Abstract

An improved approach for design of the catadioptric omnidirectional camera with horizontal scene undistorted imaging is described. In the proposed approach, the influence of the lens distortion on the imaging quality of the omnidirectional camera is taken into account. We establish a radial distortion model for an image pickup lens and retain some opposite distortion in the mirror to correct the distortion existing in the image pickup lens. A horizontal scene undistorted catadioptric omnidirectional camera is designed with an off-the-shelf TV short focus lens using our approach; the numerical simulation shows that the distortion introduced by the imaging lens is eliminated effectively.

© 2006 Optical Society of America

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  1. J. S. Chahl and M. V. Srinivasan, "Reflection surfaces for panoramic imaging," Appl. Opt. 36, 8276-8285 (1997).
    [CrossRef]
  2. S. Baker and S. K. Nayar, "A theory of single-viewpoint catadioptric image formation," Int. J. Comput. Vis. 35(2), 175-196 (1999).
    [CrossRef]
  3. J. Zeil, M. I. Hofmann, and J. S. Chahl, "Catchment areas of panoramic snapshots in outdoor scenes," J. Opt. Soc. Am. A 20, 450-469 (2003).
    [CrossRef]
  4. G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
    [CrossRef]
  5. S. K. Nayar and V. Peri, "Folded catadioptric cameras," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer, New York, 2001), pp. 103-115.
  6. K. Kamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboloidal projection," Proceedings of the 1993 IEEE International Conference on Intelligent Robots and Systems (IEEE, 1993), pp. 1029-1034.
  7. J. S. Chahl and M. V. Srinivasan, "Reflective surfaces for panoramic imaging," Appl. Opt. 36, 8275-8285 (1997).
    [CrossRef]
  8. G. Kweon, K. T. Kim, G. Kim, and H. Kim, "Folded catadioptric panoramic lens with an equidistance projection scheme," Appl. Opt. 44, 2759-2767 (2005).
    [CrossRef] [PubMed]
  9. R. A. Hicks and R. K. Perline, "Equiresolution catadioptric sensors," Appl. Opt. 44, 6108-6114 (2005).
    [CrossRef] [PubMed]
  10. R. A. Hicks and R. Bajcsy, "Reflective surfaces as computational sensors," Image Vision Comput. 19, 773-777 (2001).
    [CrossRef]
  11. R. Y. Tsai, "A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses," IEEE Trans. Rob. Autom. 3, 323-344 (1987).
    [CrossRef]
  12. Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Machine Intell. 22, 1330-1334 (2000).
    [CrossRef]

2005 (3)

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

G. Kweon, K. T. Kim, G. Kim, and H. Kim, "Folded catadioptric panoramic lens with an equidistance projection scheme," Appl. Opt. 44, 2759-2767 (2005).
[CrossRef] [PubMed]

R. A. Hicks and R. K. Perline, "Equiresolution catadioptric sensors," Appl. Opt. 44, 6108-6114 (2005).
[CrossRef] [PubMed]

2003 (1)

2001 (2)

S. K. Nayar and V. Peri, "Folded catadioptric cameras," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer, New York, 2001), pp. 103-115.

R. A. Hicks and R. Bajcsy, "Reflective surfaces as computational sensors," Image Vision Comput. 19, 773-777 (2001).
[CrossRef]

2000 (1)

Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Machine Intell. 22, 1330-1334 (2000).
[CrossRef]

1999 (1)

S. Baker and S. K. Nayar, "A theory of single-viewpoint catadioptric image formation," Int. J. Comput. Vis. 35(2), 175-196 (1999).
[CrossRef]

1997 (2)

J. S. Chahl and M. V. Srinivasan, "Reflection surfaces for panoramic imaging," Appl. Opt. 36, 8276-8285 (1997).
[CrossRef]

J. S. Chahl and M. V. Srinivasan, "Reflective surfaces for panoramic imaging," Appl. Opt. 36, 8275-8285 (1997).
[CrossRef]

1993 (1)

K. Kamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboloidal projection," Proceedings of the 1993 IEEE International Conference on Intelligent Robots and Systems (IEEE, 1993), pp. 1029-1034.

1987 (1)

R. Y. Tsai, "A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses," IEEE Trans. Rob. Autom. 3, 323-344 (1987).
[CrossRef]

Bajcsy, R.

R. A. Hicks and R. Bajcsy, "Reflective surfaces as computational sensors," Image Vision Comput. 19, 773-777 (2001).
[CrossRef]

Baker, S.

S. Baker and S. K. Nayar, "A theory of single-viewpoint catadioptric image formation," Int. J. Comput. Vis. 35(2), 175-196 (1999).
[CrossRef]

Chahl, J. S.

Coelho, C.

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

Hicks, R. A.

R. A. Hicks and R. K. Perline, "Equiresolution catadioptric sensors," Appl. Opt. 44, 6108-6114 (2005).
[CrossRef] [PubMed]

R. A. Hicks and R. Bajcsy, "Reflective surfaces as computational sensors," Image Vision Comput. 19, 773-777 (2001).
[CrossRef]

Hofmann, M. I.

Kamazawa, K.

K. Kamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboloidal projection," Proceedings of the 1993 IEEE International Conference on Intelligent Robots and Systems (IEEE, 1993), pp. 1029-1034.

Kim, G.

Kim, H.

Kim, K. T.

Kweon, G.

Marcenaro, L.

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

Nayar, S. K.

S. K. Nayar and V. Peri, "Folded catadioptric cameras," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer, New York, 2001), pp. 103-115.

S. Baker and S. K. Nayar, "A theory of single-viewpoint catadioptric image formation," Int. J. Comput. Vis. 35(2), 175-196 (1999).
[CrossRef]

Peri, V.

S. K. Nayar and V. Peri, "Folded catadioptric cameras," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer, New York, 2001), pp. 103-115.

Perline, R. K.

Regazzoni, C. S.

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

Scotti, G.

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

Selvaggi, F.

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

Srinivasan, M. V.

J. S. Chahl and M. V. Srinivasan, "Reflective surfaces for panoramic imaging," Appl. Opt. 36, 8275-8285 (1997).
[CrossRef]

J. S. Chahl and M. V. Srinivasan, "Reflection surfaces for panoramic imaging," Appl. Opt. 36, 8276-8285 (1997).
[CrossRef]

Tsai, R. Y.

R. Y. Tsai, "A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses," IEEE Trans. Rob. Autom. 3, 323-344 (1987).
[CrossRef]

Yachida, M.

K. Kamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboloidal projection," Proceedings of the 1993 IEEE International Conference on Intelligent Robots and Systems (IEEE, 1993), pp. 1029-1034.

Yagi, Y.

K. Kamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboloidal projection," Proceedings of the 1993 IEEE International Conference on Intelligent Robots and Systems (IEEE, 1993), pp. 1029-1034.

Zeil, J.

Zhang, Z.

Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Machine Intell. 22, 1330-1334 (2000).
[CrossRef]

Appl. Opt. (4)

IEE Proc. Vision (1)

G. Scotti, L. Marcenaro, C. Coelho, F. Selvaggi, and C. S. Regazzoni, "Dual camera intelligent sensor for high definition 360 degrees surveillance," IEE Proc. Vision , Image Signal Process. 152, 250-257 (2005).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

Z. Zhang, "A flexible new technique for camera calibration," IEEE Trans. Pattern Anal. Machine Intell. 22, 1330-1334 (2000).
[CrossRef]

IEEE Trans. Rob. Autom. (1)

R. Y. Tsai, "A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses," IEEE Trans. Rob. Autom. 3, 323-344 (1987).
[CrossRef]

Image Vision Comput. (1)

R. A. Hicks and R. Bajcsy, "Reflective surfaces as computational sensors," Image Vision Comput. 19, 773-777 (2001).
[CrossRef]

Int. J. Comput. Vis. (1)

S. Baker and S. K. Nayar, "A theory of single-viewpoint catadioptric image formation," Int. J. Comput. Vis. 35(2), 175-196 (1999).
[CrossRef]

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

Other (2)

S. K. Nayar and V. Peri, "Folded catadioptric cameras," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer, New York, 2001), pp. 103-115.

K. Kamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboloidal projection," Proceedings of the 1993 IEEE International Conference on Intelligent Robots and Systems (IEEE, 1993), pp. 1029-1034.

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Figures (13)

Fig. 1
Fig. 1

Catadioptric omnidirectional camera.

Fig. 2
Fig. 2

Conventional camera geometry.

Fig. 3
Fig. 3

Schematic diagram illustrating the catadioptric omnidirectional camera with horizontal scene undistorted imaging.

Fig. 4
Fig. 4

Original distortion image.

Fig. 5
Fig. 5

Distortion-free image.

Fig. 6
Fig. 6

Cross section of mirror design based on the pinhole model.

Fig. 7
Fig. 7

Change plot of the space between adjacent intersections (ignoring lens distortion; D, the coordinate of intersections; δD, the space between adjacent intersections).

Fig. 8
Fig. 8

Change plot of the space between adjacent intersections (considering lens distortion; D, the coordinate of intersections; δD, the space between adjacent intersections).

Fig. 9
Fig. 9

Cross section of mirror design based on the radial distortion model.

Fig. 10
Fig. 10

Change plot of the space between adjacent intersections (D, the coordinate of intersections; δD, the space between adjacent intersections).

Fig. 11
Fig. 11

Omnidirectional image of an indoor scene directly taken by the horizontal scene undistorted catadioptric camera including a 6 m m video lens and a mirror design based on radial distortion. In the indoor scene, the ceiling has an equal-size pane pattern and is perpendicular to the optical axis. On the omnidirectional image, the pane pattern has equal size too, so the omnidirectional image is undistorted.

Fig. 12
Fig. 12

Omnidirectional image of an indoor scene taken by the hyperboloidal catadioptric omnidirectional camera including a 12 m m video lens and a hyperboloidal mirror. It can be seen from the image of the ceiling that there is severe distortion.

Fig. 13
Fig. 13

Perspective omnidirectional image generated from Fig. 12 by reprojective calculation. The pane pattern on the image is equal size and undistorted.

Equations (25)

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

P = [ f x z , f y z , f ] T .
Q = [ x z , y z , 1 ] T .
q u = q u 0 + P x d ,
q v = q v 0 + P y d .
q = K Q ,
K = [ f pixel 0 q u 0 0 f pixel q v 0 0 0 1 ]
r d = ( 1 + K 1 r 2 + K 2 r 4 ) r ,
r d = Q d x 2 + Q d y 2 ,
r = Q x 2 + Q y 2 ,
q d = K Q d .
D ( ρ d ) = a ρ d .
ρ d = r d f = ( 1 + K 1 r 2 + K 2 r 4 ) r f .
ρ d = [ 1 + K 1 ( ρ f ) 2 + K 2 ( ρ f ) 4 ] ρ .
D ( ρ ) = a [ 1 + K 1 ( ρ f ) 2 + K 2 ( ρ f ) 4 ] ρ .
tan γ = z ( x ) ,
tan ϕ = D ( x ) - x z ( x ) - h ,
tan θ = ρ f = x z ( x ) .
2 γ = ϕ - θ .
z ( x ) + k - k 2 + 1 = 0 ,
k = z ( x ) [ z ( x ) - h ] + x [ D ( x ) - x ] z ( x ) [ D ( x ) - x ] - x [ z ( x ) - h ] .
D ( x ) = u x z ( x ) { 1 + K 1 [ x z ( x ) ] 2 + K 2 [ x z ( x ) ] 4 } ,
u = a f .
tan ϕ = u r ( 1 + K 1 r 2 + K 2 r 4 ) - r z ( x ) z ( x ) - h ,
tan ϕ = u r ( 1 + K 1 r 2 + K 2 r 4 ) - r z 0 z 0 - h .
u = ( z 0 - h ) tan ϕ max + r max z 0 r max ( 1 + K 1 r max 2 + k 2 r max 4 ) .

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