Abstract

The integral imaging (II) system combined with a single imaging lens has provided many advantages to improve system performance and to extend its application area. When a single imaging lens forms an image, a depth and size distortion is a necessary result of imaging by lens law. Therefore the II system, employing a single imaging lens, is also accompanied with similar distortions. In the II system, the depth and size of the reconstructed image can be adjusted by parallax image array (PIA) scaling. With regard to these scaling effects, the distortion correction method in the II system combined with a single imaging lens has been investigated. The differential transfer function (DTF) is defined and theoretically derived for mathematical analysis of the distortion. By using DTF, the PIA scaling conditions for the distortion correction in the II system combined with the single imaging lens is obtained. Experimental results are presented and discussed in detail.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
  2. E. Buckley, A. Cable, N. Lawrence, and T. Wilkinson, “Viewing angle enhancement for two- and three-dimensional holographic displays with random superresolution phase masks,” Appl. Opt. 45, 7334-7341 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J.-Y. Son, V. V. Saveljev, J.-S. Kim, S.-S. Kim, and B. Javidi, “Viewing zones in three-dimensional imaging systems based on lenticular, parallax-barrier, and microlens-array plates,” Appl. Opt. 43, 4985-4992 (2004).
    [CrossRef] [PubMed]
  5. J.-Y. Son and B. Javidi, “Three-dimensional imaging methods based on multiview images,” J. Display Technol. 1, 125-140(2005).
    [CrossRef]
  6. A. Schmidt and A. Grasnick, “Multi-viewpoint autostereoscopic displays from 4D-Vision,” Proc. SPIE 4660, 212-221(2002).
    [CrossRef]
  7. G. Lippmann, “La photographic integrale,” C. R. Acad. Sci. 146, 446-451 (1908).
  8. S. Yeom and B. Javidi, “Three-dimensional distortion-tolerant object recognition using integral imaging,” Opt. Express 12, 5795-5809 (2004).
    [CrossRef] [PubMed]
  9. Y. Frael, E. Tajahuerce, O. Matoba, A. Castro, and B. Javidi, “Comparison of passive ranging integral imaging and active imaging digital holography for three-dimensional object recognition,” Appl. Opt. 43, 452-462 (2004).
    [CrossRef]
  10. Y. Frael and B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41, 5488-5495 (2002).
    [CrossRef]
  11. S.-W. Min, J. Hong, and B. Lee, “Analysis of an optical depth converter used in a three-dimensional integral imaging system,” Appl. Opt. 43, 4539-4549 (2004).
    [CrossRef] [PubMed]
  12. O. Matoba and B. Javidi, “Three-dimensional polarimetric integral imaging,” Opt. Lett. 29, 2375-2377 (2004).
    [CrossRef] [PubMed]
  13. J.-S. Jang and B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express 12, 1077-1083 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. J. Arai, F. Okano, H. Hoshino, and I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034-2045(1998).
    [CrossRef]
  16. N. Davies, M. McCormick, and M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624-3633 (1994).
    [CrossRef]
  17. D.-H. Shin, B. Lee, and E.-S. Kim, “Multidirectional curved integral imaging with large depth by additional use of a large-aperture lens,” Appl. Opt. 45, 7375-7381 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  22. M. Okui, J. Arai, Y. Nojiri, and F. Okano, “Optical screen for direct projection of integral imaging,” Appl. Opt. 45, 9132-9139 (2006).
    [CrossRef] [PubMed]
  23. F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. 36, 1598-1603 (1997).
    [CrossRef] [PubMed]
  24. J. Kim, S.-W. Min, Y. Kim, B. Lee, “Analysis on viewing characteristics of an integral floating system,” Appl. Opt. 47, D80-D86 (2008).
    [CrossRef] [PubMed]

2008 (1)

2006 (3)

2005 (2)

2004 (8)

2002 (4)

2001 (1)

1998 (1)

1997 (1)

1994 (1)

N. Davies, M. McCormick, and M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624-3633 (1994).
[CrossRef]

1908 (1)

G. Lippmann, “La photographic integrale,” C. R. Acad. Sci. 146, 446-451 (1908).

Arai, J.

Brewin, M.

N. Davies, M. McCormick, and M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624-3633 (1994).
[CrossRef]

Buckley, E.

Cable, A.

Castro, A.

Davies, N.

N. Davies, M. McCormick, and M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624-3633 (1994).
[CrossRef]

Frael, Y.

Grasnick, A.

A. Schmidt and A. Grasnick, “Multi-viewpoint autostereoscopic displays from 4D-Vision,” Proc. SPIE 4660, 212-221(2002).
[CrossRef]

Hahn, M.

Harashima, H.

Hayasaki, Y.

Hong, J.

Hoshino, H.

Jang, J.-S.

Javidi, B.

J.-Y. Son and B. Javidi, “Three-dimensional imaging methods based on multiview images,” J. Display Technol. 1, 125-140(2005).
[CrossRef]

J.-S. Jang and B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express 12, 1077-1083 (2004).
[CrossRef] [PubMed]

O. Matoba and B. Javidi, “Three-dimensional polarimetric integral imaging,” Opt. Lett. 29, 2375-2377 (2004).
[CrossRef] [PubMed]

Y. Frael, E. Tajahuerce, O. Matoba, A. Castro, and B. Javidi, “Comparison of passive ranging integral imaging and active imaging digital holography for three-dimensional object recognition,” Appl. Opt. 43, 452-462 (2004).
[CrossRef]

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]

J.-Y. Son, V. V. Saveljev, J.-S. Kim, S.-S. Kim, and B. Javidi, “Viewing zones in three-dimensional imaging systems based on lenticular, parallax-barrier, and microlens-array plates,” Appl. Opt. 43, 4985-4992 (2004).
[CrossRef] [PubMed]

S. Yeom and B. Javidi, “Three-dimensional distortion-tolerant object recognition using integral imaging,” Opt. Express 12, 5795-5809 (2004).
[CrossRef] [PubMed]

Y. Frael and B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41, 5488-5495 (2002).
[CrossRef]

S.-H. Shin and B. Javidi, “Speckle-reduced three-dimensional volume holographic display by use of integral imaging,” Appl. Opt. 41, 2644-2649 (2002).
[CrossRef] [PubMed]

Jung, S.

Kim, E.-S.

Kim, J.

Kim, J.-S.

Kim, S.-S.

Kim, Y.

Kouno, M.

Lawrence, N.

Lee, B.

Lippmann, G.

G. Lippmann, “La photographic integrale,” C. R. Acad. Sci. 146, 446-451 (1908).

Matoba, O.

McCormick, M.

N. Davies, M. McCormick, and M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624-3633 (1994).
[CrossRef]

Min, S.-W.

Muguruma, S.

Naemura, T.

Nagai, Y.

Nishida, N.

Nojiri, Y.

Okano, F.

Okui, M.

Park, J.-H.

Saveljev, V. V.

Schmidt, A.

A. Schmidt and A. Grasnick, “Multi-viewpoint autostereoscopic displays from 4D-Vision,” Proc. SPIE 4660, 212-221(2002).
[CrossRef]

Shimizu, Y.

Shin, D.-H.

Shin, S.-H.

Son, J.-Y.

Tajahuerce, E.

Wilkinson, T.

Yamamoto, H.

Yeom, S.

Yoshida, T.

Yuyama, I.

Appl. Opt. (12)

F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. 36, 1598-1603 (1997).
[CrossRef] [PubMed]

J. Arai, F. Okano, H. Hoshino, and I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034-2045(1998).
[CrossRef]

S.-H. Shin and B. Javidi, “Speckle-reduced three-dimensional volume holographic display by use of integral imaging,” Appl. Opt. 41, 2644-2649 (2002).
[CrossRef] [PubMed]

Y. Frael and B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41, 5488-5495 (2002).
[CrossRef]

H. Yamamoto, M. Kouno, S. Muguruma, Y. Hayasaki, Y. Nagai, Y. Shimizu, and N. Nishida, “Enlargement of viewing area of stereoscopic full-color LED display by use of a parallax barrier,” Appl. Opt. 41, 6907-6919 (2002).
[CrossRef] [PubMed]

Y. Frael, E. Tajahuerce, O. Matoba, A. Castro, and B. Javidi, “Comparison of passive ranging integral imaging and active imaging digital holography for three-dimensional object recognition,” Appl. Opt. 43, 452-462 (2004).
[CrossRef]

S.-W. Min, J. Hong, and B. Lee, “Analysis of an optical depth converter used in a three-dimensional integral imaging system,” Appl. Opt. 43, 4539-4549 (2004).
[CrossRef] [PubMed]

J.-Y. Son, V. V. Saveljev, J.-S. Kim, S.-S. Kim, and B. Javidi, “Viewing zones in three-dimensional imaging systems based on lenticular, parallax-barrier, and microlens-array plates,” Appl. Opt. 43, 4985-4992 (2004).
[CrossRef] [PubMed]

E. Buckley, A. Cable, N. Lawrence, and T. Wilkinson, “Viewing angle enhancement for two- and three-dimensional holographic displays with random superresolution phase masks,” Appl. Opt. 45, 7334-7341 (2006).
[CrossRef] [PubMed]

D.-H. Shin, B. Lee, and E.-S. Kim, “Multidirectional curved integral imaging with large depth by additional use of a large-aperture lens,” Appl. Opt. 45, 7375-7381 (2006).
[CrossRef] [PubMed]

M. Okui, J. Arai, Y. Nojiri, and F. Okano, “Optical screen for direct projection of integral imaging,” Appl. Opt. 45, 9132-9139 (2006).
[CrossRef] [PubMed]

J. Kim, S.-W. Min, Y. Kim, B. Lee, “Analysis on viewing characteristics of an integral floating system,” Appl. Opt. 47, D80-D86 (2008).
[CrossRef] [PubMed]

C. R. Acad. Sci. (1)

G. Lippmann, “La photographic integrale,” C. R. Acad. Sci. 146, 446-451 (1908).

J. Display Technol. (1)

Opt. Eng. (1)

N. Davies, M. McCormick, and M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624-3633 (1994).
[CrossRef]

Opt. Express (4)

Opt. Express (1)

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (1)

A. Schmidt and A. Grasnick, “Multi-viewpoint autostereoscopic displays from 4D-Vision,” Proc. SPIE 4660, 212-221(2002).
[CrossRef]

Other (1)

B.Javidi and F.Okano, eds., Three-Dimensional Television, Video, and Display Technologies (Springer, 2002), pp. 101-123.

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Figures (18)

Fig. 1
Fig. 1

Definition of the position vectors of the corresponding points and displacements in an ideal imaging system.

Fig. 2
Fig. 2

Corresponding displacements for the x axis.

Fig. 3
Fig. 3

Corresponding displacements for the z axis.

Fig. 4
Fig. 4

Distortion caused by Ω 22 / Ω 11 .

Fig. 5
Fig. 5

Distortions of the image with Δ x / Δ z | Δ x > 0 (part A) and Δ z / Δ x | Δ z < 0 (part B).

Fig. 6
Fig. 6

Distortions of the image with d ( Ω 12 / Ω 22 ) / d x > 0 (part A), d ( Ω 12 / Ω 22 ) / d x < 0 (part B), d ( Ω 21 / Ω 11 ) / d z > 0 (part C), and d ( Ω 21 / Ω 11 ) / d z < 0 (part D).

Fig. 7
Fig. 7

Corresponding relation in the II system between (a) an object point and the image points on PIA plane in the pickup process and (b) the image points on the PIA plane and a reconstructed point in the reconstruction process.

Fig. 8
Fig. 8

Position vectors for object and image points in single imaging lens imaging.

Fig. 9
Fig. 9

Position vectors for the II system combined with the single imaging lens: (a) (system A) The single imaging lens is placed before the II system. (b) (system B) The single imaging lens is placed after the II system.

Fig. 10
Fig. 10

Coordinate for the PIA. ξ n represents the position of the nth image point of the PIA. The origin is located at the nth elemental lens’ center.

Fig. 11
Fig. 11

Complex imaging system 1 for the experiment. The single imaging lens is placed at a distance before the II system. The pickup lens array (lens array 1) and the reconstruction lens array (lens array 2) are relayed by the projection system.

Fig. 12
Fig. 12

Object used in the complex imaging system 1.

Fig. 13
Fig. 13

Experimental results with the complex imaging system 1: (a) the object, (b) the image with the single imaging lens, (c) reconstructed image with the complex imaging system before the correction, and (d) the reconstructed image after the correction. (a) and (b) are rearranged in a reverse order; and (b) is rotated 180 degrees for convenient comparison. Each column represents the images with 3 degree parallaxes.

Fig. 14
Fig. 14

Complex imaging system 2 for experiment. The single imaging lens is placed closely after the II system.

Fig. 15
Fig. 15

Object used in the complex imaging system 2.

Fig. 16
Fig. 16

Experimental results with the complex imaging system 2: reconstructed images (a) with the II system only, (b) with the complex imaging system before the correction, and (c) with the complex imaging system after the correction. All the images are rotated 180 degrees.

Fig. 17
Fig. 17

Complex imaging system 3 for the experiment. The single imaging lens is placed away from the lens array after the II system.

Fig. 18
Fig. 18

Experimental results with the complex imaging system 3: reconstructed images (a) with the complex imaging system before the correction and (b) with the complex imaging system after the correction.

Equations (59)

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x = f x ( ρ ) ,
z = f z ( ρ ) .
Ω ( ρ ) = ( f x ( ρ ) x f x ( ρ ) z f z ( ρ ) x f z ( ρ ) z ) ,
d ( Ω 12 / Ω 22 ) d x = ( 1 Ω 11 ) d ( Ω 12 / Ω 22 ) d x .
d ( Ω 21 / Ω 11 ) d z = ( 1 Ω 22 ) d ( Ω 21 / Ω 11 ) d z .
Ω = C ( 1 0 0 ± 1 ) ,
f ( ρ ) = h ( g ( ρ ) ) .
Ω f ( ρ ) = ( f x ( ρ ) x f x ( ρ ) z f z ( ρ ) x f z ( ρ ) z ) = ( h x ( g ( ρ ) ) x h x ( g ( ρ ) ) z h z ( g ( ρ ) ) x h z ( g ( ρ ) ) z ) .
Ω f ( ρ ) = ( h x ( g ( ρ ) ) g x ( ρ ) g x ( ρ ) x + h x ( g ( ρ ) ) g z ( ρ ) g z ( ρ ) x h x ( g ( ρ ) ) g x ( ρ ) g x ( ρ ) z + h x ( g ( ρ ) ) g z ( ρ ) g z ( ρ ) z h z ( g ( ρ ) ) g x ( ρ ) g x ( ρ ) x + h z ( g ( ρ ) ) g z ( ρ ) g z ( ρ ) x h z ( g ( ρ ) ) g x ( ρ ) g x ( ρ ) z + h z ( g ( ρ ) ) g z ( ρ ) g z ( ρ ) z ) = ( h x ( ρ ) x g x ( ρ ) x + h x ( ρ ) z g z ( ρ ) x h x ( ρ ) x g x ( ρ ) z + h x ( ρ ) z g z ( ρ ) z h z ( ρ ) x g x ( ρ ) x + h z ( ρ ) z g z ( ρ ) x h z ( ρ ) x g x ( ρ ) z + h z ( ρ ) z g z ( ρ ) z ) .
Ω f ( ρ ) = ( h x ( ρ ) x h x ( ρ ) z h z ( ρ ) x h z ( ρ ) z ) ( g x ( ρ ) x g x ( ρ ) z g z ( ρ ) x g z ( ρ ) z ) .
Ω f ( ρ ) = Ω h ( ρ ) Ω g ( ρ ) .
x en = ( 1 s 0 z 0 ) i = 0 n P i + x 0 s 0 z 0 ,
x en = ( 1 s I z I ) i = 0 n P i + x I s I z I ,
i = 0 n [ ( 1 s 0 z 0 ) P i ( 1 s I z I ) P i ] ( x I s I z I x 0 s 0 z 0 ) = 0.
( 1 s 0 z 0 ) P n ( 1 s I z I ) P n = 0 ,
x I s I z I = x 0 s 0 z 0 .
z I = s I ( s 0 z 0 1 ) P n P n + 1 .
x I = x 0 s 0 / z 0 ( s 0 z 0 1 ) P n P n + 1 .
II ( ρ ) = ( i · ρ ) s 0 / ( k · ρ ) ( s 0 ( k · ρ ) 1 ) P n P n + 1 i + s I ( s 0 ( k · ρ ) 1 ) P n P n + 1 k .
Ω II = ( 1 0 0 ± 1 ) .
1 f = 1 z L 1 z s .
z L = f z s + f z s ,
x L = z L z s x s = f z s + f x s .
L ( ρ ) = f z s + f x s i + f z s + f z s k .
Ω L = ( L x ( ρ ) x L x ( ρ ) z L z ( ρ ) x L z ( ρ ) z ) = ( f z s + f f x s ( z s + f ) 2 0 f 2 ( z s + f ) 2 ) .
Ω H A ( ρ ) = Ω II ( ρ ) Ω L ( ρ ) = ( f z s d z + f f ( x s d x ) ( z s d z + f ) 2 0 ± f 2 ( z s d z + f ) 2 ) ,
Ω H B ( ρ ) = Ω L ( ρ ) Ω II ( ρ ) = ( f z s d z + f f ( x s d x ) ( z s d z + f ) 2 0 ± f 2 ( z s d z + f ) 2 ) .
Ω H B ( ρ ) = ( f ± z 0 d z + f f ( x 0 d x ) ( ± z 0 d z + f ) 2 0 ± f 2 ( ± z 0 d z + f ) 2 ) .
x en = M x en + ϕ ( 1 M ) ,
i = 0 n [ ( 1 s 0 z 0 ) M P i ( 1 s I z I ) P i ] + ( M x 0 s 0 z 0 x I s I z I ) + ϕ ( 1 M ) = 0 ,
II ( ρ ) = M x 0 s 0 + ϕ ( 1 M ) z 0 z 0 ( 1 M ) + M s 0 i + s I z 0 z 0 ( 1 M ) + M s 0 k .
Ω II M = ( M s 0 z 0 ( 1 M ) + M s 0 ( 1 M ) M s 0 ( ϕ x 0 ) [ z 0 ( 1 M ) + M s 0 ] 2 0 s I s 0 M [ z 0 ( 1 M ) + M s 0 ] 2 ) .
x en = i = 0 n P i + ξ n .
x en = i = 0 n P i + M ξ n = i = 0 n P i + M s 0 z 0 ( x 0 i = 0 n P i ) .
II ( ρ ) = x 0 i + s I M s 0 z 0 k .
Ω Π M = ( 1 0 0 s I M s 0 ) ,
Ω II ( ρ ) = Ω II M ( ρ ) Ω II M ( ρ )
Ω II M ( ρ ) = ( 1 0 0 1 M ) .
Ω II ( ρ ) = ( 1 0 0 1 M ) ( s 0 M z 0 ( 1 M ) + M s 0 ( 1 M ) M s 0 ( ϕ x 0 ) [ z 0 ( 1 M ) + M s 0 ] 2 0 s I s 0 M [ z 0 ( 1 M ) + M s 0 ] 2 ) = ( s 0 M z 0 ( 1 M ) + M s 0 ( 1 M ) M s 0 ( ϕ x 0 ) [ z 0 ( 1 M ) + M s 0 ] 2 0 1 M s I s 0 M [ z 0 ( 1 M ) + M s 0 ] 2 ) .
Ω H A ( ρ ) = Ω II ( ρ ) Ω L ( ρ ) = ( s 0 M z 0 ( 1 M ) + M s 0 ( 1 M ) M s 0 ( ϕ x 0 ) [ z 0 ( 1 M ) + M s 0 ] 2 0 1 M s I s 0 M [ z 0 ( 1 M ) + M s 0 ] 2 ) ( f z s d z + f f ( x s d x ) ( z s d z + f ) 2 0 f 2 ( z s d z + f ) 2 ) ,
x 0 = x L = ( x s d x ) f z s d z + f + d x ,
z 0 = z L = ( z s d z ) f z s d z + f + d z .
Ω H A ( ρ ) = ( f M s 0 z s a b f M ( 1 M ) s 0 c ( z s a b ) 2 0 f 2 M s 0 s I M ( z s a b ) 2 ) .
a = ( 1 M ) ( f + d z ) + M s 0 ,
b = ( 1 M ) d z 2 M s 0 ( f d z ) ,
c = ϕ f ( x s d x ) ( d z + s 0 M 1 M ) f x s .
M = f + d z f + d z s 0 .
M = ± s I s 0 f ( f + d z s 0 ) .
α = | f + d z f | .
Ω H B ( ρ ) = Ω L ( ρ ) Ω II ( ρ ) .
x s = x I = M x 0 s 0 + ϕ ( 1 M ) z 0 z 0 ( 1 M ) + M s 0 ,
z s = z I = s I z 0 M [ z 0 ( 1 M ) + M s 0 ] .
Ω H B ( ρ ) = ( f M M s 0 z 0 a + b f M M s 0 c ( z 0 a + b ) 2 0 f 2 M M s 0 s I ( z 0 a + b ) 2 ) ,
a = M ( 1 M ) ( f d z ) + s I ,
b = M M s 0 ( f d z ) ,
c = ϕ ( 1 M ) M ( f d z ) ( 1 M ) M ( f d z ) x 0 + s I ( d x x 0 ) .
M = f f s 0 ,
M = ± s I ( f s 0 ) s 0 ( f d z ) .
α = | f f d z | .

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