) ] } = { | Φ α ( 1 ) | 2 , n m , 1 , n = m ,
{ W B 2 } = | Φ α ( 1 ) | 2 ( W 1 2 W 2 2 ) + W 2 2 ,
F ( α 0 , . . . , α N 1 ) = n = 0 N 1 ( B n 2 B ̂ n 2 ) 2 .
B n p = ( n = 0 N 1 | B n | p ) 1 / p ,
F ( α 0 , . . . , α N 1 ) = B n p + σ ( W B 2 W 0 2 + θ ) 2 .
E = 15.72 , B sup = 3.18 , W ¯ B 2 = W ¯ 1 2 = 6.96 , P ¯ B = 1.699 ,
W C 2 W B ̂ 2 = I u 2 [ C 2 ( u ) E 2 3 W ¯ B ̂ ] d u + I u 2 C 2 ( u ) d u ,
W C 2 W B ̂ 2 3 W ¯ B ̂ 2 I [ C 2 ( u ) E 2 3 W ¯ B ̂ ] d u + 3 W ¯ B ̂ 2 I C 2 ( u ) d u = 3 W ¯ B ̂ 2 [ C 2 ( u ) d u E ] = 0 .
F α k = n = 0 N 1 F B n B n α k .
F B n = 4 ( B n 2 B ̂ n 2 ) B n .
F B n = ( B n B n p ) p 1 + 4 σ ( W B 2 W 0 2 + θ ) W n B n .
B n 2 = A n A n * = 1 N Σ Σ l , m = 0 N 1 b l b m × exp { i [ ( α l α m ) 2 π N ( l m ) n ] } .
B n 2 α k = 2 N b k B n sin ( 2 π N k n + β n α k ) .
B n α k = 1 2 B n B n 2 α k
F α k = Im { c k a k * } ,
{ c n } n = 0 N 1 = DFT 1 [ F B n exp ( i β n ) ] n = 0 N 1 ,

Metrics