Abstract

Error diffusion has been proven to be a valuable tool in the calculation of computer-generated holograms. Real-valued error diffusion has been used to calculate the transmission functions for real-valued holograms with off-axis reconstructions. We demonstrate the use of a complex-valued error-diffusion algorithm on real-valued hologram data in order to achieve larger flexibility in the shaping of the noise spectrum.

© 1993 Optical Society of America

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References

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  1. H. Inoye, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
    [CrossRef]
  2. J. C. Candy, O. J. Benjamin, “The structure of quantization noise for sigma–delta modulation,” IEEE Trans. Commun. C-29, 1316–1323 (1981).
    [CrossRef]
  3. R. M. Gray, W. Chou, P. W. Wong, “Quantization noise in single-loop sigma–delta modulation with sinusoidal inputs,” IEEE Trans. Commun. 37, 956–968 (1989).
    [CrossRef]
  4. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).
  5. J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).
  6. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. A 1, 5–10 (1984).
    [CrossRef]
  7. J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comp. Vis. Graph. Image Process. 5, 13–40 (1976).
    [CrossRef]
  8. J. C. Stoffel, J. F. Moreland, “A survey of electronic techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
    [CrossRef]
  9. C. Billotet-Hoffmann, O. Bryngdahl, “On the error diffusion technique for electronic halftoning,” Proc. Soc. Inf. Disp. 24, 253–258 (1983).
  10. R. Eschbach, K. T. Knox, “Error-diffusion algorithm with edge enhancement,” J. Opt. Soc. Am. A 8, 1844–1850 (1991).
    [CrossRef]
  11. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  12. R. Eschbach, R. Hauck, “A pulse density modulation by iteration for halftoning,” Opt. Commun. 62, 300–304 (1987).
    [CrossRef]
  13. M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
    [CrossRef]
  14. M. R. Schroeder, “Images from computers,” IEEE Spectrum 6(3), 66–78 (1969).
    [CrossRef]
  15. R. Eschbach, “Comparison of error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710 (1991).
    [CrossRef] [PubMed]
  16. S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and application in optical signal processing,” Appl. Opt. 31, 2518–2534(1992).
    [CrossRef] [PubMed]
  17. Z. Fan, “Analysis of error diffusion,” in Proceedings of the 44th Annual Conference of the IS&T, (Society for Imaging Science and Technology, Springfield, Va., 1991).
  18. K. T. Knox, “Error image in error diffusion,” in Image Processing Algorithms and Techniques III, J. R. Sullivan, M. Rabbani, B. M. Dawson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1657, 268–279 (1992).

1992

1991

1989

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
[CrossRef]

R. M. Gray, W. Chou, P. W. Wong, “Quantization noise in single-loop sigma–delta modulation with sinusoidal inputs,” IEEE Trans. Commun. 37, 956–968 (1989).
[CrossRef]

1987

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

R. Eschbach, R. Hauck, “A pulse density modulation by iteration for halftoning,” Opt. Commun. 62, 300–304 (1987).
[CrossRef]

1984

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

1983

C. Billotet-Hoffmann, O. Bryngdahl, “On the error diffusion technique for electronic halftoning,” Proc. Soc. Inf. Disp. 24, 253–258 (1983).

1981

J. C. Stoffel, J. F. Moreland, “A survey of electronic techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

J. C. Candy, O. J. Benjamin, “The structure of quantization noise for sigma–delta modulation,” IEEE Trans. Commun. C-29, 1316–1323 (1981).
[CrossRef]

1976

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comp. Vis. Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

1969

M. R. Schroeder, “Images from computers,” IEEE Spectrum 6(3), 66–78 (1969).
[CrossRef]

1963

H. Inoye, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Allebach, J. P.

Benjamin, O. J.

J. C. Candy, O. J. Benjamin, “The structure of quantization noise for sigma–delta modulation,” IEEE Trans. Commun. C-29, 1316–1323 (1981).
[CrossRef]

Billotet-Hoffmann, C.

C. Billotet-Hoffmann, O. Bryngdahl, “On the error diffusion technique for electronic halftoning,” Proc. Soc. Inf. Disp. 24, 253–258 (1983).

Broja, M.

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
[CrossRef]

Bryngdahl, O.

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
[CrossRef]

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

C. Billotet-Hoffmann, O. Bryngdahl, “On the error diffusion technique for electronic halftoning,” Proc. Soc. Inf. Disp. 24, 253–258 (1983).

Candy, J. C.

J. C. Candy, O. J. Benjamin, “The structure of quantization noise for sigma–delta modulation,” IEEE Trans. Commun. C-29, 1316–1323 (1981).
[CrossRef]

Chou, W.

R. M. Gray, W. Chou, P. W. Wong, “Quantization noise in single-loop sigma–delta modulation with sinusoidal inputs,” IEEE Trans. Commun. 37, 956–968 (1989).
[CrossRef]

Eschbach, R.

Fan, Z.

Z. Fan, “Analysis of error diffusion,” in Proceedings of the 44th Annual Conference of the IS&T, (Society for Imaging Science and Technology, Springfield, Va., 1991).

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

Gray, R. M.

R. M. Gray, W. Chou, P. W. Wong, “Quantization noise in single-loop sigma–delta modulation with sinusoidal inputs,” IEEE Trans. Commun. 37, 956–968 (1989).
[CrossRef]

Hale, J. A. G.

J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).

Hauck, R.

R. Eschbach, R. Hauck, “A pulse density modulation by iteration for halftoning,” Opt. Commun. 62, 300–304 (1987).
[CrossRef]

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

Inoye, H.

H. Inoye, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Jarvis, J. F.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comp. Vis. Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Judice, C. N.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comp. Vis. Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Knox, K. T.

R. Eschbach, K. T. Knox, “Error-diffusion algorithm with edge enhancement,” J. Opt. Soc. Am. A 8, 1844–1850 (1991).
[CrossRef]

K. T. Knox, “Error image in error diffusion,” in Image Processing Algorithms and Techniques III, J. R. Sullivan, M. Rabbani, B. M. Dawson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1657, 268–279 (1992).

Moreland, J. F.

J. C. Stoffel, J. F. Moreland, “A survey of electronic techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

Ninke, W. H.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comp. Vis. Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Schroeder, M. R.

M. R. Schroeder, “Images from computers,” IEEE Spectrum 6(3), 66–78 (1969).
[CrossRef]

Seldowitz, M. A.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

Stoffel, J. C.

J. C. Stoffel, J. F. Moreland, “A survey of electronic techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

Sweeney, D. W.

Weissbach, S.

Wong, P. W.

R. M. Gray, W. Chou, P. W. Wong, “Quantization noise in single-loop sigma–delta modulation with sinusoidal inputs,” IEEE Trans. Commun. 37, 956–968 (1989).
[CrossRef]

Wyrowski, F.

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

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
[CrossRef]

Yasuda, Y.

H. Inoye, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Appl. Opt.

Comp. Vis. Graph. Image Process.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comp. Vis. Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

IEEE Spectrum

M. R. Schroeder, “Images from computers,” IEEE Spectrum 6(3), 66–78 (1969).
[CrossRef]

IEEE Trans. Commun.

J. C. Stoffel, J. F. Moreland, “A survey of electronic techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

J. C. Candy, O. J. Benjamin, “The structure of quantization noise for sigma–delta modulation,” IEEE Trans. Commun. C-29, 1316–1323 (1981).
[CrossRef]

R. M. Gray, W. Chou, P. W. Wong, “Quantization noise in single-loop sigma–delta modulation with sinusoidal inputs,” IEEE Trans. Commun. 37, 956–968 (1989).
[CrossRef]

J. Opt. Soc. A

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

J. Opt. Soc. Am. A

Opt. Commun.

R. Eschbach, R. Hauck, “A pulse density modulation by iteration for halftoning,” Opt. Commun. 62, 300–304 (1987).
[CrossRef]

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
[CrossRef]

Proc. IEEE

H. Inoye, Y. Yasuda, “A unity bit coding method by negative feedback,” Proc. IEEE 51, 1524–1535 (1963).
[CrossRef]

Proc. Soc. Inf. Disp.

C. Billotet-Hoffmann, O. Bryngdahl, “On the error diffusion technique for electronic halftoning,” Proc. Soc. Inf. Disp. 24, 253–258 (1983).

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).

Other

Z. Fan, “Analysis of error diffusion,” in Proceedings of the 44th Annual Conference of the IS&T, (Society for Imaging Science and Technology, Springfield, Va., 1991).

K. T. Knox, “Error image in error diffusion,” in Image Processing Algorithms and Techniques III, J. R. Sullivan, M. Rabbani, B. M. Dawson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1657, 268–279 (1992).

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Figures (7)

Fig. 1
Fig. 1

One-dimensional error-diffusion algorithm for quantization of the continuous-tone function i(n) into a discrete set b(n), with the use of an error signal e(n) in order to maintain average density.

Fig. 2
Fig. 2

Filter function F(r, s) generating zero noise at location (r0, s0) = (N/4, N/4).

Fig. 3
Fig. 3

Filter function F(r, s) generated by shifting the filter function generated with the use of the error-diffusion weight suggested in Ref. 4 to the location (r0, s0) = (N/4, N/4).

Fig. 4
Fig. 4

Binary holograms obtained with the use of the error-diffusion weight suggested in Ref. 6. The results of applying the weights of Ref. 6 with the following shifts are shown: (a) no additional shift, (b) a shift of (r0, s0) = (0, N/8), (c) a shift of (r0, s0) = (0, N/4).

Fig. 5
Fig. 5

Simulated reconstructions of the corresponding binary holograms of Fig. 4 (a nonlinear renormalization was performed).

Fig. 6
Fig. 6

Binary holograms obtained with the use of the error-diffusion weight suggested in Ref. 4. The results of applying the weights of Ref. 6 with the following shifts are shown: (a) (r0, s0) = (N/4, 0), (b) (r0, s0) = (N/4, N/8), (c)(r0, s0) = (N/4, N/4).

Fig. 7
Fig. 7

Simulated reconstructions of the corresponding binary holograms of Fig. 6 (a nonlinear renormalization was performed).

Equations (23)

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u ( r , s ) = o ( r , s ) exp [ i ϕ ( r , s ) ] ,
H ( m , n ) = U ( m , n ) × cos [ ( 2 π / N ) ( r 0 m + s 0 n ) + θ ( m , n ) ] ,
H ( m , n ) = U ( m , n ) cos [ ( 2 π / N ) ( r 0 m + s 0 n ) + θ ( m , n ) ] + bias ,
i mod ( m , n ) = i ( m , n ) - k , l S a k , l [ e ( m - k , n - l ) ] ,
b ( m , n ) = step [ i mod ( m , n ) - t ] ,
e ( m , n ) = b ( m , n ) - i mod ( m , n ) .
b ( m , n ) = i ( m , n ) - k , k l S a k , l [ e ( m - k , n - l ) ] + e ( m , n )
B ( r , s ) = I ( r , s ) + ( 1 - k , l S a k , l { exp [ - i ( 2 π / N ) ( k r + l s ) } ) × E ( r , s ) ,
B ( r , s ) = I ( r , s ) + F ( r , s ) E ( r , s ) ,
F ( r , s ) = 1 - k , l S a k , l { exp [ - i ( 2 π / N ) ( k r + l s ) ] } .
k , l S a k , l = 1 ,
k , l S a k , l { exp [ - i ( 2 π / N ) ( k r 0 + l s 0 ) ] } = 1
a 1 , 1 = - 1 , a 0 , 2 = 1 , a - 1 , 1 = 1.
i ( m , n ) , b ( m , n ) , e ( m , n ) R a k , l R ,
i ( m , n ) , b ( m , n ) R ,             e ( m , n ) , a k , l C
a k , l = a k , l exp [ + i ( 2 π / N ) ( k r 0 + l s 0 ) ] , k , l S a k , l = 1 ,
a 1 , 0 = / 16 7 exp [ + i ( 2 π / N ) r 0 ] , a 1 , 1 = / 16 1 exp [ + i ( 2 π / N ) ( r 0 + s 0 ) ] , a 0 , 1 = / 16 5 exp [ + i ( 2 π / N ) s 0 ] , a - 1 , 1 = / 16 3 exp [ + i ( 2 π / N ) ( - r 0 + s 0 ) ] .
F ( r 0 , r , s 0 , s ) = 1 - k , l S a k , l exp [ + i ( 2 π / N ) ( k r 0 + l s 0 ) ] × exp [ - i ( 2 π / N ) ( k r + l s ) ] ,
F ( r 0 , r , s 0 , s ) = 1 - k , l S a k , l exp { - i ( 2 π / N ) [ k ( r - r 0 ) + l ( s - s 0 ) ] }
F ( r 0 , r , s 0 , s ) = F 0 ( r , s ) δ ( r 0 , s 0 ) ,
b ( m , n ) = step { Re [ i mod ( m , n ) ] - t } ,
e ( m , n ) = { Re [ e ( m , n ) ] 2 + Im [ e ( m , n ) ] 2 } 1 / 2 ,
a 1 , 0 = a 1 , 1 = a - 1 , 1 = 0 , a 0 , 1 = exp [ i ( 2 π / N ) s 0 ] ,

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