Abstract

A composite hologram to reconstruct a three-dimensional object is synthesized with a computer using the technique of holographic stereogram. In the first step of the method, a sequence of perspective projections of the three-dimensional object is calculated, and their Fourier transform holograms are also generated with a computer. These holograms are assembled into a final hologram in the second step. This method requires a computation time far less than the conventional method of producing a computer-generated hologram and makes it possible to reconstruct an image with a wide view angle. Some characteristics of the reconstructed image are described, and optimum parameters for the synthesis and reconstruction process are discussed. Finally, an experimental example is presented.

© 1976 Optical Society of America

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References

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  1. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  2. J. P. Waters, Appl. Opt. 9, 405 (1966).
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    [CrossRef]
  4. Y. Ichioka, M. Izumi, T. Suzuki, Appl. Opt. 10, 403 (1971).
    [CrossRef] [PubMed]
  5. M. C. King, A. M. Noll, D. H. Berry, Appl. Opt. 9, 471 (1970).
    [CrossRef] [PubMed]
  6. T. Yatagai, Opt. Commun. 12, 43 (1974).
    [CrossRef]
  7. J. T. McCrickerd, N. George, Appl. Phys. Lett. 12, 10 (1968).
    [CrossRef]
  8. T. Kasahara, Y. Kimura, M. Kawai, Applications of Holography, E. S. Barrekette et al., Eds. (Plenum, New York, 1971), p. 178.
  9. J. C. Gray, in Proc. 22nd ACM National Conference (1967), p. 355.
  10. P. P. Loutrel, IEEE Trans. Comput. C-19, 205 (1970).
    [CrossRef]
  11. M. Idesawa, Bull. JSME 16, 216 (1973).
    [CrossRef]
  12. J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
    [CrossRef]
  13. E. N. Leith, D. B. Brumm, S. S. H. Hsiao, Appl. Opt. 11, 2016 (1972).
    [CrossRef] [PubMed]
  14. L. Rosen, Appl. Phys. Lett. 9, 337 (1966).
    [CrossRef]
  15. J. D. Redman, J. Phys. E 1, 821 (1968).
    [CrossRef]
  16. R. K. Mueller, Proc. IEEE 59, 1319 (1971).
    [CrossRef]
  17. G. G. Goetz, Appl. Phys. Lett. 17, 63 (1970).
    [CrossRef]
  18. D. Casasent, W. M. Sterling, IEEE Trans. Comput. C-24, 348 (1975).
    [CrossRef]

1975 (1)

D. Casasent, W. M. Sterling, IEEE Trans. Comput. C-24, 348 (1975).
[CrossRef]

1974 (1)

T. Yatagai, Opt. Commun. 12, 43 (1974).
[CrossRef]

1973 (1)

M. Idesawa, Bull. JSME 16, 216 (1973).
[CrossRef]

1972 (1)

1971 (2)

1970 (3)

M. C. King, A. M. Noll, D. H. Berry, Appl. Opt. 9, 471 (1970).
[CrossRef] [PubMed]

P. P. Loutrel, IEEE Trans. Comput. C-19, 205 (1970).
[CrossRef]

G. G. Goetz, Appl. Phys. Lett. 17, 63 (1970).
[CrossRef]

1968 (3)

J. D. Redman, J. Phys. E 1, 821 (1968).
[CrossRef]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 611 (1968).
[CrossRef]

J. T. McCrickerd, N. George, Appl. Phys. Lett. 12, 10 (1968).
[CrossRef]

1967 (1)

1966 (2)

J. P. Waters, Appl. Opt. 9, 405 (1966).

L. Rosen, Appl. Phys. Lett. 9, 337 (1966).
[CrossRef]

1965 (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Berry, D. H.

Brumm, D. B.

Casasent, D.

D. Casasent, W. M. Sterling, IEEE Trans. Comput. C-24, 348 (1975).
[CrossRef]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

George, N.

J. T. McCrickerd, N. George, Appl. Phys. Lett. 12, 10 (1968).
[CrossRef]

Goetz, G. G.

G. G. Goetz, Appl. Phys. Lett. 17, 63 (1970).
[CrossRef]

Gray, J. C.

J. C. Gray, in Proc. 22nd ACM National Conference (1967), p. 355.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 611 (1968).
[CrossRef]

Hsiao, S. S. H.

Ichioka, Y.

Idesawa, M.

M. Idesawa, Bull. JSME 16, 216 (1973).
[CrossRef]

Izumi, M.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 611 (1968).
[CrossRef]

Kasahara, T.

T. Kasahara, Y. Kimura, M. Kawai, Applications of Holography, E. S. Barrekette et al., Eds. (Plenum, New York, 1971), p. 178.

Kawai, M.

T. Kasahara, Y. Kimura, M. Kawai, Applications of Holography, E. S. Barrekette et al., Eds. (Plenum, New York, 1971), p. 178.

Kimura, Y.

T. Kasahara, Y. Kimura, M. Kawai, Applications of Holography, E. S. Barrekette et al., Eds. (Plenum, New York, 1971), p. 178.

King, M. C.

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 611 (1968).
[CrossRef]

Lohmann, A. W.

Loutrel, P. P.

P. P. Loutrel, IEEE Trans. Comput. C-19, 205 (1970).
[CrossRef]

McCrickerd, J. T.

J. T. McCrickerd, N. George, Appl. Phys. Lett. 12, 10 (1968).
[CrossRef]

Mueller, R. K.

R. K. Mueller, Proc. IEEE 59, 1319 (1971).
[CrossRef]

Noll, A. M.

Paris, D. P.

Redman, J. D.

J. D. Redman, J. Phys. E 1, 821 (1968).
[CrossRef]

Rosen, L.

L. Rosen, Appl. Phys. Lett. 9, 337 (1966).
[CrossRef]

Sterling, W. M.

D. Casasent, W. M. Sterling, IEEE Trans. Comput. C-24, 348 (1975).
[CrossRef]

Suzuki, T.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Waters, J. P.

Yatagai, T.

T. Yatagai, Opt. Commun. 12, 43 (1974).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (3)

L. Rosen, Appl. Phys. Lett. 9, 337 (1966).
[CrossRef]

G. G. Goetz, Appl. Phys. Lett. 17, 63 (1970).
[CrossRef]

J. T. McCrickerd, N. George, Appl. Phys. Lett. 12, 10 (1968).
[CrossRef]

Bull. JSME (1)

M. Idesawa, Bull. JSME 16, 216 (1973).
[CrossRef]

Commun. ACM (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 611 (1968).
[CrossRef]

IEEE Trans. Comput. (2)

D. Casasent, W. M. Sterling, IEEE Trans. Comput. C-24, 348 (1975).
[CrossRef]

P. P. Loutrel, IEEE Trans. Comput. C-19, 205 (1970).
[CrossRef]

J. Phys. E (1)

J. D. Redman, J. Phys. E 1, 821 (1968).
[CrossRef]

Math. Comp. (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Opt. Commun. (1)

T. Yatagai, Opt. Commun. 12, 43 (1974).
[CrossRef]

Proc. IEEE (1)

R. K. Mueller, Proc. IEEE 59, 1319 (1971).
[CrossRef]

Other (2)

T. Kasahara, Y. Kimura, M. Kawai, Applications of Holography, E. S. Barrekette et al., Eds. (Plenum, New York, 1971), p. 178.

J. C. Gray, in Proc. 22nd ACM National Conference (1967), p. 355.

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Figures (12)

Fig. 1
Fig. 1

Geometry for calculating the perspective projections of a three-dimensional object viewed from a viewpoint.

Fig. 2
Fig. 2

Optical setup for reconstructing the images from the composite hologram by a point source.

Fig. 3
Fig. 3

Diagram for calculating the image size.

Fig. 4
Fig. 4

Diagram for calculating the flipping of the image.

Fig. 5
Fig. 5

No flipping area described by Eq. (13) in the case when the size D of the elementary hologram is 4 mm, and the diameter α of the smallest pupil in the observing system is 2 mm.

Fig. 6
Fig. 6

Depth of the image vs the size of the elementary hologram: (a) and (b) correspond to the pupil size a = 2 mm and a = 4 mm, respectively.

Fig. 7
Fig. 7

Ratio of the image size h and depth ΔZ vs the distance Z from the hologram plane to the image plane for the case a = 2 mm as a function of the number N of the sample points.

Fig. 8
Fig. 8

Computation time ratio R′ vs the number of the sample points described by Eq. (19).

Fig. 9
Fig. 9

Optical arrangement for observing a three-dimensional image by which the size and the position of the image can be controlled.

Fig. 10
Fig. 10

Elementary hologram whose original size is 20 cm × 20 cm.

Fig. 11
Fig. 11

A typical composite hologram, the size of which is 11.8 cm × 2.3 cm.

Fig. 12
Fig. 12

Reconstructed images from the composite hologram shown in Fig. 11. These images (a)–(e) of a two-storied temple correspond to the angle ϕ of the viewpoint 41°, 27°, −3°, −15°, and −35°, respectively. The depression angle θ is equally 20° in these projections.

Equations (19)

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r = M 1 M 2 r ,
M 1 = ( 1 0 0 0 cos θ sin θ 0 - sin θ cos θ )
M 2 = ( cos ϕ 0 - sin ϕ 0 1 0 sin ϕ 0 cos ϕ )
u = Z v x / ( Z 0 - z )
v = Z v y / ( Z 0 - z ) .
ν = sin γ / λ = γ / λ ,
W = 2 h / 2 λ Z = h / λ Z ,
ν c = N / D ,
h = λ Z N / D .
s = Z d D / ( Z + Z d ) .
s β ,
β = 1.22 λ Z / α .
- 1.22 λ Z 2 D α + 1.22 λ Z Z d 1.22 λ Z 2 D α - 1.22 λ Z
Δ Z = 1.22 λ Z 2 D α - 1.22 λ Z + 1.22 λ Z 2 D α + 1.22 λ Z 2.44 λ Z 2 / D α .
Δ Z = 2.44 λ Z 2 / D a             when a D
Δ Z = 2.44 λ Z 2 / D 2             when a > D .
h / Δ Z = N a / 2.44 Z
R = log 2 N / ( log 2 M + log 2 N ) .
R = R / M .

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