Abstract

The iterative interlacing error-diffusion technique is a combination of the error-diffusion and the modified iterative interlacing techniques for synthesizing computer-generated holograms. The iterative interlacing error-diffusion technique leads to a dramatic improvement in the quality of reconstructed images, provided that the two constant parameters involved in iterations are chosen properly.

© 1993 Optical Society of America

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References

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  1. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grey-scale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).
  2. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  3. 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]
  4. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” Opt. Eng. 28, 629–637 (1989).
  5. E. Barnard, “Optimal error diffusion for computer-generated holograms,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
    [CrossRef]
  6. R. Eschbach, “Comparison on error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710 (1991).
    [CrossRef] [PubMed]
  7. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  8. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  9. O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt.31, 6894–6901 (1992); Tech. Rep. TR-EE-92-2 (Purdue University, West Lafayette, Indiana, 1992).
    [CrossRef]
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
    [CrossRef]
  12. M. Broja, K. Michalowski, O. Bryngdahl, “Error diffusion concept for multi-level quantization,” Opt. Commun. 79, 280–284 (1990).
    [CrossRef]

1991 (1)

1990 (2)

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
[CrossRef]

M. Broja, K. Michalowski, O. Bryngdahl, “Error diffusion concept for multi-level quantization,” Opt. Commun. 79, 280–284 (1990).
[CrossRef]

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]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” Opt. Eng. 28, 629–637 (1989).

1988 (1)

1987 (1)

1986 (1)

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

1984 (1)

1982 (1)

1976 (1)

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

Allebach, J. P.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” Opt. Eng. 28, 629–637 (1989).

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]

Barnard, E.

Brede, J.

O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt.31, 6894–6901 (1992); Tech. Rep. TR-EE-92-2 (Purdue University, West Lafayette, Indiana, 1992).
[CrossRef]

Broja, M.

M. Broja, K. Michalowski, O. Bryngdahl, “Error diffusion concept for multi-level quantization,” Opt. Commun. 79, 280–284 (1990).
[CrossRef]

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

Bryngdahl, O.

M. Broja, K. Michalowski, O. Bryngdahl, “Error diffusion concept for multi-level quantization,” Opt. Commun. 79, 280–284 (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]

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

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

Ersoy, O. K.

O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt.31, 6894–6901 (1992); Tech. Rep. TR-EE-92-2 (Purdue University, West Lafayette, Indiana, 1992).
[CrossRef]

Eschbach, R.

R. Eschbach, “Comparison on error diffusion methods for computer-generated holograms,” Appl. Opt. 30, 3702–3710 (1991).
[CrossRef] [PubMed]

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

Fienup, J. R.

Floyd, R. W.

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

Hauck, R.

Jennison, B. K.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” Opt. Eng. 28, 629–637 (1989).

Michalowski, K.

M. Broja, K. Michalowski, O. Bryngdahl, “Error diffusion concept for multi-level quantization,” Opt. Commun. 79, 280–284 (1990).
[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).

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” Opt. Eng. 28, 629–637 (1989).

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]

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.

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (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]

Zhuang, J. Y.

O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt.31, 6894–6901 (1992); Tech. Rep. TR-EE-92-2 (Purdue University, West Lafayette, Indiana, 1992).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

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, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

M. Broja, K. Michalowski, O. Bryngdahl, “Error diffusion concept for multi-level quantization,” Opt. Commun. 79, 280–284 (1990).
[CrossRef]

Opt. Eng. (1)

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” Opt. Eng. 28, 629–637 (1989).

Proc. Soc. Inf. Disp. (1)

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

Other (1)

O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt.31, 6894–6901 (1992); Tech. Rep. TR-EE-92-2 (Purdue University, West Lafayette, Indiana, 1992).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Geometry used in constructing two interlaced holograms.

Fig. 2
Fig. 2

Error diffusion: T, constant threshold; n, pixel number.

Fig. 3
Fig. 3

Placement of the images and zero regions in the ED and IIED techniques.

Fig. 4
Fig. 4

Generation of twin images owing to decimation into two holograms: (a) no decimation (b) odd-row hologram only, (c) even-row hologram only. Xd, desired image; Xd*, Hermitian image; background initialized with zeros (shaded area); object field (white squares).

Fig. 5
Fig. 5

Direction of error diffusion in the two holograms with the IIED technique.

Fig. 6
Fig. 6

Flow chart of the IED technique.

Fig. 7
Fig. 7

Reconstruction error versus the ED coefficient with the three techniques used with the girl image: 1 iteration and 1 hologram with the ED technique (curve A); 20 sweeps and 2 holograms with the unmodified IIT and the ED technique (curve B); 20 sweeps and 2 holograms with the IIED technique and β = 1.0 (curve C); 20 sweeps and 2 holograms with the lIED technique and β = 1.5 (curve D).

Fig. 8
Fig. 8

(a) Reconstruction error versus the iteration number in the IIED technique, (b) enlarged right-hand side of Fig. 9(a).

Fig. 9
Fig. 9

Reconstructed images obtained with the three techniques: case A, ED; case B, IIED with β = 1.0; case D, IIED with β = 1.5.

Fig. 10
Fig. 10

Total reconstructed image at the focal plane with the ED technique.

Fig. 11
Fig. 11

Total reconstructed image at the focal plane with the IIED technique.

Fig. 12
Fig. 12

Binary image used in the second set of computer experiments.

Fig. 13
Fig. 13

Reconstruction error versus the ED coefficient ω in the three techniques used with the binary image. The curves have the same meaning as in Fig. 7 except that curves B–D denote ten sweeps.

Tables (1)

Tables Icon

Table 1 Relative Reconstruction Errors Associated with the Iterative Interlacing Error-Diffusiona Method and with the Error-Diffusion Method

Equations (24)

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E 1 ( n 1 , n 2 ) = X ( n 1 , n 2 ) - λ 1 X rec 1 ( n 1 , n 2 ) ,
E 2 ( n 1 , n 2 ) = X ( n 1 , n 2 ) - λ 2 [ X rec 1 ( n 1 , n 2 ) + X rec 2 ( n 1 , n 2 ) ] ,
X 2 , 1 ( n 1 , n 2 ) = X rec 1 ( n 1 , n 2 ) + E 2 ( n 1 , n 2 ) / λ 2
X 2 , 2 ( n 1 , n 2 ) = X rec 2 ( n 1 , n 2 ) + E 2 , 1 ( n 1 , n 2 ) / λ 2 , 1 ,
X i , 1 ( n 1 , n 2 ) = β X i - 1 , 1 ( n 1 , n 2 ) + E i - 1 , 2 ( n 1 , n 2 ) / λ i - 1 , 2 ,
X i , 2 ( n 1 , n 2 ) = β X i - 1 , 2 ( n 1 , n 2 ) + E i , 1 ( n 1 , n 2 ) / λ i , 1 .
E 1 = H 1 - B 1 .
H ˜ m = H m + ω E m - 1 ,
B m = f ( H ˜ m - T ) ,
E m = H ˜ m - B m ,
E m = ω E m - 1 + ( H m - B m ) .
E m = ω E m - 1 = ω m - 1 E 1 .
H ( k 1 , k 2 ) = H ( k 1 , k 2 ) for k 1 odd = 0 otherwise ,
H ( k 1 , k 2 ) = 0.5 [ ( - 1 ) k 1 + 1 ] H ( k 1 , k 2 ) .
X ( n 1 , n 2 ) = 0.5 [ X ( n 1 , n 2 ) + X ( n 1 + N / 2 , n 2 ) ] ;
X i , j ( n 1 , n 2 ) = X ( n 1 , n 2 ) i = 1 , j = 1 = X ( n 1 , n 2 ) / λ i , j - X r e c 1 ( n 1 , n 2 ) i = 1 , j = 2 = β X i - 1 , j ( n 1 , n 2 ) + E i - 1 , 2 ( n 1 , n 2 ) / λ i - 1 , 2 i > 1 , j = 1 = β X i - 1 , j ( n 1 , n 2 ) + E i - 1 , 1 ( n 1 , n 2 ) / λ i - 1 , 1 i > 1 , j = 2 .
H j ( k 1 , k 2 ) = H j ( k 1 , k 2 ) for k 1 even = 0 otherwise .
H j ( k 1 , k 2 ) = H j ( k 1 , k 2 ) for k 1 odd = 0 otherwise .
H j ( k 1 , k 2 ) = [ H j ( k 1 , k 2 ) - H min ( k 1 , k 2 ) ] / [ H max ( k 1 , k 2 ) - H max ( k 1 , k 2 ) ] ,
E i , j ( n 1 , n 2 ) = X ( n 1 , n 2 ) - λ i , j [ X rec 1 ( n 1 , n 2 ) + X rec 2 ( n 1 , n 2 ) ] .
MSE = 1 A B m n X m n - λ X ˜ m n 2 , ( m , n ) R ,
λ = λ r + i λ i .
λ opt = λ r opt + i λ i opt ,
λ r opt = n m Re ( X m n X ˜ m n * ) n m X ˜ m n 2 , λ i opt = n m Im ( X m n X ˜ m n * ) n m X ˜ m n 2 , ( m , n ) R .

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