Abstract

A closed-form solution of describing perceived image in stereoscopic imaging systems with radial recording and projecting geometry for arbitrary viewer positions is presented. This solution is derived by finding a condition for making the heights of homolog points in both left and right images projected on the screen in the geometry equal. The solution has the same equation form as that of the parallel geometry except that it has a constant shifting term in the horizontal direction. This term is a main source for distortions in the perceived image. The condition of eliminating the term makes the solution the same as that for stereoscopic imaging systems with parallel recording and projecting geometry.

© 2007 Optical Society of America

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References

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  1. I.Takehiro, ed., Fundamentals of 3D Images, (NHK Broadcasting Technology Research Center, 1995), pp. 72-73.
  2. J. T. Rule, "The shape of stereoscopic images," J. Opt. Soc. Am. 31, 124-129 (1941).
    [Crossref]
  3. B. G. Saunders, "Stereoscopic drawing by computer-is it orthoscopic?" Appl. Opt. 7, 1459-1504 (1968).
    [Crossref]
  4. L. F. Hodge, "Time-multiplexed stereoscopic computer graphics," IEEE Comput. Graphics Appl. 12, 20-30 (1992).
    [Crossref]
  5. H. Yamanoue, "The relation between size distortion and shooting conditions for stereoscopic images," J. SMPTE 106, 225-232 (1997).
    [Crossref]
  6. A. Woods, T. Docherty, and R. Koch, "Image distortions in stereoscopic video systems," in Proc. SPIE 1915, 36-48 (1993).
    [Crossref]
  7. D. L. MacAdam, "Stereoscopic perception of size, shape, distance and direction," J. SMPTE 62, 271-293 (1954).
    [Crossref]
  8. H. Dewhurst, "Photography, viewing and projection of pictures with stereoscopic effect," U.S. patent 2,693,128 (November 2, 1954).
  9. J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
    [Crossref]
  10. L. Lipton, Foundations of the Stereoscopic Cinema, A Study in Depth (Van Nostrand Reinhold, 1982).

2002 (1)

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

1997 (1)

H. Yamanoue, "The relation between size distortion and shooting conditions for stereoscopic images," J. SMPTE 106, 225-232 (1997).
[Crossref]

1993 (1)

A. Woods, T. Docherty, and R. Koch, "Image distortions in stereoscopic video systems," in Proc. SPIE 1915, 36-48 (1993).
[Crossref]

1992 (1)

L. F. Hodge, "Time-multiplexed stereoscopic computer graphics," IEEE Comput. Graphics Appl. 12, 20-30 (1992).
[Crossref]

1968 (1)

B. G. Saunders, "Stereoscopic drawing by computer-is it orthoscopic?" Appl. Opt. 7, 1459-1504 (1968).
[Crossref]

1954 (1)

D. L. MacAdam, "Stereoscopic perception of size, shape, distance and direction," J. SMPTE 62, 271-293 (1954).
[Crossref]

1941 (1)

Bahn, J.-E.

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Bobrinev, V. I.

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Choi, Y.-J.

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Chun, J.-H.

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Dewhurst, H.

H. Dewhurst, "Photography, viewing and projection of pictures with stereoscopic effect," U.S. patent 2,693,128 (November 2, 1954).

Docherty, T.

A. Woods, T. Docherty, and R. Koch, "Image distortions in stereoscopic video systems," in Proc. SPIE 1915, 36-48 (1993).
[Crossref]

Gruts, Y.

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Hodge, L. F.

L. F. Hodge, "Time-multiplexed stereoscopic computer graphics," IEEE Comput. Graphics Appl. 12, 20-30 (1992).
[Crossref]

Koch, R.

A. Woods, T. Docherty, and R. Koch, "Image distortions in stereoscopic video systems," in Proc. SPIE 1915, 36-48 (1993).
[Crossref]

Lipton, L.

L. Lipton, Foundations of the Stereoscopic Cinema, A Study in Depth (Van Nostrand Reinhold, 1982).

MacAdam, D. L.

D. L. MacAdam, "Stereoscopic perception of size, shape, distance and direction," J. SMPTE 62, 271-293 (1954).
[Crossref]

Rule, J. T.

Saunders, B. G.

B. G. Saunders, "Stereoscopic drawing by computer-is it orthoscopic?" Appl. Opt. 7, 1459-1504 (1968).
[Crossref]

Son, J.-Y.

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Woods, A.

A. Woods, T. Docherty, and R. Koch, "Image distortions in stereoscopic video systems," in Proc. SPIE 1915, 36-48 (1993).
[Crossref]

Yamanoue, H.

H. Yamanoue, "The relation between size distortion and shooting conditions for stereoscopic images," J. SMPTE 106, 225-232 (1997).
[Crossref]

Appl. Opt. (1)

B. G. Saunders, "Stereoscopic drawing by computer-is it orthoscopic?" Appl. Opt. 7, 1459-1504 (1968).
[Crossref]

IEEE Comput. Graphics Appl. (1)

L. F. Hodge, "Time-multiplexed stereoscopic computer graphics," IEEE Comput. Graphics Appl. 12, 20-30 (1992).
[Crossref]

J. Opt. Soc. Am. (1)

J. SMPTE (2)

D. L. MacAdam, "Stereoscopic perception of size, shape, distance and direction," J. SMPTE 62, 271-293 (1954).
[Crossref]

H. Yamanoue, "The relation between size distortion and shooting conditions for stereoscopic images," J. SMPTE 106, 225-232 (1997).
[Crossref]

Opt. Eng. (Bellingham) (1)

J.-Y. Son, Y. Gruts, J.-H. Chun, Y.-J. Choi, J.-E. Bahn, and V. I. Bobrinev, "Distortion analysis in stereoscopic images," Opt. Eng. (Bellingham) 41, 680-685 (2002).
[Crossref]

Proc. SPIE (1)

A. Woods, T. Docherty, and R. Koch, "Image distortions in stereoscopic video systems," in Proc. SPIE 1915, 36-48 (1993).
[Crossref]

Other (3)

H. Dewhurst, "Photography, viewing and projection of pictures with stereoscopic effect," U.S. patent 2,693,128 (November 2, 1954).

I.Takehiro, ed., Fundamentals of 3D Images, (NHK Broadcasting Technology Research Center, 1995), pp. 72-73.

L. Lipton, Foundations of the Stereoscopic Cinema, A Study in Depth (Van Nostrand Reinhold, 1982).

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Figures (3)

Fig. 1
Fig. 1

Three-dimensional view of a stereoimage pair recording geometry for both parallel- and radial-type layouts.

Fig. 2
Fig. 2

Top view of the Fig. 1 geometry.

Fig. 3
Fig. 3

Image projection geometry in the x z plane.

Equations (68)

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x C = x P = l 2 ,
y C = y P = h 2 .
x C = l 2 , x P = m ( l 2 ) ,
y C = h 2 , y P = m ( h 2 ) .
s I = A C V I ( z C Z I ) ,
s I = [ x C L i x C R i y C L R i ] , V I = [ X I Y I Z I ] , A C = [ z C 0 c l 2 z C 0 c l 2 0 z C h 2 ]
x C R = X I z C Z I ( l 2 + c ) z C Z I ,
x C L i = X I z C Z I ( l 2 c ) z C Z I ,
y C L R i = Y I z C Z I ( h 2 ) z C Z I .
x C L i l 2 , α R = α A C R C L , α L = A C L C R α ,
A C R C L = D A C R = tan 1 ( z C D A ¯ ) , D A ¯ = A ¯ F ¯ + c = ( l 2 + c ) x C R i ,
A C L C R = 180 ° D A C L = 180 ° tan 1 ( z C D A ¯ ) , D A ¯ = A ¯ F ¯ c = ( l 2 c ) x C L i for x C L i ( l 2 c ) ,
A C L C R = D A C L = tan 1 ( z C D A ¯ ) , D A ¯ = c A ¯ F ¯ = { ( l 2 c ) x C L i } for ( l 2 c ) < x C L i l 2 ,
x C L i > l 2 , α R = A C R C L α , α L = α A C L C R ,
A C R C L = D A C R = tan 1 ( z C D A ¯ ) , D A ¯ = c A ¯ F ¯ = ( l 2 + c ) x C R i for l 2 < x C R i l 2 + c ,
A C R C L = 180 ° D A C R = 180 ° tan 1 ( z C D A ¯ ) , D A ¯ = A ¯ F ¯ c = { ( l 2 + c ) x C R i } for l 2 + c < x C R i ,
A C L C R = D A C L = tan 1 ( z C D A ¯ ) , D A ¯ = c + A ¯ F ¯ = { ( l 2 c ) x C L i } ,
α R = tan 1 ( z C c ) tan 1 [ z C Z I l 2 + c X I ] ,
α L = tan 1 ( z C c ) + tan 1 [ z C Z I ( l 2 c ) X I ] .
B ¯ F ¯ = F C ¯ R tan α R = c 2 + z C 2 tan { tan 1 ( z C c ) tan 1 ( z C Z I l 2 + c X I ) } ,
B ¯ F ¯ = F C ¯ L tan α L = c 2 + z C 2 tan { tan 1 ( z C c ) + tan 1 ( z C Z I l 2 c X I ) } ,
B ¯ F ¯ = F C ¯ R tan α R = c 2 + z C 2 z C ( l 2 X I ) + c Z I c ( l 2 + c X I ) + z C ( z C Z I ) ,
B ¯ F ¯ = F C ¯ L tan α L = c 2 + z C 2 z C ( l 2 X I ) c Z I c ( l 2 c X I ) z C ( z C Z I ) ,
y C R i = h 2 ( h 2 y C L R i ) z C B ¯ F ¯ sin φ C z C = y C L R i + ( h 2 y C L R i ) B ¯ F ¯ sin φ C z C ,
y C L i = h 2 ( h 2 y C L R i ) z C + B ¯ F ¯ sin φ C z C = y C L R i ( h 2 y C L R i ) B ¯ F ¯ sin φ C z C .
β R = tan 1 m B ¯ F ¯ p 2 + z P 2 = tan 1 ( m c 2 + z C 2 p 2 + z P 2 z C ( l 2 X I ) + c Z I c ( c + l 2 X I ) + z C ( z C Z I ) ) ,
β L = tan 1 m B ¯ F ¯ p 2 + z P 2 = tan 1 ( m c 2 + z C 2 p 2 + z P 2 z C ( l 2 X I ) c Z I c ( l 2 c X I ) z C ( z C Z I ) ) .
S F ¯ = m B ¯ F ¯ cos β R cos ( β R + φ P ) = m c 2 + z C 2 z C ( l 2 X I ) + c Z I c ( c + l 2 X I ) + z C ( z C Z I ) cos β R cos ( β R + φ P ) ,
S ¯ F ¯ = m B ¯ F ¯ cos β L cos ( β L φ P ) = m c 2 + z C 2 z C ( l 2 X I ) c Z I c ( l 2 c X I ) z C ( z C Z I ) cos β L cos ( β L φ P ) ,
S F ¯ = m c 2 + z C 2 p 2 + z P 2 z C ( l 2 X I ) + c Z I p 2 + z P 2 c ( c + l 2 X I ) + z C ( z C Z I ) cos φ P m c 2 + z C 2 z C ( l 2 X I ) + c Z I sin φ P ,
S ¯ F ¯ = m c 2 + z C 2 p 2 + z P 2 z C ( l 2 X I ) c Z I p 2 + z P 2 c ( l 2 c X I ) z C ( z C Z I ) cos φ P + m c 2 + z C 2 z C ( l 2 X I ) c Z I sin φ P .
x P R i = m ( l 2 ) S F ¯ , x P L i = m ( l 2 ) S ¯ F ¯ .
y P R i = m h 2 z P z P m B ¯ F ¯ sin φ P m ( h 2 y C R i ) ,
y P L i = m h 2 z P z P + m B ¯ F ¯ sin φ P m ( h 2 y C L i ) .
sin φ C sin φ P = m z C z P .
m h 2 y P R i = m h 2 y P R i = m ( h 2 y C L R i ) .
z P c p 2 + z P 2 = m z C p c 2 + z C 2 .
S F ¯ = c ( p 2 + z P 2 ) z C ( l 2 X I ) + c Z I p { z c c ( c + l 2 X I ) + z C ( z C Z I ) c z C ( l 2 X I ) + c Z I } ,
S ¯ F ¯ = c ( p 2 + z P 2 ) z C ( l 2 X I ) c Z I p { z c c ( c + l 2 X I ) z C ( z C Z I ) ± c z C ( l 2 X I ) c Z I } .
m ( l 2 x C R i ) S F ¯ = c ( c + l 2 X I ) + z C ( z C z I ) c ( l 2 X I ) + c Z I z C p 2 + z P 2 c 2 + z C 2 ( z C Z I ) z P ,
m ( l 2 x C L i ) S ¯ F ¯ = c ( c + l 2 X I ) z C ( z C z I ) + c ( l 2 X I ) c Z I z C p 2 + z P 2 c 2 + z C 2 ( z C Z I ) z P .
m ( l 2 X C R i ) S F ¯ = c 2 + z C ( z C Z I ) c 2 Z I z C p 2 + z P 2 c 2 + z C 2 ( z C Z I ) z P = c 2 + z C 2 z C p 2 + z P 2 z P = cos φ P cos φ C ,
m ( l 2 X C L i ) S ¯ F ¯ = c 2 + z C ( z C Z I ) c 2 Z I z C p 2 + z P 2 c 2 + z C 2 ( z C Z I ) z P = cos φ P cos φ C .
z P 2 = m p z C 2 c .
[ x I y I z I ] = 1 ( 2 b + x P L i x P R i ) [ b + x V b x V 0 y V y V 2 b z V z V 0 ] [ x P L i x P R i y P L R i ] ,
x I = ( b + x V ) x P L i + ( b x V ) x P R i 2 b + x P L i x P R i = b { m l ( S F ¯ + S ¯ F ¯ ) } + x V ( S F ¯ S ¯ F ¯ ) 2 b + S F ¯ S ¯ F ¯ ,
y I = y V x P L i y V x P R i + 2 b y P L R i 2 b + x P L i x P R i = y V ( S F ¯ S ¯ F ¯ ) + 2 b m y C L R i 2 b + ( S F ¯ S ¯ F ¯ ) ,
= y V ( z c Z I ) ( S F ¯ S ¯ F ¯ ) + 2 b m ( z c Y I Z I ( h 2 ) ) ( z c Z I ) { 2 b + ( S F ¯ S ¯ F ¯ ) } ,
z I = z V x P L i z V x P R i 2 b + x P L i x P R i = z v ( S F ¯ S ¯ F ¯ ) 2 b + ( S F ¯ S ¯ F ¯ ) .
x I = b { m l ( S F ¯ + S ¯ F ¯ ) } + x V ( S F ¯ S ¯ F ¯ ) 2 b + ( S F ¯ S ¯ F ¯ ) = b [ m l p { z C 2 A C c z C ( B C A D ) c 2 B D } c z C ( p 2 + z P 2 ) ( A D + B C ) ] + x V c ( p 2 + z P 2 ) { z C ( B C A D ) + 2 c B D } 2 b p { z C 2 A C c z C ( B C A D ) c 2 B D } + c ( p 2 + z P 2 ) { z C ( B C A D ) + 2 c B D } ,
y I = y V ( z C Z I ) ( S F ¯ S ¯ F ¯ ) + 2 b m ( z c Y I Z I ( h 2 ) ) ( z C Z I ) { 2 b + ( S F ¯ S ¯ F ¯ ) } = y V ( z C Z I ) c ( p 2 + z P 2 ) { z C ( B C A D ) + 2 c B D } + 2 b m p ( z c Y I Z I ( h 2 ) ) { z C A C c z C ( B C A D ) c 2 B D } ( z C Z I ) [ 2 b p { z C 2 A C c z C ( B C A D ) c 2 B D } + c ( p 2 + z P 2 ) { z C ( B C A D ) + 2 c B D } ] ,
z I = z V ( S F ¯ S ¯ F ¯ ) 2 b + ( S F ¯ S ¯ F ¯ ) = z V c ( p 2 + z P 2 ) { z C ( B C A D ) + 2 c B D } 2 b p { z C 2 A C c z C ( B C A D ) c 2 B D } + c ( p 2 + z P 2 ) { z C ( B C A D ) + 2 c B D } ,
A C = z C 2 ( z C Z I ) 2 + 2 c 2 z C ( z C Z I ) + c 2 { c ( l 2 X I ) } ( c + l 2 X I ) ,
A D = ( c 2 + z C 2 ) ( l 2 X I ) ( z C Z I ) + z C c ( l 2 X I ) 2 c Z I { c 2 + z C ( z C Z I ) } ,
B D = z C 2 ( l 2 X I ) 2 c 2 Z I 2 ,
B C = [ ( c 2 + z C 2 ) ( l 2 X I ) ( z C Z I ) z C c ( l 2 X I ) 2 + c Z I { c 2 + z C ( z C Z I ) } ] .
B C A D = 2 c Z I { c 2 + z C ( z C Z I ) } 2 z C c ( l 2 X I ) 2 ,
B C + A D = 2 ( c 2 + z C 2 ) ( l 2 X I ) ( z C Z I ) .
z C ( B C A D ) + 2 c B D = 2 c Z I ( z C Z I ) ( c 2 + z C 2 ) ,
z C 2 A C c z C ( B C A D ) c 2 B D = ( z C Z I ) 2 ( z C 2 + c 2 ) 2 ,
z C 2 + c 2 = ( z P c m z C p ) 2 ( z P 2 + p 2 ) .
x I = b m ( l 2 ) ( z C Z I ) z P 2 m 2 b p z C 3 ( l 2 X I ) c + x V m 2 p z C 2 Z I b z P 2 ( z C Z I ) + m 2 p z C 2 Z I = m 2 b p z C 3 X I c + Z I ( x V m 2 p z C 2 ( l 2 ) b m z P 2 ) + ( l 2 ) ( b m z C z P 2 m 2 b p z C 3 c ) b z P 2 z C + Z I ( m 2 p z C 2 b z P 2 ) ,
y I = y V m 2 p z C 2 Z I + b m z P 2 ( z C Y I Z I ( h 2 ) ) b z P 2 ( z C Z I ) + m 2 p z C 2 Z I = b m z P 2 z C Y I + Z I ( y V m 2 p z C 2 ( h 2 ) b m z P 2 ) b z P 2 z C + Z I ( m 2 p z C 2 b z P 2 ) ,
z I = z V m 2 p z C 2 Z I b z P 2 ( z C Z I ) + m 2 p z C 2 Z I = z V m 2 p z C 2 Z I b z P 2 z C + Z I ( m 2 p z C 2 b z P 2 ) .
x I = m b z C X I + Z I ( x V c b ( l 2 ) ) b z C + Z I ( m c b ) ,
y I = m b z C Y I + Z I ( y V c b ( h 2 ) ) b z C + Z I ( m c b ) ,
z I = m c z V Z I b z C + Z I ( m c b ) .
x I = b c X I + l 2 ( m b c ) , y I = m Y I , z I = z V z C Z I .

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