Abstract

A new method called gradual and random binarization to binarize gray-scale holograms, based on an iterative algorithm, is proposed. The binarization process is performed gradually, and the pixels to be binarized are chosen randomly. Errors caused by this operation are spatially diffused. A comparison with other established methods based on error diffusion, direct binary search, and iterative stepwise quantization shows that the gradual and random binarization method achieves a very good compromise between computational complexity and reconstruction quality. Optical reconstructions are presented.

© 1995 Optical Society of America

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References

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1993

1992

1991

1989

1988

E. Barnard, “Optimal error diffusion for computer-generated holograms,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

1987

1984

1982

1976

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 12, 55–77 (1976).

1974

T. C. Strand, “Signal/noise in analog and binary holograms,” Opt. Eng. 13, 219–227 (1974).

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Allebach, J. P.

Barnard, E.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys.1481–1571 (1991).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
[CrossRef]

Casasent, D. P.

Chen, C.-L.

Davidson, N.

Dietrich, C. H.

E. Zhang, J. Hesser, C. H. Dietrich, S. Noehte, R. Männer, “Mathematical analysis of computer generated binary Fourier transform holograms,” J. Opt. Soc. Am. A (to be published).

Eschbach, R.

Fiddy, M. A.

Fienup, J. R.

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 12, 55–77 (1976).

Friesem, A. A.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gmitro, A. F.

Guest, C. C.

Hasman, E.

Hauck, R.

Hesser, J.

E. Zhang, J. Hesser, C. H. Dietrich, S. Noehte, R. Männer, “Mathematical analysis of computer generated binary Fourier transform holograms,” J. Opt. Soc. Am. A (to be published).

Jennison, B. K.

Keller, P. E.

Kim, M. S.

Lee, A. J.

Männer, R.

E. Zhang, J. Hesser, C. H. Dietrich, S. Noehte, R. Männer, “Mathematical analysis of computer generated binary Fourier transform holograms,” J. Opt. Soc. Am. A (to be published).

McHugh, T. J.

W. B. Veldkamp, T. J. McHugh, “Binary optics,” Sci. Am. 266, 92–97 (1992).
[CrossRef]

Noehte, S.

E. Zhang, J. Hesser, C. H. Dietrich, S. Noehte, R. Männer, “Mathematical analysis of computer generated binary Fourier transform holograms,” J. Opt. Soc. Am. A (to be published).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Seldowitz, M. A.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 12, 55–77 (1976).

Strand, T. C.

T. C. Strand, “Signal/noise in analog and binary holograms,” Opt. Eng. 13, 219–227 (1974).

Sweeney, D. W.

Veldkamp, W. B.

W. B. Veldkamp, T. J. McHugh, “Binary optics,” Sci. Am. 266, 92–97 (1992).
[CrossRef]

Weissbach, S.

S. Weissbach, F. Wyrowski, “Error-diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
[CrossRef] [PubMed]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

Wu, C.-H.

Wyrowski, F.

S. Weissbach, F. Wyrowski, “Error-diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys.1481–1571 (1991).
[CrossRef]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3869 (1989).
[CrossRef] [PubMed]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

Zhang, E.

E. Zhang, J. Hesser, C. H. Dietrich, S. Noehte, R. Männer, “Mathematical analysis of computer generated binary Fourier transform holograms,” J. Opt. Soc. Am. A (to be published).

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

Opt. Eng.

T. C. Strand, “Signal/noise in analog and binary holograms,” Opt. Eng. 13, 219–227 (1974).

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. Soc. Inf. Disp.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 12, 55–77 (1976).

Rep. Prog. Phys.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys.1481–1571 (1991).
[CrossRef]

Sci. Am.

W. B. Veldkamp, T. J. McHugh, “Binary optics,” Sci. Am. 266, 92–97 (1992).
[CrossRef]

Other

E. Zhang, J. Hesser, C. H. Dietrich, S. Noehte, R. Männer, “Mathematical analysis of computer generated binary Fourier transform holograms,” J. Opt. Soc. Am. A (to be published).

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Figures (12)

Fig. 1
Fig. 1

Flow chart of the GRB algorithm.

Fig. 2
Fig. 2

(a) Object (letter F) with three gray values and (b) its gray-scale hologram (256 gray values).

Fig. 3
Fig. 3

Illustration of the binarization process: (a) the original gray-scale hologram and (a′) its histogram; (b)–(e) Partly binarized holograms for S 1 = 8, S 2 = 4, S 3 = 2, and S 4 = 1; (b′)–(e′) their gray-value distributions.

Fig. 4
Fig. 4

MSE changes with iterative cycles k (1–7, 8–14, 15–19, and 20–22) in each iteration and with iterations i (1, 2, 3, 4).

Fig. 5
Fig. 5

(a) Binary hologram generated by the GRB method proposed here and (b) its computer simulated reconstruction.

Fig. 6
Fig. 6

Same as Fig. 5 but for hard clipping.

Fig. 7
Fig. 7

Same as Fig. 5 but for error diffusion.

Fig. 8
Fig. 8

Same as Fig. 5 but for direct binary search.

Fig. 9
Fig. 9

Same as Fig. 5 but for iterative stepwise quantization.

Fig. 10
Fig. 10

Emblem of the University of Mannheim as the object.

Fig. 11
Fig. 11

Optical reconstruction of the binary hologram generated by iterative stepwise quantization.

Fig. 12
Fig. 12

Optical reconstruction of the binary hologram generated by GRB.

Tables (3)

Tables Icon

Table 1 Changes of the Mean-Square Error in Each Iteration for the Example (Letter F)

Tables Icon

Table 2 Results of Simulation of Binary Holograms Generated by Different Methods and Computational Complexity of These Methods

Tables Icon

Table 3 Calculated MSE, η, and Measured η′ from the Reconstruction of Binary Holograms

Equations (13)

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

H ( μ , ν ) = | U ( μ , ν ) | cos [ Φ ( μ , ν ) 2 π ( x 0 μ + y 0 ν ) ] Bs 1 Bs ,
U 1 ( μ , ν ) = FT { u ( x , y ) } .
H k ( μ , ν ) = C [ U k ( μ , ν ) ] , k = 1 , 2 , 3 , .
H k ( μ , ν ) L H k ( μ , ν ) .
h k ( x , y ) = IFT { H k ( μ , ν ) } .
h k ( x , y ) l h k + 1 ( x , y ) .
MSE = 1 M 2 x y w | f x y f ¯ σ f g x y σ g | 2 ,
f ¯ = 1 M 2 x y w f x y , = 1 M 2 x y w g x y , σ f 2 = 1 M 2 x y w | f x y f ¯ | 2 , σ g 2 = 1 M 2 x y w | g x y | 2 .
U 1 ( μ , ν ) = FT { u ( x , y ) } .
H k ( μ , ν ) = { 0 if H k ( μ , ν ) < T threshold 1 otherwise
h k ( x , y ) = IFT { H k ( μ , ν ) } .
| h k + 1 ( x , y ) | = { | u ( x , y ) | / c k if | h k ( x , y ) | | u ( x , y ) | / c k | h k ( x , y ) | otherwise ,
c k = ( x , y ) w | u ( x , y ) | | h k ( x , y ) | ( x , y ) w | h k ( x , y ) | 2 .

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