Abstract

An interlacing technique algorithm is proposed for the synthesis of kinoforms. The conventional iterative methods are quite powerful for optimizing kinoforms, but there is still a large reconstruction error for a quantized kinoform. We suggest the use of a number of subkinoforms interlaced together to synthesize a multikinoform for reconstructing the desired image. The idea of our interlacing technique is to increase the size of a kinoform to reduce the reconstruction error. The first subkinoform is generated from the desired image. Other subkinoforms are generated from the error images between the desired image and the image reconstructed from the previous subkinoforms. A theoretical analysis shows that the reconstruction error will be reduced as the number of subkinoforms is increased. Simulation results show that our interlacing method can reduce the reconstruction error more than do the conventional iterative methods and that the reconstructed image can be improved.

© 1996 Optical Society of America

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References

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  1. L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [Crossref]
  2. T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).
  3. F. Mok, J. Diep, H. K. Liu, D. Psaltis, “Real-time computer-generated hologram by means of liquid-crystal television spatial light modulator,” Opt. Lett. 11, 748–750 (1986).
    [Crossref] [PubMed]
  4. J. A. Davis, G. M. Heissenberg, R. A. Lilly, D. M. Cottrell, M. F. Brownell, “High efficiency optical reconstruction of binary phase-only filter using the Hughes liquid crystal light valve,” Appl. Opt. 26, 929–933 (1987).
    [Crossref] [PubMed]
  5. M. R. Feldman, C. C. Guest, “Computer generated holographic optical elements for optical interconnection of very large scale integrated circuits,” Appl. Opt. 26, 4377–4384 (1987).
    [Crossref] [PubMed]
  6. J. Amako, T. Sonehara, “Kinoform using an electrically controlled birefringent liquid-crystal spatial light modulator,” Appl. Opt. 30, 4622–4628 (1991).
    [Crossref] [PubMed]
  7. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik (Stuttgart) 35, 237–246 (1972).
  8. N. C. Gallagher, B. Liu, “Method for computing kinoform that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [Crossref] [PubMed]
  9. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated hologram,” Opt. Eng. 19, 297–305 (1980).
  10. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  11. G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
    [Crossref]
  12. N. Yoshikawa, T. Yatagai, “Phase optimization of a kinoform by simulated annealing,” Appl. Opt. 33, 863–868 (1994).
    [Crossref] [PubMed]
  13. N. Yoshikawa, M. Itoh, T. Yatagai, “Quantized phase optimization of two-dimensional Fourier kinoform by a genetic algorithm,” Opt. Lett. 20, 752–754 (1995).
    [Crossref] [PubMed]
  14. O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt. 31, 6894–6901 (1992).
    [Crossref] [PubMed]

1995 (1)

1994 (2)

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[Crossref]

N. Yoshikawa, T. Yatagai, “Phase optimization of a kinoform by simulated annealing,” Appl. Opt. 33, 863–868 (1994).
[Crossref] [PubMed]

1992 (1)

1991 (1)

1987 (2)

1986 (1)

1982 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated hologram,” Opt. Eng. 19, 297–305 (1980).

1975 (1)

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

1973 (1)

1972 (1)

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

1969 (1)

L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Amako, J.

Brede, J.

Brownell, M. F.

Cottrell, D. M.

Davis, J. A.

Diep, J.

Ersoy, O. K.

Feldman, M. R.

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated hologram,” Opt. Eng. 19, 297–305 (1980).

Gallagher, N. C.

Gerchberg, R. W.

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

Guest, C. C.

Heissenberg, G. M.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Itoh, M.

Jordan, T. A.

L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Lilly, R. A.

Liu, B.

Liu, H. K.

Lu, G.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[Crossref]

Mok, F.

Psaltis, D.

Saxton, W. O.

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

Sonehara, T.

Takeda, M.

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

Tanone, A.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[Crossref]

Yatagai, T.

Yoshikawa, N.

Yu, F. T. S.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[Crossref]

Zhang, Z.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[Crossref]

Zhuang, J. Y.

Appl. Opt. (7)

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Opt. Eng. (2)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated hologram,” Opt. Eng. 19, 297–305 (1980).

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[Crossref]

Opt. Lett. (2)

Optik (Stuttgart) (2)

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

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

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Figures (6)

Fig. 1
Fig. 1

Geometric configuration of the IT approach: ○, (0, 0)th subkinoform; △, (1, 0)th subkinoform; ⋄, (0, 1)th subkinoform; □, (1, 1)th subkinoform.

Fig. 2
Fig. 2

Schematic of the NITA. FFT, fast Fourier transform; IFFT, inverse FFT.

Fig. 3
Fig. 3

MSE dependence on the number of subkinoforms.

Fig. 4
Fig. 4

MSE dependence on the number of iterations: (a) with different quantization levels and (b) with a different number of subkinoforms.

Fig. 5
Fig. 5

Simulated reconstructed image.

Fig. 6
Fig. 6

Comparison of the NITA and conventional iterative methods.

Equations (38)

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G ( 0 , 0 ) ( u , v ) = G ( 0 , 0 ) ( u , v ) exp [ j Φ ( 0 , 0 ) ( u , v ) ] = x = 0 N - 1 y = 0 N - 1 f ( x , y ) exp [ j η ( 0 , 0 ) ( x , y ) ] × exp [ - j 2 π N ( x u + y v ) ] ,
W ( 0 , 0 ) ( u , v ) = A exp [ j Φ ( 0 , 0 ) , L ( u , v ) ] ,
A 2 = x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 .
g ( 0 , 0 ) ( x , y ) = 1 N 2 x = 0 N - 1 y = 0 N - 1 W ( 0 , 0 ) ( u , v ) × exp [ j 2 π N ( x u + y v ) ] .
g ( p , q ) ( x , y ) = { λ ( p , q ) f ( x , y ) exp [ j η ( p , q ) - 1 ( x , y ) ] - g ( p , q ) - 1 ( x , y ) } exp [ - j 2 π N ( p x P + q y Q ) ] ,
g ( p , q ) - 1 ( x , y ) = g ( p , q ) - 1 ( x , y ) exp [ j θ ( p , q ) - 1 ( x , y ) ] = ( p , q ) = 0 ( p , q ) - 1 g ( p , q ) ( x , y ) .
G ( p , q ) ( u , v ) = G ( p , q ) ( u , v ) exp [ j Φ ( p , q ) ( u , v ) ] = x = 0 N - 1 y = 0 N - 1 g ( p , q ) ( x , y ) exp [ - j 2 π N ( x u + y v ) ] ,
W ( p , q ) ( u , v ) = A exp [ j Φ ( p , q ) , L ( u , v ) ] .
g ( p , q ) ( x , y ) = { 1 N 2 x = 0 N - 1 y = 0 N - 1 W ( p , q ) ( u , v ) × exp [ j 2 π N ( x u + y v ) ] } × exp [ j 2 π N ( p x P + q y Q ) ] .
W ( u , v ) = W ( u P + p , v Q + q ) = W ( p , q ) ( u , v ) ,
g ( x , y ) = g ( x , y ) exp [ j θ ( x , y ) ] = 1 P N Q N u = 0 P × N - 1 v = 0 Q × N - 1 W ( u , v ) × exp [ j 2 π ( x u P N + y v Q N ) ] ,
g ( x + r N , y + s N ) = 1 P Q p = 0 P - 1 q = 0 Q - 1 ( 1 N 2 x = 0 N - 1 y = 0 N - 1 { W ( p , q ) ( u , v ) × exp [ j 2 π N ( x u + y v ) ] × exp [ j 2 π N ( p x P + q y Q ) ] } exp [ j 2 π ( p r P + q s Q ) ] ) = 1 P Q p = 0 P - 1 q = 0 Q - 1 { g ( p , q ) ( x , y ) exp [ j 2 π ( p r p + q s Q ) ] } .
g ( x , y ) = 1 P Q p = 0 P - 1 q = 0 Q - 1 g ( p , q ) ( x , y ) .
g ( p , q ) ( x , y ) = { λ ( p , q ) f ( x , y ) exp [ j θ ( p , q ) ( x , y ) ] - g ( p , q ) - 1 ( x , y ) } exp [ - j 2 π N ( p x P + q y Q ) ] .
E mse = 10 log [ x = 0 N - 1 y = 0 N - 1 I obj ( x , y ) - α I rec ( x , y ) 2 ] ,
I obj ( x , y ) = f ( x , y ) 2 ,
I rec ( x , y ) = g ( x , y ) 2 .
α = x = 0 N - 1 y = 0 N - 1 I obj ( x , y ) I rec ( x , y ) x = 0 N - 1 y = 0 N - 1 I rec ( x , y ) 2 .
x = 0 N - 1 y = 0 N - 1 | λ ( p , q ) f ( x , y ) exp [ j η ( p , q ) - 1 ( x , y ) ] - g ( p , q ) - 1 ( x , y ) | 2 = x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 .
λ ( p , q ) 2 x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 - 2 λ ( p , q ) x = 0 N - 1 y = 0 N - 1 f ( x , y ) g ( p , q ) - 1 ( x , y ) × cos [ η ( p , q ) - 1 ( x , y ) - θ ( p , q ) - 1 ( x , y ) ] + x = 0 N - 1 y = 0 N - 1 g ( p , q ) - 1 ( x , y ) 2 - x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 = 0 .
Δ = 4 { x = 0 N - 1 y = 0 N - 1 f ( x , y ) g ( p , q ) - 1 ( x , y ) × cos [ η ( p , q ) - 1 ( x , y ) - θ ( p , q ) - 1 ( x , y ) ] } 2 - 4 x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 { x = 0 N - 1 y = 0 N - 1 g ( p , q ) - 1 ( x , y ) 2 - x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 } 0 .
η ( p , q ) - 1 ( x , y ) θ ( p , q ) - 1 ( x , y ) ,
g ( p , q ) - 1 ( x , y ) ( p + q P ) f ( x , y ) ,
Δ 4 ( p + q P ) 2 [ x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 ] 2 - 4 [ ( p + q P ) 2 - 1 ] [ x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 ] 2 = 4 [ x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 ] 2 > 0.
λ ( p , q ) , 1 = x = 0 N - 1 y = 0 N - 1 f ( x , y ) g ( p , q ) - 1 ( x , y ) cos [ η ( p , q ) - 1 ( x , y ) - θ ( p , q ) - 1 ( x , y ) ] + ( Δ / 2 ) x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 λ ( p , q ) , 2 = x = 0 N - 1 y = 0 N - 1 f ( x , y ) g ( p , q ) - 1 ( x , y ) cos [ η ( p , q ) - 1 ( x , y ) - θ ( p , q ) - 1 ( x , y ) ] - ( Δ / 2 ) x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 .
λ ( p , q ) = x = 0 N - 1 y = 0 N - 1 f ( x , y ) g ( p , q ) - 1 ( x , y ) cos [ η ( p , q ) - 1 ( x , y ) - θ ( p , q ) - 1 ( x , y ) ] + ( Δ / 2 ) x = 0 N - 1 y = 0 N - 1 f ( x , y ) 2 .
λ ( p , q ) p + q × P + 1 ,
λ ( P - 1 , Q - 1 ) < λ ( P - 1 , Q - 1 ) + 1 .
f ( x , y ) exp [ j η ( P - 1 , Q - 1 ) - 1 ( x , y ) ] - 1 λ ( P - 1 , Q - 1 ) × [ P Q g P Q ( x , y ) - g ( P - 1 , Q - 1 ) ( x , y ) × ( 1 - exp { j 2 π N [ ( P - 1 ) x P + ( Q - 1 ) y Q ] } ) ] = 1 λ ( P - 1 , Q - 1 ) { g ( P - 1 , Q - 1 ) ( x , y ) - g ( P - 1 , Q - 1 ) ( x , y ) } .
P Q g P Q ( x , y ) > g ( P - 1 , Q - 1 ) ( x , y ) .
| P Q g P Q ( x , y ) - g ( P - 1 , Q - 1 ) ( x , y ) × ( 1 - exp { j 2 π N [ ( P - 1 ) x P + ( Q - 1 ) y Q ] } ) | P Q g P Q ( x , y ) .
θ ( P - 1 , Q - 1 ) ( x , y ) θ ( P - 1 , Q - 1 ) ( x , y ) ,
x = 0 N - 1 y = 0 N - 1 | f ( x , y ) - P Q λ ( P - 1 , Q - 1 ) g P Q ( x , y ) | 2 1 λ ( P - 1 , Q - 1 ) 2 x = 0 N - 1 y = 0 N - 1 | g ( P - 1 , Q - 1 ) ( x , y ) - g ( P - 1 , Q - 1 ) ( x , y ) | 2 .
x = 0 N - 1 y = 0 N - 1 [ g ( P - 1 , Q - 1 ) ( x , y ) - g ( P - 1 , Q - 1 ) ( x , y ) ] 2 x = 0 N - 1 y = 0 N - 1 [ g ( P - 1 , Q - 1 ) + 1 ( x , y ) - g ( P - 1 , Q - 1 ) + 1 ( x , y ) ] 2 .
x = 0 N - 1 y = 0 N - 1 [ f ( x , y ) - g P Q + 1 ( x , y ) ] 2 < x = 0 N - 1 y = 0 N - 1 [ f ( x , y ) - g P Q ( x , y ) ] 2 .
g ( p , q ) ( x , y ) g ( p , q ) ( x , y ) .
g ( x + r N , y + s N ) 1 P Q g ( p , q ) ( x , y ) × p = 0 P - 1 q = 0 Q - 1 exp [ j 2 π ( p r P + q s Q ) ] .
p = 0 P - 1 q = 0 Q - 1 exp [ j 2 π ( p r P + q s Q ) ] = 0 ,

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