Abstract

Error diffusion (ED) is a powerful tool for the generation of binary computer-generated holograms (CGH's). Several modifications of the original ED algorithm have been proposed to incorporate special requirements and assumptions present in CGH's. This paper compares different versions of the algorithm for their application to computer-generated holography with respect to reconstruction errors and the overall brightness of the reconstruction.

© 1991 Optical Society of America

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References

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  1. 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]
  2. M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1989).
    [CrossRef]
  3. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).
  4. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  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. Am. A 1, 5–10 (1984).
    [CrossRef]
  7. E. Barnard, “Optimal error diffusion for computer-generated holograms,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
    [CrossRef]
  8. R. Eschbach, K. T. Knox, “Error diffusion algorithm with edge enhancement,” to be submitted to J. Opt. Soc. A.
  9. K. T. Knox, “Measurement of edge enhancement in error diffusion,” presented at the Society for Imaging Science and Technology, SPSE Annual Meeting, Rochester, N.Y., May 1990.
  10. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
    [CrossRef]

1989

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

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

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

1976

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

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

Allebach, J. P.

Barnard, E.

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]

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

Eschbach, R.

R. Eschbach, K. T. Knox, “Error diffusion algorithm with edge enhancement,” to be submitted to J. Opt. Soc. A.

Floyd, R. W.

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

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.

Knox, K. T.

R. Eschbach, K. T. Knox, “Error diffusion algorithm with edge enhancement,” to be submitted to J. Opt. Soc. A.

K. T. Knox, “Measurement of edge enhancement in error diffusion,” presented at the Society for Imaging Science and Technology, SPSE Annual Meeting, Rochester, N.Y., May 1990.

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).

Sweeney, D. W.

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]

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

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

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

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

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

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

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

Proc. Soc. Inf. Disp.

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

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

Other

R. Eschbach, K. T. Knox, “Error diffusion algorithm with edge enhancement,” to be submitted to J. Opt. Soc. A.

K. T. Knox, “Measurement of edge enhancement in error diffusion,” presented at the Society for Imaging Science and Technology, SPSE Annual Meeting, Rochester, N.Y., May 1990.

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Figures (7)

Fig. 1
Fig. 1

Generation of the output pulse sequence from the continuous tone input using ED. The input pixel is compared to a threshold (here 1/2), and the output is set to 1 if the input exceeds the threshold and to 0 otherwise. The error generated in this operation is subtracted from the next pixel, and this new value is again compared to the threshold.

Fig. 2
Fig. 2

Weighting coefficients for (a) the original ED algorithm and (b) Barnard's method.

Fig. 3
Fig. 3

Edge-enhanced ED algorithm. The constant threshold of Fig. 1 is replaced by a varying threshold T of the form T = (1 − K)I + ½K (here K = 2) with I representing the input. The resultant pulse distributions change depending on K, with the original ED algorithm reproduced for K = 1.

Fig. 4
Fig. 4

Gray tone holograms for use with (a) the original ED algorithm, (b) Hauck's algorithm, and (c) Barnard's algorithm. The different position of the object inside the transformation array is reflected in the different carrier structure.

Fig. 5
Fig. 5

Binary holograms resulting from the gray tone data of Fig. 4.

Fig. 6
Fig. 6

Simulated reconstructions of a, the original ED; b, the original ED with K = 3; c, Hauck's method; d, Hauck's method with K = 3; e, Barnard's method; and f, Barnard's method with K = 3. The increase in the brightness of the reconstruction in relation to the noise clouds can be seen by the reconstructions with K = 3.

Fig. 7
Fig. 7

Magnified portions of the reconstructions of Fig. 6. Hauck's method in Fig. 6c gives the best subjective image quality, which is in perfect agreement with the data in Tables IIIV.

Tables (11)

Tables Icon

Table I Relative Reconstruction Errors and Brightnesses for the Three-Hologram Calculation Schemes for an Object That Is Spatially Separable from the dc Peak

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Table II Relative Reconstruction Errors and Brightnesses for the Three-Hologram Calculation Schemes but Calculated over a Larger Window Size

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Table III Relative MSE and Relative Brightness for the Original Error Diffusion Method, Varying the Edge Enhancement Introduced by a Threshold Modulationa

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Table IV Relative MSE and Relative Brightness for Hauck's Methoda

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Table V Relative MSE and Relative Brightness for Barnard's Methoda

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Table VI Relative MSE and Relative Brightness for the Original Error Diffusion Method, Varying the Edge Enhancement Introduced by Threshold Modulationa

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Table VII Relative MSE and Relative Brightness for Hauck's Methoda

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Table VIII Relative MSE and Relative Brightness for Barnard's Method, Varying the Edge Enhancement Introduced by a Threshold Modulationa

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Table IX Experimental Measurement of the Reconstruction Brightness for the Original Error Diffusion Method

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Table X Experimental Measurement of the Reconstruction Brightness for Hauck's Method

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Table XI Experimental Measurement of the Reconstruction Brightness for Barnard's Method

Equations (5)

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

u ( x , y ) = o ( x , y ) exp [ i Φ ( x , y ) ] ,
H ( μ , ν ) = | U ( μ , ν ) | cos [ 2 π ( x s μ + y s ν ) + Φ ( μ , ν ) ] + B ,
T = ( 1 K ) H + ½ K .
MSE m n | f m n f m n σ f g m n g m n σ g | 2 ,
B = ( x , y ) R B ( x , y ) d x d y .

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