Abstract

There are two kinds of method that utilize the redundancy in kinoform domains for reducing the reconstruction errors of quantized kinoforms. One is the iterative-dummy area (IDA) method, which increases the kinoform size indirectly by the addition of a dummy area to the desired image. The other is the interlacing technique (IT), which increases the kinoform size directly by the interlacing of a number of subkinoforms whose sizes are the same as the desired image. We compare the error reduction of quantized kinoforms between these two methods. Simulation results show that reconstruction errors from the IT method can be reduced further and faster than those from the IDA method when the kinoform size is increased to larger than 4 × 4 times the size of the desired image.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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1996 (1)

1994 (1)

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]

1990 (1)

1989 (1)

1986 (1)

1982 (1)

1980 (1)

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

1975 (1)

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

1973 (2)

1972 (1)

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]

Akahori, H.

Bryngdahl, O.

O. Bryngdahl, F. Wyrowski, “Digital holography: computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 1–86.
[CrossRef]

Chu, D. C.

Fienup, J. R.

Gallagher, N. C.

Goodman, J. W.

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]

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]

Liu, B.

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]

Merzlyakov, N. S.

L. P. Yaroslavskii, N. S. Merzlyakov, Digital Holography (Consultants Bureau, New York, 1980), Appendix 2, pp. 167–168.

Shimomura, 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]

Wyrowski, F.

Yang, S.

Yaroslavskii, L. P.

L. P. Yaroslavskii, N. S. Merzlyakov, Digital Holography (Consultants Bureau, New York, 1980), Appendix 2, pp. 167–168.

Yatagai, T.

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

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]

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]

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

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

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]

Optik (Stuttgart) (1)

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

Other (2)

L. P. Yaroslavskii, N. S. Merzlyakov, Digital Holography (Consultants Bureau, New York, 1980), Appendix 2, pp. 167–168.

O. Bryngdahl, F. Wyrowski, “Digital holography: computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 1–86.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Configuration of a dummy area added to the desired image.

Fig. 2
Fig. 2

Schematic of the IT method. DFT, discrete Fourier transform; IDFT, inverse discrete Fourier transform.

Fig. 3
Fig. 3

Diagram of the algorithm of the IT method with combined iterations and circulations. FFT, fast Fourier transform; IFFT, inverse fast Fourier transform.

Fig. 4
Fig. 4

Desired images.

Fig. 5
Fig. 5

Dependence of the MSE on the number of iterations when the IDA method is used: △ Fig. 4(a), ◇ Fig. 4(b), ○ Fig. 4(c), × Fig. 4(d).

Fig. 6
Fig. 6

Dependence of the MSE on the number of iterations when the IT method is used: △ Fig. 4(a), ◇ Fig. 4(b), ○ Fig. 4(c), × Fig. 4(d).

Fig. 7
Fig. 7

Comparison of the IDA method with stepwise quantization and the IT method by means of increasing the kinoform size. The solid curves represent the IDA method, and the dotted curves represent the IT method: △ Fig. 4(a), ◇ Fig. 4(b), ○ Fig. 4(c), × Fig. 4(d).

Equations (14)

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F u ,   v = x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   | f x ,   y | exp j η x ,   y × exp - j   2 π N xu + yv ,
F u ,   v = x = - N / 2 + x 0 N / 2 - 1 + x 0 y = - N / 2 + y 0 N / 2 - 1 + y 0   | f x - x 0 ,   y - y 0 | × exp j η x - x 0 ,   y - y 0 × exp - j 2 π x u PN + y v QN ,
W u ,   v = A   exp arg   F u ,   v ,
g k x ,   y = | f x - x 0 ,   y - y 0 | exp j   arg   g k - 1 x ,   y   for   x - N / 2 + x 0 ,   N / 2 - 1 + x 0 y - N / 2 + y 0 ,   N / 2 - 1 + y 0 α k g k - 1 x ,   y   otherwise ,
e k x ,   y = | f x - x 0 ,   y - y 0 | exp j   arg   g k - 1 x ,   y - α k g k - 1 x ,   y .
G k u ,   v = G k uP + p ,   vQ + q = x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   e k x ,   y exp - j   2 π N p x + x 0 P + q y + y 0 Q exp - j   2 π N xu + yv × exp - j   2 π N x 0 u + y 0 v + W k - 1 u ,   v .
F p , q u ,   v = F uP + p ,   vQ + q = x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   | f x ,   y | exp j η x ,   y × exp - j   2 π N p x + x 0 P + q y + y 0 Q × exp - j   2 π N xu + yv × exp - j   2 π N x 0 u + y 0 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 + x 0 P + q y + y 0 Q .
g p , q , k x ,   y = λ p , q , k | f x ,   y | exp j   arg   g p , q , k - 1 x ,   y - g p , q - 1 x ,   y × exp - j   2 π N p x + x 0 P + q y + y 0 Q .
g p , q x ,   y = λ p , q | f x ,   y | exp j η p , q - 1 x ,   y - g x ,   y + g p , q x ,   y × exp - j   2 π N p x + x 0 P + q y + y 0 Q ,
E MSE = 10   log x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   | I 0 x ,   y - α I r x ,   y | 2 x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   I 0 x ,   y 2 ,
I 0 x ,   y = | f x ,   y | 2 ,
I r x ,   y = | g x ,   y | 2 .
α = x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   I 0 x ,   y I r x ,   y x = - N / 2 N / 2 - 1 y = - N / 2 N / 2 - 1   I r x ,   y 2 .

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