Abstract

We present an explanation of the minimum-average-error- (MAE-) based error diffusion algorithm for computer-generated hologram (CGH) calculation. This leads to a direct and straightforward link between the CGH reconstruction plane signal windows and the MAE diffusion weights: the diffusion weights should be the Fourier transform of the signal window function. A MAE algorithm based on these results is described and used to calculate test CGH’s, whose computer-simulated and experimental reconstructions confirm our analysis by generating true, low-error signal windows akin to those obtained with iterative algorithms. Comparisons made with an iterative algorithm show that the new algorithm is a powerful, low-computation-load, CGH binarization tool and that, when combined with random or image-independent diffusers, it makes possible the calculation of acceptable-performance, high-space–bandwidth-product CGH’s whose calculation would be unfeasible with iterative algorithms.

© 1998 Optical Society of America

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References

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  1. M. A. Seldowitz, J. P. Allebach, D. W. Sweeny, “A synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  2. B. K. Jennison, J. P. Allebach, D. W. Sweeny, “Efficient design of direct binary search computer-generated holograms,” J. Opt. Soc. Am. A 8, 652–660 (1991).
    [CrossRef]
  3. This popular and powerful algorithm has been used, in several domains and under different names, by many researchers, for example P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971); R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); N. C. Gallagher, B. Lui, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973); J. R. Fenup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef] [PubMed]
  4. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  5. E. Zhang, S. Noeht, C. H. Dietrich, R. Männer, “Gradual and random binarization of gray-scale holograms,” Appl. Opt. 34, 5987–5995 (1995).
    [CrossRef] [PubMed]
  6. D. Just, D. T. Ling, “Neural networks for binarizing computer-generated holograms,” Opt. Commun. 81, 1–5 (1991).
    [CrossRef]
  7. R. Hauck, O. Bryngdahl, “Computer generated holograms with pulse density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  8. V. Boutenko, R. Chevallier, “Second order direct binary search algorithm for the synthesis of computer-generated holograms,” Opt. Commun. 125, 43–47 (1996).
    [CrossRef]
  9. A. Kirk, K. Powell, T. Hall, “A generalisation of the error diffusion method for binary computer generated hologram design,” Opt. Commun. 92, 12–18 (1992).
    [CrossRef]
  10. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
    [CrossRef]
  11. R. Eschbach, “Comparison of error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710 (1991).
    [CrossRef] [PubMed]
  12. E. Bernard, “Optimal error diffusion for computer-generated holograms with pulse density modulation,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
    [CrossRef]
  13. S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
    [CrossRef] [PubMed]
  14. R. Eschbach, Z. Fan, “Complex valued error diffusion for off-axis computer-generated holograms,” Appl. Opt. 32, 3130–3136 (1993).
    [CrossRef] [PubMed]
  15. A. Kirk, K. Powell, T. Hall, “Error diffusion and the representation problem in computer generated hologram design,” Opt. Comput. Process. 2, 199–212 (1992).
  16. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).
  17. M. R. Schroeder, “Images from computers,” IEEE Spectr. (March) 66–78 (1969).
    [CrossRef]
  18. F. Fetthauer, S. Weissbach, O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
    [CrossRef]
  19. F. Wyrowski, “Diffractive efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
    [CrossRef]
  20. G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-generated holograms: carrier of polar geometry,” Appl. Opt. 24, 777–784 (1985).
    [CrossRef] [PubMed]
  21. R. Nagarajan, R. Easton, R. Eschbach, “Using adaptive quantization in cell-oriented holograms,” Opt. Commun. 144, 370–374 (1995).
    [CrossRef]
  22. P. Thorston, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
    [CrossRef]
  23. P. W. Wong, J. Allebach, “Optimum error diffusion kernel design,” in Color Imaging:Device-Independent Color, Color Hard Copy and Graphic Arts II, G. B. Beretta, R. Eschbach, eds., Proc. SPIE3018, 236–243 (1997).
    [CrossRef]
  24. Special issue on Diffractive and Micro-Optics, J. Jahns, A. Cox, M. G. Moharam, eds., Appl. Opt. 36, 4633–4771 (1997).
    [CrossRef]
  25. F. Fetthauer, S. Weissbach, O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
    [CrossRef]
  26. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  27. M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
    [CrossRef]

1997 (1)

1996 (1)

V. Boutenko, R. Chevallier, “Second order direct binary search algorithm for the synthesis of computer-generated holograms,” Opt. Commun. 125, 43–47 (1996).
[CrossRef]

1995 (4)

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

R. Nagarajan, R. Easton, R. Eschbach, “Using adaptive quantization in cell-oriented holograms,” Opt. Commun. 144, 370–374 (1995).
[CrossRef]

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

E. Zhang, S. Noeht, C. H. Dietrich, R. Männer, “Gradual and random binarization of gray-scale holograms,” Appl. Opt. 34, 5987–5995 (1995).
[CrossRef] [PubMed]

1994 (1)

M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

1993 (2)

P. Thorston, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[CrossRef]

R. Eschbach, Z. Fan, “Complex valued error diffusion for off-axis computer-generated holograms,” Appl. Opt. 32, 3130–3136 (1993).
[CrossRef] [PubMed]

1992 (3)

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

A. Kirk, K. Powell, T. Hall, “A generalisation of the error diffusion method for binary computer generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

A. Kirk, K. Powell, T. Hall, “Error diffusion and the representation problem in computer generated hologram design,” Opt. Comput. Process. 2, 199–212 (1992).

1991 (3)

1990 (1)

1989 (2)

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

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef] [PubMed]

1988 (2)

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

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

1987 (1)

1985 (1)

1984 (1)

1976 (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

1969 (1)

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

Allebach, J.

P. W. Wong, J. Allebach, “Optimum error diffusion kernel design,” in Color Imaging:Device-Independent Color, Color Hard Copy and Graphic Arts II, G. B. Beretta, R. Eschbach, eds., Proc. SPIE3018, 236–243 (1997).
[CrossRef]

Allebach, J. P.

Bernard, E.

Boutenko, V.

V. Boutenko, R. Chevallier, “Second order direct binary search algorithm for the synthesis of computer-generated holograms,” Opt. Commun. 125, 43–47 (1996).
[CrossRef]

Bryngdahl, O.

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

P. Thorston, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[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]

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-generated holograms: carrier of polar geometry,” Appl. Opt. 24, 777–784 (1985).
[CrossRef] [PubMed]

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

Chevallier, R.

V. Boutenko, R. Chevallier, “Second order direct binary search algorithm for the synthesis of computer-generated holograms,” Opt. Commun. 125, 43–47 (1996).
[CrossRef]

Dietrich, C. H.

Easton, R.

R. Nagarajan, R. Easton, R. Eschbach, “Using adaptive quantization in cell-oriented holograms,” Opt. Commun. 144, 370–374 (1995).
[CrossRef]

Eschbach, R.

Fan, Z.

Fetthauer, F.

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Gale, M. T.

M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Hall, T.

A. Kirk, K. Powell, T. Hall, “Error diffusion and the representation problem in computer generated hologram design,” Opt. Comput. Process. 2, 199–212 (1992).

A. Kirk, K. Powell, T. Hall, “A generalisation of the error diffusion method for binary computer generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

Hauck, R.

Hirsch, P. M.

This popular and powerful algorithm has been used, in several domains and under different names, by many researchers, for example P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971); R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); N. C. Gallagher, B. Lui, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973); J. R. Fenup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef] [PubMed]

Jennison, B. K.

Jordan, J. A.

This popular and powerful algorithm has been used, in several domains and under different names, by many researchers, for example P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971); R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); N. C. Gallagher, B. Lui, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973); J. R. Fenup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef] [PubMed]

Just, D.

D. Just, D. T. Ling, “Neural networks for binarizing computer-generated holograms,” Opt. Commun. 81, 1–5 (1991).
[CrossRef]

Kirk, A.

A. Kirk, K. Powell, T. Hall, “Error diffusion and the representation problem in computer generated hologram design,” Opt. Comput. Process. 2, 199–212 (1992).

A. Kirk, K. Powell, T. Hall, “A generalisation of the error diffusion method for binary computer generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

Lesem, L. B.

This popular and powerful algorithm has been used, in several domains and under different names, by many researchers, for example P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971); R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); N. C. Gallagher, B. Lui, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973); J. R. Fenup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef] [PubMed]

Ling, D. T.

D. Just, D. T. Ling, “Neural networks for binarizing computer-generated holograms,” Opt. Commun. 81, 1–5 (1991).
[CrossRef]

Männer, R.

Nagarajan, R.

R. Nagarajan, R. Easton, R. Eschbach, “Using adaptive quantization in cell-oriented holograms,” Opt. Commun. 144, 370–374 (1995).
[CrossRef]

Neugebauer, G.

Noeht, S.

Pedersen, J.

M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Powell, K.

A. Kirk, K. Powell, T. Hall, “Error diffusion and the representation problem in computer generated hologram design,” Opt. Comput. Process. 2, 199–212 (1992).

A. Kirk, K. Powell, T. Hall, “A generalisation of the error diffusion method for binary computer generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

Rossi, K.

M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Schroeder, M. R.

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

Schütz, H.

M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Seldowitz, M. A.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Sweeny, D. W.

Thorston, P.

P. Thorston, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[CrossRef]

Weissbach, S.

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[CrossRef]

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, “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]

Wong, P. W.

P. W. Wong, J. Allebach, “Optimum error diffusion kernel design,” in Color Imaging:Device-Independent Color, Color Hard Copy and Graphic Arts II, G. B. Beretta, R. Eschbach, eds., Proc. SPIE3018, 236–243 (1997).
[CrossRef]

Wyrowski, F.

P. Thorston, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[CrossRef]

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

F. Wyrowski, “Diffractive efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
[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]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (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.

Appl. Opt. (8)

IEEE Spectr. (1)

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

J. Mod. Opt. (1)

P. Thorston, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (8)

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Computer-generated Fresnel holograms: quantization with the error diffusion algorithm,” Opt. Commun. 114, 230–234 (1995).
[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]

F. Fetthauer, S. Weissbach, O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

R. Nagarajan, R. Easton, R. Eschbach, “Using adaptive quantization in cell-oriented holograms,” Opt. Commun. 144, 370–374 (1995).
[CrossRef]

V. Boutenko, R. Chevallier, “Second order direct binary search algorithm for the synthesis of computer-generated holograms,” Opt. Commun. 125, 43–47 (1996).
[CrossRef]

A. Kirk, K. Powell, T. Hall, “A generalisation of the error diffusion method for binary computer generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

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

D. Just, D. T. Ling, “Neural networks for binarizing computer-generated holograms,” Opt. Commun. 81, 1–5 (1991).
[CrossRef]

Opt. Comput. Process. (1)

A. Kirk, K. Powell, T. Hall, “Error diffusion and the representation problem in computer generated hologram design,” Opt. Comput. Process. 2, 199–212 (1992).

Opt. Eng. (1)

M. T. Gale, K. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresists,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Proc. Soc. Inf. Disp. (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Other (2)

P. W. Wong, J. Allebach, “Optimum error diffusion kernel design,” in Color Imaging:Device-Independent Color, Color Hard Copy and Graphic Arts II, G. B. Beretta, R. Eschbach, eds., Proc. SPIE3018, 236–243 (1997).
[CrossRef]

This popular and powerful algorithm has been used, in several domains and under different names, by many researchers, for example P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971); R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972); N. C. Gallagher, B. Lui, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973); J. R. Fenup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef] [PubMed]

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Figures (18)

Fig. 1
Fig. 1

ED direction weightings d(i, j) and diffusion domain A.

Fig. 2
Fig. 2

Differences in ED pixel processing orders and the subsequent differences in direction weightings and diffusion domains.

Fig. 3
Fig. 3

Diagrammatic representation of the minimization of in MAE diffusion.

Fig. 4
Fig. 4

Test target images used in the simulations.

Fig. 5
Fig. 5

Signal window weighting function W(I, J).

Fig. 6
Fig. 6

DFT w(i, j) of the signal window weighting function W(I, J) for two different image screen normalizations.

Fig. 7
Fig. 7

Simulated reconstructions for increasing numbers of diffusion weights. The example shown is for a 32×32-pixel target image in a 128×128-pixel hologram.

Fig. 8
Fig. 8

Simulated diffraction efficiency and reconstruction error for increasing numbers of diffusion weights. The example is the same as that for Fig. 7.

Fig. 9
Fig. 9

Simulated reconstruction of a hologram calculated with SWMAE weights corresponding to incorrectly positioned signal windows.

Fig. 10
Fig. 10

Simulated reconstruction of a hologram calculated with a lexicographic MAE processing (96 weights).

Fig. 11
Fig. 11

Different ways of diffusing quantization error from hologram edge pixels.

Fig. 12
Fig. 12

(a) Smooth signal window function, (b) and (c) its DFT for two different image screen normalizations.

Fig. 13
Fig. 13

Simulated reconstruction of a SWMAE-calculated CGH with the use of a smooth signal window function.

Fig. 14
Fig. 14

Simulated reconstruction of a SWMAE-calculated CGH with a target-optimized diffuser on the target image. The same reconstruction is shown for three different image screen normalizations to make the quantization noise visible.

Fig. 15
Fig. 15

Simulated reconstruction of a SWMAE-calculated hologram with a random phase diffuser. The target image contains 450×490 pixels, and the CGH contains 2048×2048 pixels. The screen normalization has been adjusted to show the quantization error clouds.

Fig. 16
Fig. 16

Signal window area of the simulated reconstruction of the SWMAE-calculated hologram shown in Fig. 15.

Fig. 17
Fig. 17

Experimental view of the optical reconstruction of the 2048×2048-pixel SWMAE-calculated hologram corresponding to the simulations shown in Fig. 15.

Fig. 18
Fig. 18

Experimental view of the signal window area in the optical reconstruction shown in Fig. 17.

Tables (2)

Tables Icon

Table 1 Values for the Central MAE Diffusion Weights

Tables Icon

Table 2 Comparison of Reconstruction Error (Err), Diffraction Efficiency (Eff), and Calculation Time (CPU) in Simulated Reconstructions for Different-Sized Test CGH’s with the IFT and SWMAE Algorithms

Equations (16)

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

e(i, j)=f(i, j)-b(i, j).
b(i, j)=Θf(i, j)+r,sAd(r, s)e(i-r, j-s),
e(i, j)=f(i, j)+r,sAd(r, s)e(i-r, j-s)-b(i, j).
e(i, j)=f(i, j)-b(i, j).
=f(i, j)+r,sAd(r, s)e(i-r, j-s)-b(i, j).
=f(i, j)-b(i, j)+r,sAd(r, s)e(i-r, j-s)
=r,sA0d(r, s)e(i-r, j-s)
=d * e,
E2(I, J)=[F(I, J)-B(I, J)]2,
E2=I,J=0N-1W2(F-B)2,
E2=i,j=0N-1[w * (f-b)][w * (f-b)]
=i,j=0N-1[w * (f-b)]2
=i,j=0N-1(w * e)2,
d(i, j)=w(i, j),
Error=1NΔ2I,JΔ[|F¯(I, J)|2-λ|B(I, J)|2]21/2,
λ=Δ|F¯(I, J)|2|B(I, J)|2Δ|B(I, J)|4

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