Abstract

A method for accelerating the synthesis of computer-generated three-dimensional (3-D) holograms, based on conventional ray tracing, is proposed. In ray tracing, computers expend almost all of their resources in calculating the 3-D distances between each one of the point sources composing an object and a sampling point on the hologram. We present recurrence formulas that precisely compute the distances and reduce the computation time for synthesizing holograms to one half to one quarter, depending on the processor type. We demonstrate that a full-parallax hologram with an area of 4800 × 4800 pixels, synthesized for a 3-D object containing 966 point sources of light, is computed within 17 min and is optically reconstructed.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
    [CrossRef]

1999 (1)

1998 (1)

1997 (2)

1996 (1)

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

1994 (1)

T. Ito, T. Yabe, M. Okazaki, M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, 104–110 (1994).
[CrossRef]

1993 (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
[CrossRef]

1992 (1)

A. D. Stein, Z. Wang, J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

1989 (1)

1988 (1)

1987 (2)

1971 (1)

T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1347 (1971).
[CrossRef]

1966 (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Allebach, J. P.

Ambs, P.

Benton, S. A.

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., SPIE Proc.1212, 174–182 (1990).

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Real-time holographic display: improvements using a multichannel acoustooptic modulator and holographic optical elements,” Practical Holography V, S. A. Benton, ed., SPIE Proc.1461, 254–261 (1991).

Berriel-Valdos, R.

Böttger, J.

Bryndahl, O.

Deussen, O.

Eldeib, H.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

Huang, T. S.

T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1347 (1971).
[CrossRef]

Ito, T.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

T. Ito, T. Yabe, M. Okazaki, M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, 104–110 (1994).
[CrossRef]

Itoh, M.

Juárez-Pérez, J. L.

König, M.

Kunugi, T.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

Legeard, L.

Leigh, J. J. S.

A. D. Stein, Z. Wang, J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Lucente, M.

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
[CrossRef]

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Real-time holographic display: improvements using a multichannel acoustooptic modulator and holographic optical elements,” Practical Holography V, S. A. Benton, ed., SPIE Proc.1461, 254–261 (1991).

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., SPIE Proc.1212, 174–182 (1990).

Okazaki, M.

T. Ito, T. Yabe, M. Okazaki, M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, 104–110 (1994).
[CrossRef]

Olivares-Pérez, A.

Réfrégier, P.

Ritter, A.

Seldwitz, M. A.

St.-Hilaire, P.

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Real-time holographic display: improvements using a multichannel acoustooptic modulator and holographic optical elements,” Practical Holography V, S. A. Benton, ed., SPIE Proc.1461, 254–261 (1991).

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., SPIE Proc.1212, 174–182 (1990).

Stein, A. D.

A. D. Stein, Z. Wang, J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Strothotte, T.

Sweedney, D. W.

Takahashi, S.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

Tricoles, G.

Underkoffler, J.

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., SPIE Proc.1212, 174–182 (1990).

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Real-time holographic display: improvements using a multichannel acoustooptic modulator and holographic optical elements,” Practical Holography V, S. A. Benton, ed., SPIE Proc.1461, 254–261 (1991).

Wang, Z.

A. D. Stein, Z. Wang, J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Waters, J. P.

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Wyrowski, F.

Yabe, T.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

T. Ito, T. Yabe, M. Okazaki, M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, 104–110 (1994).
[CrossRef]

Yanagi, M.

T. Ito, T. Yabe, M. Okazaki, M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, 104–110 (1994).
[CrossRef]

Yatagai, T.

Yoshida, K.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

Yoshikawa, H.

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Real-time holographic display: improvements using a multichannel acoustooptic modulator and holographic optical elements,” Practical Holography V, S. A. Benton, ed., SPIE Proc.1461, 254–261 (1991).

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., SPIE Proc.1212, 174–182 (1990).

Yoshikawa, N.

Appl. Opt. (6)

Appl. Phys. Lett. (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Comput. Phys. (1)

A. D. Stein, Z. Wang, J. J. S. Leigh, “Computer-generated holograms: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Comput. Phys. Commun. (2)

T. Ito, T. Yabe, M. Okazaki, M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, 104–110 (1994).
[CrossRef]

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, T. Kunugi, “Special purpose computer for holography HORN-2,” Comput. Phys. Commun. 93, 13–20 (1996).
[CrossRef]

J. Electron. Imag. (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
[CrossRef]

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

Opt. Lett. (1)

Proc. IEEE (1)

T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1347 (1971).
[CrossRef]

Other (2)

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Electronic display system for computational holography,” in Practical Holography IV, S. A. Benton, ed., SPIE Proc.1212, 174–182 (1990).

P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler, H. Yoshikawa, “Real-time holographic display: improvements using a multichannel acoustooptic modulator and holographic optical elements,” Practical Holography V, S. A. Benton, ed., SPIE Proc.1461, 254–261 (1991).

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Figures (8)

Fig. 1
Fig. 1

Diagram of reference coordinate system.

Fig. 2
Fig. 2

2-D coordinates.

Fig. 3
Fig. 3

Numerical error of recurrence formula and binomial approximation; d 0 = 105λ.

Fig. 4
Fig. 4

Error contours given by numerical calculations of recurrence formula for sampling pitches of (a) δx ≤ λ/2 and (b) δx > λ/2.

Fig. 5
Fig. 5

Error of recurrence formula with error control; d 0 = 105λ. The sampling pitches and permissible errors are (a) δx = 2λ, E lim = 10-3λ; (b) δx = 3λ, E lim = 5 × 10-3λ; (c) δx = λ, E lim = 10-4λ.

Fig. 6
Fig. 6

Error of scaled-divisionless recurrence formula. d 0 = 105λ.

Fig. 7
Fig. 7

Performance of recurrence formulas.

Fig. 8
Fig. 8

Photograph of optical reconstruction of an in-line hologram calculated with divisionless recurrence formula. For an object with 966 point sources, calculation takes 16 min 45 s on the Alpha processor.

Equations (23)

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

OXp, Yq=j=0N-1ajrjXp, YqexpikrjXp, Yq,
rjXp, Yq=Xp-Xjo2+Yq-Yjo2+Zjo1/2.
rxn=xn2+d021/2, d0=Yp-Y02+Z021/2,
rxnd0+xn22d0.
rxn+1=rxn+δxrxn+δx22 rxn+=rxn+xn+δx2δxrxn-xn22rxn×δxrxn2+.
rxn+1rxn+xn+δx/2δxrxn.
rn+1=rn+sn/rn, sn+1=sn+c1.
r0d02+x021/2, s0x0+δx/2δx, c1δx2.
rxn=d0+xn22d0-xn48d03+.
EBA-δx48d03 n4.
k-12δx2xk2rxk3.
ERFn=0n kdk,
-δx6r03x0+nδx3-x03,
ERF-δx46d03 n3.
nlim=x03-6r03Elimδx1/3-x0δx-1
ξ=ξ0-fξ0fξ0=ξ02-χξ0.
pn+1=pn2-rn+1pn.
rn+1=rn+snpn, sn+1=sn+c1.
qn=rn/r0-1.
qn+1=qn+snpn, pn+1=pn2-qn+1pn+pn, sn+1=sn+c2.
q0=0,  p0=1,  s0=x0+δx/2δx/d02+x02, c2δx2/d02+x02.
Ttotal=τaMN,
IXp, Yq=ReOXp, Yq j=0N-1ajZjocos krjXp, Yq,

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