Abstract

I investigate the optical transfer function of three-dimensional display systems. Moreover, I obtain an average sampled modulation transfer function describing discrete, sampled display systems and show that in the proper limit of geometrical optics it is equivalent to the shift-invariant optical transfer function. I apply the theory to describe holographic stereograms and discuss the effects of amplitude and phase filters on the optical resolution.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Lippmann, "Epreuves reversibles," C. R. Acad. Sci. 146, 446-451 (1908).
  2. H. E. Ives, "Optical properties of a Lippmann lenticulated sheet," J. Opt. Soc. Am. 21, 171-176 (1931).
    [CrossRef]
  3. S. Pastoor and M. Wöpking, "3-D displays: a review of current technologies," Displays 17, 100-110 (1996).
    [CrossRef]
  4. M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, "Phase-added stereogram: calculation of hologram using computer graphics technique," Proc. SPIE 1914, 25-31 (1993).
    [CrossRef]
  5. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  6. L. E. Helseth and I. Singstad, "Diffusers for holographic stereography," Opt. Commun. 193, 81-86 (2001).
    [CrossRef]
  7. B. Lee, S. W. Min, and B. Javidi, "Theoretical analysis for three-dimensional integral imaging systems with double devices," Appl. Opt. 41, 4856-4865 (2002).
    [CrossRef] [PubMed]
  8. J. S. Jang and B. Javidi, "Improvement of viewing angle in integral imaging by use of moving lenslet arrays with low fill factor," Appl. Opt. 42, 1996-2002 (2003).
    [CrossRef] [PubMed]
  9. J. S. Jang and B. Javidi, "Depth and lateral size control of three-dimensional images in projection integral imaging," Opt. Express 12, 3778-3790 (2004).
    [CrossRef] [PubMed]
  10. T. Kawai, "3D displays and applications," Displays 23, 49-56 (2002).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  12. P. St. Hilaire, "Modulation transfer function and optimum sampling of holographic stereograms," Appl. Opt. 33, 768-774 (1994).
    [CrossRef] [PubMed]
  13. H. Hoshino, F. Okano, H. Isono, and Y. Yuyama, "Analysis of resolution limitation of integral photography," J. Opt. Soc. Am. A 15, 2059-2065 (1998).
    [CrossRef]
  14. J. Arai, H. Hoshino, M. Okui, and F. Okano, "Effects of focusing on the resolution characteristics of integral photography," J. Opt. Soc. Am. A 20, 996-1004 (2003).
    [CrossRef]
  15. S. A. Benton, "The principles of reflection holographic stereograms," in Proceedings of the Third International Symposium on Display Holography (Lake Forest College, Lake Forest, Illinois1988), pp. 593-608.
  16. W. S. Stiles and B. H. Crawford, "The luminous efficiency of rays entering the eye pupil at different points," Proc. R. Soc. London, Ser. B 112, 428-450 (1933).
    [CrossRef]
  17. D. A. Atchison, A. Joblin, and G. Smith, "Influence of Stiles-Crawford effect apodization on spatial visual performance," J. Opt. Soc. Am. A 15, 2545-2551 (1998).
    [CrossRef]
  18. X. Zhang, M. Ye, A. Bradley, and L. Thibos, "Apodization by the Stiles-Crawford effect moderates the visual impact of retinal image defocus," J. Opt. Soc. Am. A 16, 812-820 (1999).
    [CrossRef]

2004

2003

2002

2001

L. E. Helseth and I. Singstad, "Diffusers for holographic stereography," Opt. Commun. 193, 81-86 (2001).
[CrossRef]

1999

1998

1997

1996

S. Pastoor and M. Wöpking, "3-D displays: a review of current technologies," Displays 17, 100-110 (1996).
[CrossRef]

1994

1993

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, "Phase-added stereogram: calculation of hologram using computer graphics technique," Proc. SPIE 1914, 25-31 (1993).
[CrossRef]

1933

W. S. Stiles and B. H. Crawford, "The luminous efficiency of rays entering the eye pupil at different points," Proc. R. Soc. London, Ser. B 112, 428-450 (1933).
[CrossRef]

1931

1908

G. Lippmann, "Epreuves reversibles," C. R. Acad. Sci. 146, 446-451 (1908).

Arai, J.

Atchison, D. A.

Benton, S. A.

S. A. Benton, "The principles of reflection holographic stereograms," in Proceedings of the Third International Symposium on Display Holography (Lake Forest College, Lake Forest, Illinois1988), pp. 593-608.

Bradley, A.

Crawford, B. H.

W. S. Stiles and B. H. Crawford, "The luminous efficiency of rays entering the eye pupil at different points," Proc. R. Soc. London, Ser. B 112, 428-450 (1933).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Helseth, L. E.

L. E. Helseth and I. Singstad, "Diffusers for holographic stereography," Opt. Commun. 193, 81-86 (2001).
[CrossRef]

Hilaire, P. St.

Honda, T.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, "Phase-added stereogram: calculation of hologram using computer graphics technique," Proc. SPIE 1914, 25-31 (1993).
[CrossRef]

Hoshino, H.

Isono, H.

Ives, H. E.

Jang, J. S.

Javidi, B.

Joblin, A.

Kawai, T.

T. Kawai, "3D displays and applications," Displays 23, 49-56 (2002).
[CrossRef]

Lee, B.

Lippmann, G.

G. Lippmann, "Epreuves reversibles," C. R. Acad. Sci. 146, 446-451 (1908).

Min, S. W.

Ohyama, N.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, "Phase-added stereogram: calculation of hologram using computer graphics technique," Proc. SPIE 1914, 25-31 (1993).
[CrossRef]

Okano, F.

Okui, M.

Pastoor, S.

S. Pastoor and M. Wöpking, "3-D displays: a review of current technologies," Displays 17, 100-110 (1996).
[CrossRef]

Singstad, I.

L. E. Helseth and I. Singstad, "Diffusers for holographic stereography," Opt. Commun. 193, 81-86 (2001).
[CrossRef]

Smith, G.

Stiles, W. S.

W. S. Stiles and B. H. Crawford, "The luminous efficiency of rays entering the eye pupil at different points," Proc. R. Soc. London, Ser. B 112, 428-450 (1933).
[CrossRef]

Thibos, L.

Wöpking, M.

S. Pastoor and M. Wöpking, "3-D displays: a review of current technologies," Displays 17, 100-110 (1996).
[CrossRef]

Yamaguchi, I.

Yamaguchi, M.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, "Phase-added stereogram: calculation of hologram using computer graphics technique," Proc. SPIE 1914, 25-31 (1993).
[CrossRef]

Ye, M.

Yuyama, Y.

Zhang, T.

Zhang, X.

Appl. Opt.

C. R. Acad. Sci.

G. Lippmann, "Epreuves reversibles," C. R. Acad. Sci. 146, 446-451 (1908).

Displays

S. Pastoor and M. Wöpking, "3-D displays: a review of current technologies," Displays 17, 100-110 (1996).
[CrossRef]

T. Kawai, "3D displays and applications," Displays 23, 49-56 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

L. E. Helseth and I. Singstad, "Diffusers for holographic stereography," Opt. Commun. 193, 81-86 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. R. Soc. London, Ser. B

W. S. Stiles and B. H. Crawford, "The luminous efficiency of rays entering the eye pupil at different points," Proc. R. Soc. London, Ser. B 112, 428-450 (1933).
[CrossRef]

Proc. SPIE

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, "Phase-added stereogram: calculation of hologram using computer graphics technique," Proc. SPIE 1914, 25-31 (1993).
[CrossRef]

Other

S. A. Benton, "The principles of reflection holographic stereograms," in Proceedings of the Third International Symposium on Display Holography (Lake Forest College, Lake Forest, Illinois1988), pp. 593-608.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Single pixel of width d moves a distance x s between two perspective pictures, and thus appears to be located at the apparent plane (dashed line).

Fig. 2
Fig. 2

Radius of curvature of images located at the projection plane and the apparent plane (dashed vertical line) have different curvatures. Thus a viewer trying to locate a point in the image to the apparent plane ( x 1 , z a ) will experience a defocused image due to the error in curvature.

Fig. 3
Fig. 3

Dashed curves show the OTF for a rectangular slit when s = 1 mm (thick dashed curve) and s = 3 mm (thin dashed curve). The solid curves show the OTF when s = 1 mm (thick solid curve) and s = 3 mm (thin solid curve). Here we have used z p = 0.3 m , z a = 0.4 m , and λ = 550 nm .

Fig. 4
Fig. 4

SMTF a for d = 0 (solid curve) and d = 100 μ m (dashed curve). Here we have used z p = 0.3 m , z a = 0.31 m , s = 3 mm , and λ = 550 nm .

Equations (28)

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

x s = z p z a z a s ,
R p = ( x x 1 ) 2 + z p 2 ,
R a = ( x x 1 ) 2 + z a 2 .
R p = z p + ( x x 1 ) 2 2 z p ,
R a = z a + ( x x 1 ) 2 2 z a .
R a R p = ( z a z p ) + ( z p z a ) ( x x 1 ) 2 2 z p z a .
P ( x ) = A ( x ) exp ( i k z a z p 2 z p z a x 2 ) ,
OTF ( f x ) = P ( x λ z p f x 2 ) P * ( x + λ z p f x 2 ) d x P ( x ) 2 d x ,
P ( x ) = rect ( x s ) exp ( i k z p z a 2 z p z a x 2 ) ,
OTF s ( f x ) = ( 1 λ z p f x s ) sin [ π z p z a z a s f x ( 1 λ z p f x s ) ] [ π z p z a z a s f x ( 1 λ z p f x s ) ] .
OTF s ( f x ) = sinc ( z p z a z a s f x ) ,
sinc ( u ) = sin ( π u ) π u .
P ( x ) = rect ( x s ) exp [ i ϕ ( x ) ] exp ( i k z a z p 2 z p z a x 2 ) .
ϕ ( x ) = k z b z p 2 z p z b x 2 ,
P ( x ) = rect ( x s ) exp ( i k z a z b 2 z b z a x 2 ) .
P ( x ) = exp ( α x 2 ) exp ( i k z a z p 2 z p z a x 2 ) .
OTF G ( f x ) = exp { α ( λ z p f x 2 ) 2 [ 1 + 2 ( π λ α ) 2 ( z a z p z p z a ) 2 ] } .
OTF G ( f x ) = exp [ 1 8 ( π s f x z a z p z a ) 2 ] .
POTF ( f x ) = H ( λ ) OTF ( f x , λ ) d λ H ( λ ) d λ ,
E ( λ ) = exp [ γ ( λ λ 0 ) 2 ] ,
POTF ( f x ) = γ γ + ( z p f x s ) 2 exp [ γ λ 0 2 ( z p f x s ) 2 γ + ( z p f x s ) 2 ] exp [ 1 8 ( π s f x z a z p z a ) 2 ] .
I ¯ max = 1 d Δ x 1 + d 2 Δ x 1 d 2 [ 1 + cos ( 2 π f x x ) ] d x = 1 + sinc ( f x d ) cos ( 2 π f x Δ x 1 ) .
I ¯ min = 1 d Δ x 2 + d 2 Δ x 2 d 2 { 1 + cos [ 2 π f x ( x + 1 2 f x ) ] } d x = 1 sinc ( f x d ) cos ( 2 π f x Δ x 2 ) .
SMTF ( f x ) = I ¯ max I ¯ min I ¯ max + I ¯ min = sinc ( f x d ) [ cos ( 2 π f x Δ x 2 ) + cos ( 2 π f x Δ x 2 ) ] .
I ¯ max = 1 x s x s 2 x s 2 I ¯ max d Δ x 1 = 1 + sinc ( f x d ) sinc ( f x x s ) ,
I ¯ min = 1 x s x s 2 x s 2 I ¯ min d Δ x 2 = 1 sinc ( f x d ) sinc ( f x x s ) .
SMTF a ( f x ) = I ¯ max I ¯ min I ¯ max + I ¯ min = sinc ( f x d ) sinc ( f x x s ) .
SMTF a ( f x ) = sinc ( f x d ) sinc ( z a z p z a s f x ) .

Metrics