Abstract

An approach for optimizing computer-generated holograms is discussed. The approach can be summarized most generally as hierarchically designing a number of holograms to add up coherently to a single desired reconstruction. In the case of binary holograms, this approach results in the interlacing technique (IT) and the iterative interlacing technique (IIT). In the IT, a number of subholograms are designed and interlaced together to generate the total binary hologram. The first subhologram is designed to reconstruct the desired image. The succeeding subholograms are designed to correct the remaining error image. In the IIT, the remaining error image after the last subhologram is circulated back to the first subhologram, and the process is continued a number of sweeps until convergence. Both techniques can be used together with most computer-generated-hologram synthesis algorithms and result in a substantial reduction in reconstruction error as well as an increased speed of convergence in the case of iterative algorithms.

© 1992 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

1991 (2)

J. Amako, T. Sonehara, “Kinoform using an electrically controlled birefringent liquid-crystal spatial light modulator,” Appl. Opt. 30, 4622–4628 (1991).
[Crossref] [PubMed]

S. Aghagolzadeh, O. K. Ersoy, “Optimal multistage transform image coding,” IEEE Trans. Circuits Syst. Video Technol. 1, 308–317 (1991).
[Crossref]

1990 (2)

O. K. Ersoy, Y. Yoon, N. Keshavo, D. Zimmerman, “Nonlinear matched filtering II,” Opt. Eng. 29, 1002–1012 (1990).
[Crossref]

S. J. Walker, J. Jahns, “Array generation with multilevel phase gratings,” J. Opt. Soc. Am. A 7, 1509–1513 (1990).
[Crossref]

1989 (1)

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

1987 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

1980 (1)

H. Kitajima, “A symmetric cosine transform,” IEEE Trans. Comput. C-29, 317–323 (1980).
[Crossref]

1976 (1)

O. K. Ersoy, “Construction of point images with the scanning electron microscope: a simple algorithm,” Optik (Stuttgart) 46, 61–66 (1976).

1974 (1)

1973 (1)

1966 (1)

Aghagolzadeh, S.

S. Aghagolzadeh, O. K. Ersoy, “Optimal multistage transform image coding,” IEEE Trans. Circuits Syst. Video Technol. 1, 308–317 (1991).
[Crossref]

Allebach, J.

J. Allebach, School of Electrical Engineering, Purdue University, West Lafayette, Ind. 47907 (personal communication).

Allebach, J. P.

Amako, J.

Brown, B. R.

Bryngdahl, O.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

Deng, S.-W.

O. K. Ersoy, S.-W. Deng, “Parallel, self-organizing, hierarchical neural networks with continuous inputs and outputs,” Rep. TR-EE-91-51 (Purdue University, West Lafayette, Ind., 1991); IEEE Trans. Neural Net. (to be published).

Ersoy, O. K.

S. Aghagolzadeh, O. K. Ersoy, “Optimal multistage transform image coding,” IEEE Trans. Circuits Syst. Video Technol. 1, 308–317 (1991).
[Crossref]

O. K. Ersoy, Y. Yoon, N. Keshavo, D. Zimmerman, “Nonlinear matched filtering II,” Opt. Eng. 29, 1002–1012 (1990).
[Crossref]

O. K. Ersoy, “Construction of point images with the scanning electron microscope: a simple algorithm,” Optik (Stuttgart) 46, 61–66 (1976).

O. K. Ersoy, S.-W. Deng, “Parallel, self-organizing, hierarchical neural networks with continuous inputs and outputs,” Rep. TR-EE-91-51 (Purdue University, West Lafayette, Ind., 1991); IEEE Trans. Neural Net. (to be published).

Gallagher, N. C.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Hirsch, P. M.

P. M. Hirsch, L. B. Lesem, J. A. Jordan, “Method of making an object-dependent diffuser,” U. S. patent3,619,022 (1971).

Jahns, J.

Jordan, J. A.

P. M. Hirsch, L. B. Lesem, J. A. Jordan, “Method of making an object-dependent diffuser,” U. S. patent3,619,022 (1971).

Keshavo, N.

O. K. Ersoy, Y. Yoon, N. Keshavo, D. Zimmerman, “Nonlinear matched filtering II,” Opt. Eng. 29, 1002–1012 (1990).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Kitajima, H.

H. Kitajima, “A symmetric cosine transform,” IEEE Trans. Comput. C-29, 317–323 (1980).
[Crossref]

Lesem, L. B.

P. M. Hirsch, L. B. Lesem, J. A. Jordan, “Method of making an object-dependent diffuser,” U. S. patent3,619,022 (1971).

Liu, B.

Lohmann, A. W.

Luenberger, D. G.

D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

Naqvi, S.

S. Naqvi, Department of Electrical Engineering, University of New Mexico, Albuquerque, N. Mex. 87131 (personal communication).

Seldowitz, M. A.

Sonehara, T.

Swanson, G. J.

W. B. Veldkamp, G. J. Swanson, “Developments in fabrication of binary optical elements,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 54–59 (1983).

Sweeney, D. W.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Veldkamp, W. B.

W. B. Veldkamp, G. J. Swanson, “Developments in fabrication of binary optical elements,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 54–59 (1983).

Walker, S. J.

Weissbach, S.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

Wyrowski, F.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

Yoon, Y.

O. K. Ersoy, Y. Yoon, N. Keshavo, D. Zimmerman, “Nonlinear matched filtering II,” Opt. Eng. 29, 1002–1012 (1990).
[Crossref]

Zimmerman, D.

O. K. Ersoy, Y. Yoon, N. Keshavo, D. Zimmerman, “Nonlinear matched filtering II,” Opt. Eng. 29, 1002–1012 (1990).
[Crossref]

Appl. Opt. (5)

IEEE Trans. Circuits Syst. Video Technol. (1)

S. Aghagolzadeh, O. K. Ersoy, “Optimal multistage transform image coding,” IEEE Trans. Circuits Syst. Video Technol. 1, 308–317 (1991).
[Crossref]

IEEE Trans. Comput. (1)

H. Kitajima, “A symmetric cosine transform,” IEEE Trans. Comput. C-29, 317–323 (1980).
[Crossref]

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

Opt. Commun. (1)

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

Opt. Eng. (1)

O. K. Ersoy, Y. Yoon, N. Keshavo, D. Zimmerman, “Nonlinear matched filtering II,” Opt. Eng. 29, 1002–1012 (1990).
[Crossref]

Optik (Stuttgart) (1)

O. K. Ersoy, “Construction of point images with the scanning electron microscope: a simple algorithm,” Optik (Stuttgart) 46, 61–66 (1976).

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Other (7)

D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

O. K. Ersoy, S.-W. Deng, “Parallel, self-organizing, hierarchical neural networks with continuous inputs and outputs,” Rep. TR-EE-91-51 (Purdue University, West Lafayette, Ind., 1991); IEEE Trans. Neural Net. (to be published).

S. Naqvi, Department of Electrical Engineering, University of New Mexico, Albuquerque, N. Mex. 87131 (personal communication).

J. Allebach, School of Electrical Engineering, Purdue University, West Lafayette, Ind. 47907 (personal communication).

W. B. Veldkamp, G. J. Swanson, “Developments in fabrication of binary optical elements,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 54–59 (1983).

H. Stark, ed., Image Recovery: Theory and Applications (Academic, New York, 1986).

P. M. Hirsch, L. B. Lesem, J. A. Jordan, “Method of making an object-dependent diffuser,” U. S. patent3,619,022 (1971).

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Figures (10)

Fig. 1
Fig. 1

Hermitian symmetry used in the POCS-CT algorithm.

Fig. 2
Fig. 2

Edge-enhanced cat-brain cross-section image of size 128 × 128.

Fig. 3
Fig. 3

Convergence of the POCS-CT algorithm.

Fig. 4
Fig. 4

Comparison of two POCS algorithms.

Fig. 5
Fig. 5

Interlacing geometry used in the IT and the IIT.

Fig. 6
Fig. 6

Another interlacing geometry used in the IT and the IIT.

Fig. 7
Fig. 7

Mean-square reconstruction error with the interlacing technique.

Fig. 8
Fig. 8

Binary hologram generated by the interlacing technique.

Fig. 9
Fig. 9

Optical reconstruction of the cat-brain image when the hologram is designed by the POCS-CT algorithm.

Fig. 10
Fig. 10

Optical reconstruction of the cat-brain image when the hologram is designed by the IT together with the POCS-CT algorithm and the number of subholograms is eight.

Tables (3)

Tables Icon

Table 1 MSRE With the IIT As a Function of the Number of Sweeps and Subholograms When the Subholograms Are Designed by the POCS-CT Algorithma

Tables Icon

Table 2 MSRE With the IIT As a Function of the Number of Sweeps and Subholograms When the Subholograms Are Designed by Hirsch’s Methoda

Tables Icon

Table 3 MSRE and Computation Time with the IIT As a Function of the Number of Sweeps and Subholograms and the Computation Time When the Subholograms Are Designed by the DBS Algorithm or the POCS-CT Algorithma

Equations (18)

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x ( n 1 , n 2 ) = x ( N 1 - n 1 , N 2 - n 2 ) ,
x ( n 1 , n 2 ) = x ( N 1 + n 1 , N 2 + n 2 ) .
x ( n 1 , n 2 ) = x ( - n 1 , - n 2 ) .
X ( n 1 , n 2 ) = 4 N 1 N 2 k 1 = 0 N 1 - 1 k 2 = 0 N 2 / 2 x ( k 1 , k 2 ) v ( k 2 ) × cos [ 2 π ( n 1 k 1 N 1 + n 2 k 2 N 2 ) ] ,
v ( k 2 ) = ½ k 2 = 0 , N 2 / 2 = 1 otherwise .
x ( n 1 , n 2 ) = k 1 = 0 N 1 - 1 k 2 = 0 N 2 / 2 X ( k 1 , k 2 ) cos [ 2 π ( n 1 k 1 N 1 + n 2 k 2 N 2 ) ] .
X ( n 1 , n 2 ) = 1 X ( n 1 , n 2 ) 0 , = 0 otherwise .
MSRE ( x ) = 1 N 2 ( n 1 , n 2 ) l x ( n 1 , n 2 ) - λ x ( n 1 , n 2 ) 2 ,
x ˜ ( n 1 , n 2 ) = x ( n 1 , n 2 ) for n 1 , n 2 l , = x ( n 1 , n 2 ) otherwise .
Δ i = n 1 = 0 N 1 - 1 n 2 = 0 N 2 - 1 x i ( n 1 , n 2 ) - x i - 1 ( n 1 , n 2 ) ,
e 1 ( n 1 , n 2 ) = x ( n 1 , n 2 ) - λ 1 x rec 1 ( n 1 , n 2 ) .
X ( n 1 , n 2 ) = X ( n 1 , n 2 ) both ( n 1 , n 2 ) are even or odd , = 0 otherwise ,
X ( n 1 , n 2 ) = ½ [ ( - 1 ) n 1 + n 2 + 1 ] X ( n 1 , n 2 ) .
x ( n 1 , n 2 ) = 1 2 [ x ( n 1 , n 2 ) + x ( n 1 + N 1 2 , n 2 + N 2 2 ) ] .
x 1 ( n 1 , n 2 ) = x rec 1 ( n 1 , n 2 ) - e f ( n 1 , n 2 ) / λ f .
e 1 ( n 1 , n 2 ) = x 1 ( n 1 , n 2 ) - λ 1 x rec 1 ( n 1 , n 2 ) .
MSRE ( x ) = 1 N 2 ( n 1 , n 2 ) x ( n 1 , n 2 ) - λ x ( n 1 , n 2 ) 2 ,
λ = ( n 1 , n 2 ) x ( n 1 , n 2 ) [ x ( n 1 , n 2 ) ] * ( n 1 , n 2 ) x ( n 1 , n 2 ) 2 .

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