Abstract

A central catadioptric-perspective camera system is widely used nowadays. A critical problem is that current calibration methods cannot determine the extrinsic parameters between the central catadioptric camera and a perspective camera effectively. We present a novel calibration method for a central catadioptric-perspective camera system, in which the central catadioptric camera has a hyperbolic mirror. Two cameras are used to capture images of one calibration pattern at different spatial positions. A virtual camera is constructed at the origin of the central catadioptric camera and faced toward the calibration pattern. The transformation between the virtual camera and the calibration pattern could be computed first and the extrinsic parameters between the central catadioptric camera and the calibration pattern could be obtained. Three-dimensional reconstruction results of the calibration pattern show a high accuracy and validate the feasibility of our method.

© 2012 Optical Society of America

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References

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  1. A. Voigtlander, S. Lange, M. Lauer, and M. Riedmiller, “Real-time 3D ball recognition using perspective and catadioptric cameras,” in Proceedings of the Third European Conference on Mobile Robots (Elsevier, 2007), pp. 1–6.
  2. M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
    [CrossRef]
  3. F. Dornaika and J. Elder, “Image registration for foveated omnidirectional sensing,” in Proceedings of the 7th European Conference on Computer Vision (Springer, 2002), pp. 606–620.
  4. D. Chen and J. Yang, “Image registration with uncalibrated cameras in hybrid vision systems,” in Proceedings of the 7th IEEE Workshops on Application of Computer Vision (IEEE, 2005), pp. 427–432.
  5. E. Damien, C. Demonceaux, and P. Vasseur, “UAV motion estimation using hybrid stereoscopic vision,” in Proceedings of the 12th IAPR Conference on Machine Vision Applications (IEEE, 2011), pp. 340–343.
  6. O. Faugeras and S. Laveau, “Representing three-dimensional data as a collection of images and fundamental matrices for image synthesis,” in Proceedings of the 12th International Conference on Pattern Recognition (Springer, 1994), pp. 689–691.
  7. B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
    [CrossRef]
  8. P. Beardsley and D. Murray, “Camera calibration using vanishing points,” in Proceedings of the British Machine Vision Conference (Springer, 1992), pp. 416–425.
  9. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334(2002).
    [CrossRef]
  10. X. Zhang and L. M. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48, 117208 (2009).
    [CrossRef]
  11. C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223–243 (2001).
    [CrossRef]
  12. X. Ying and Z. Hu, “Catadioptric camera calibration using geometric invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1260–1271, (2004).
    [CrossRef]
  13. D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the 4th IEEE International Conference on Computer Vision Systems (IEEE, 2006), pp. 45–52.
  14. C. Mei and P. Rives, “Single view point omnidirectional camera calibration from planar grids,” in Proceedings of the 2007 IEEE International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.
  15. B. Zhang and Y. Li, “Homography-based method for calibrating an omnidirectional vision system,” J. Opt. Soc. Am. A 25, 1389–1394 (2008).
    [CrossRef]
  16. G. Caron and D. Eynard, “Multiple camera types simultaneous stereo calibration,” in Proceedings of 2011 IEEE International Conference on Robotics and Automation (IEEE, 2011), pp. 2933–2938.
  17. X. Chen, J. Yang, and A. Waibel, “Calibration of a hybrid camera network,” in Proceedings of the 9th IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 1, pp. 150–155.
  18. N. D. Jankovic and M. D. Naish, “Calibrating an active omnidirectional vision system,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 3093–3098.
  19. S. Cagnoni, M. Mordonini, L. Mussi, and G. Adorni, “Hybrid stereo sensor with omnidirectional vision capabilities: overview and calibration procedures,” in Proceedings of the 14th International Conference on Image Analysis and Processing (IEEE, 2007), pp. 99–104.
  20. J.-Y. Bouguet, “Camera calibration toolbox for Matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/ .
  21. X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
    [CrossRef]
  22. F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

2012

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

2011

M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
[CrossRef]

2009

X. Zhang and L. M. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48, 117208 (2009).
[CrossRef]

2008

2004

X. Ying and Z. Hu, “Catadioptric camera calibration using geometric invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1260–1271, (2004).
[CrossRef]

2002

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334(2002).
[CrossRef]

2001

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223–243 (2001).
[CrossRef]

1990

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

Adorni, G.

S. Cagnoni, M. Mordonini, L. Mussi, and G. Adorni, “Hybrid stereo sensor with omnidirectional vision capabilities: overview and calibration procedures,” in Proceedings of the 14th International Conference on Image Analysis and Processing (IEEE, 2007), pp. 99–104.

Beardsley, P.

P. Beardsley and D. Murray, “Camera calibration using vanishing points,” in Proceedings of the British Machine Vision Conference (Springer, 1992), pp. 416–425.

Cagnoni, S.

S. Cagnoni, M. Mordonini, L. Mussi, and G. Adorni, “Hybrid stereo sensor with omnidirectional vision capabilities: overview and calibration procedures,” in Proceedings of the 14th International Conference on Image Analysis and Processing (IEEE, 2007), pp. 99–104.

Caprile, B.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

Carelli, R.

F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

Caron, G.

G. Caron and D. Eynard, “Multiple camera types simultaneous stereo calibration,” in Proceedings of 2011 IEEE International Conference on Robotics and Automation (IEEE, 2011), pp. 2933–2938.

Chang, L.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

Chen, D.

D. Chen and J. Yang, “Image registration with uncalibrated cameras in hybrid vision systems,” in Proceedings of the 7th IEEE Workshops on Application of Computer Vision (IEEE, 2005), pp. 427–432.

Chen, X.

X. Chen, J. Yang, and A. Waibel, “Calibration of a hybrid camera network,” in Proceedings of the 9th IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 1, pp. 150–155.

Damien, E.

E. Damien, C. Demonceaux, and P. Vasseur, “UAV motion estimation using hybrid stereoscopic vision,” in Proceedings of the 12th IAPR Conference on Machine Vision Applications (IEEE, 2011), pp. 340–343.

Daniilidis, K.

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223–243 (2001).
[CrossRef]

Demonceaux, C.

E. Damien, C. Demonceaux, and P. Vasseur, “UAV motion estimation using hybrid stereoscopic vision,” in Proceedings of the 12th IAPR Conference on Machine Vision Applications (IEEE, 2011), pp. 340–343.

Deng, X.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

Dornaika, F.

F. Dornaika and J. Elder, “Image registration for foveated omnidirectional sensing,” in Proceedings of the 7th European Conference on Computer Vision (Springer, 2002), pp. 606–620.

Duan, F.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

Elder, J.

F. Dornaika and J. Elder, “Image registration for foveated omnidirectional sensing,” in Proceedings of the 7th European Conference on Computer Vision (Springer, 2002), pp. 606–620.

Eynard, D.

G. Caron and D. Eynard, “Multiple camera types simultaneous stereo calibration,” in Proceedings of 2011 IEEE International Conference on Robotics and Automation (IEEE, 2011), pp. 2933–2938.

Faugeras, O.

O. Faugeras and S. Laveau, “Representing three-dimensional data as a collection of images and fundamental matrices for image synthesis,” in Proceedings of the 12th International Conference on Pattern Recognition (Springer, 1994), pp. 689–691.

Geyer, C.

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223–243 (2001).
[CrossRef]

Hu, Z.

X. Ying and Z. Hu, “Catadioptric camera calibration using geometric invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1260–1271, (2004).
[CrossRef]

Jankovic, N. D.

N. D. Jankovic and M. D. Naish, “Calibrating an active omnidirectional vision system,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 3093–3098.

Lange, S.

M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
[CrossRef]

A. Voigtlander, S. Lange, M. Lauer, and M. Riedmiller, “Real-time 3D ball recognition using perspective and catadioptric cameras,” in Proceedings of the Third European Conference on Mobile Robots (Elsevier, 2007), pp. 1–6.

Lauer, M.

M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
[CrossRef]

A. Voigtlander, S. Lange, M. Lauer, and M. Riedmiller, “Real-time 3D ball recognition using perspective and catadioptric cameras,” in Proceedings of the Third European Conference on Mobile Robots (Elsevier, 2007), pp. 1–6.

Laveau, S.

O. Faugeras and S. Laveau, “Representing three-dimensional data as a collection of images and fundamental matrices for image synthesis,” in Proceedings of the 12th International Conference on Pattern Recognition (Springer, 1994), pp. 689–691.

Li, Y.

Martinelli, A.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the 4th IEEE International Conference on Computer Vision Systems (IEEE, 2006), pp. 45–52.

Mei, C.

C. Mei and P. Rives, “Single view point omnidirectional camera calibration from planar grids,” in Proceedings of the 2007 IEEE International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.

Mordonini, M.

S. Cagnoni, M. Mordonini, L. Mussi, and G. Adorni, “Hybrid stereo sensor with omnidirectional vision capabilities: overview and calibration procedures,” in Proceedings of the 14th International Conference on Image Analysis and Processing (IEEE, 2007), pp. 99–104.

Murray, D.

P. Beardsley and D. Murray, “Camera calibration using vanishing points,” in Proceedings of the British Machine Vision Conference (Springer, 1992), pp. 416–425.

Mussi, L.

S. Cagnoni, M. Mordonini, L. Mussi, and G. Adorni, “Hybrid stereo sensor with omnidirectional vision capabilities: overview and calibration procedures,” in Proceedings of the 14th International Conference on Image Analysis and Processing (IEEE, 2007), pp. 99–104.

Naish, M. D.

N. D. Jankovic and M. D. Naish, “Calibrating an active omnidirectional vision system,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 3093–3098.

Riedmiller, M.

A. Voigtlander, S. Lange, M. Lauer, and M. Riedmiller, “Real-time 3D ball recognition using perspective and catadioptric cameras,” in Proceedings of the Third European Conference on Mobile Robots (Elsevier, 2007), pp. 1–6.

Rives, P.

C. Mei and P. Rives, “Single view point omnidirectional camera calibration from planar grids,” in Proceedings of the 2007 IEEE International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.

Roberti, F.

F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

Scaramuzza, D.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the 4th IEEE International Conference on Computer Vision Systems (IEEE, 2006), pp. 45–52.

Schönbein, M.

M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
[CrossRef]

Siegwart, R.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the 4th IEEE International Conference on Computer Vision Systems (IEEE, 2006), pp. 45–52.

Soria, C.

F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

Toibero, J. M.

F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

Torre, V.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

Vassallo, R. F.

F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

Vasseur, P.

E. Damien, C. Demonceaux, and P. Vasseur, “UAV motion estimation using hybrid stereoscopic vision,” in Proceedings of the 12th IAPR Conference on Machine Vision Applications (IEEE, 2011), pp. 340–343.

Voigtlander, A.

A. Voigtlander, S. Lange, M. Lauer, and M. Riedmiller, “Real-time 3D ball recognition using perspective and catadioptric cameras,” in Proceedings of the Third European Conference on Mobile Robots (Elsevier, 2007), pp. 1–6.

Waibel, A.

X. Chen, J. Yang, and A. Waibel, “Calibration of a hybrid camera network,” in Proceedings of the 9th IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 1, pp. 150–155.

Wang, H.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

Welker, S.

M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
[CrossRef]

Wu, F.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

Wu, Y.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

Yang, J.

X. Chen, J. Yang, and A. Waibel, “Calibration of a hybrid camera network,” in Proceedings of the 9th IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 1, pp. 150–155.

D. Chen and J. Yang, “Image registration with uncalibrated cameras in hybrid vision systems,” in Proceedings of the 7th IEEE Workshops on Application of Computer Vision (IEEE, 2005), pp. 427–432.

Ying, X.

X. Ying and Z. Hu, “Catadioptric camera calibration using geometric invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1260–1271, (2004).
[CrossRef]

Zhang, B.

Zhang, X.

X. Zhang and L. M. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48, 117208 (2009).
[CrossRef]

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334(2002).
[CrossRef]

Zhu, L. M.

X. Zhang and L. M. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48, 117208 (2009).
[CrossRef]

Comput. Vis. Image Underst.

X. Deng, F. Wu, Y. Wu, F. Duan, L. Chang, and H. Wang, “Self-calibration of hybrid central catadioptric and perspective cameras,” Comput. Vis. Image Underst. 116, 715–729 (2012).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334(2002).
[CrossRef]

X. Ying and Z. Hu, “Catadioptric camera calibration using geometric invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1260–1271, (2004).
[CrossRef]

Int. J. Comput. Vis.

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223–243 (2001).
[CrossRef]

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–139 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Mechatronics

M. Lauer, M. Schönbein, S. Lange, and S. Welker, “3D-object tracking with a mixed omnidirectional stereo camera system,” Mechatronics 21, 390–398 (2011).
[CrossRef]

Opt. Eng.

X. Zhang and L. M. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48, 117208 (2009).
[CrossRef]

Other

P. Beardsley and D. Murray, “Camera calibration using vanishing points,” in Proceedings of the British Machine Vision Conference (Springer, 1992), pp. 416–425.

F. Dornaika and J. Elder, “Image registration for foveated omnidirectional sensing,” in Proceedings of the 7th European Conference on Computer Vision (Springer, 2002), pp. 606–620.

D. Chen and J. Yang, “Image registration with uncalibrated cameras in hybrid vision systems,” in Proceedings of the 7th IEEE Workshops on Application of Computer Vision (IEEE, 2005), pp. 427–432.

E. Damien, C. Demonceaux, and P. Vasseur, “UAV motion estimation using hybrid stereoscopic vision,” in Proceedings of the 12th IAPR Conference on Machine Vision Applications (IEEE, 2011), pp. 340–343.

O. Faugeras and S. Laveau, “Representing three-dimensional data as a collection of images and fundamental matrices for image synthesis,” in Proceedings of the 12th International Conference on Pattern Recognition (Springer, 1994), pp. 689–691.

G. Caron and D. Eynard, “Multiple camera types simultaneous stereo calibration,” in Proceedings of 2011 IEEE International Conference on Robotics and Automation (IEEE, 2011), pp. 2933–2938.

X. Chen, J. Yang, and A. Waibel, “Calibration of a hybrid camera network,” in Proceedings of the 9th IEEE International Conference on Computer Vision (IEEE, 2003), Vol. 1, pp. 150–155.

N. D. Jankovic and M. D. Naish, “Calibrating an active omnidirectional vision system,” in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 3093–3098.

S. Cagnoni, M. Mordonini, L. Mussi, and G. Adorni, “Hybrid stereo sensor with omnidirectional vision capabilities: overview and calibration procedures,” in Proceedings of the 14th International Conference on Image Analysis and Processing (IEEE, 2007), pp. 99–104.

J.-Y. Bouguet, “Camera calibration toolbox for Matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/ .

A. Voigtlander, S. Lange, M. Lauer, and M. Riedmiller, “Real-time 3D ball recognition using perspective and catadioptric cameras,” in Proceedings of the Third European Conference on Mobile Robots (Elsevier, 2007), pp. 1–6.

D. Scaramuzza, A. Martinelli, and R. Siegwart, “A flexible technique for accurate omnidirectional camera calibration and structure from motion,” in Proceedings of the 4th IEEE International Conference on Computer Vision Systems (IEEE, 2006), pp. 45–52.

C. Mei and P. Rives, “Single view point omnidirectional camera calibration from planar grids,” in Proceedings of the 2007 IEEE International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.

F. Roberti, R. F. Vassallo, J. M. Toibero, C. Soria, and R. Carelli, “Geometry of a hybrid stereo vision system for robotics applications,” presented at V Jornadas Argentinas de Robotica, Universidad Nacional del Sur, Bahía Blanca, Argentina, 12–14 November 2008.

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Figures (14)

Fig. 1.
Fig. 1.

Central catadioptric-perspective camera system used in this study.

Fig. 2.
Fig. 2.

Perspective camera model.

Fig. 3.
Fig. 3.

Central catadioptric model.

Fig. 4.
Fig. 4.

Illustration of the calibration principle of the central catadioptric-perspective camera system.

Fig. 5.
Fig. 5.

Illustration of the reprojection of the chessboard corners onto the hyperbolic mirror.

Fig. 6.
Fig. 6.

Determination of the direction of the Xmaxis in the virtual camera system.

Fig. 7.
Fig. 7.

Determination of the direction of the Zmaxis in the virtual camera system.

Fig. 8.
Fig. 8.

Illustration of the virtual camera and the virtual image plane.

Fig. 9.
Fig. 9.

Diagram of the calibration principle.

Fig. 10.
Fig. 10.

(a), (b) Captured images for calibrating the perspective camera. (c), (d) Captured images for calibrating the central catadioptric camera.

Fig. 11.
Fig. 11.

(a) Image taken by the perspective camera, (b) image taken by the central catadioptric camera, and (c) 3D scene representation of the central catadioptric-perspective camera system.

Fig. 12.
Fig. 12.

(a)–(d) Mean values and standard deviations of the translation matrix. (e)–(h) Mean values and standard deviations of the rotation angles.

Fig. 13.
Fig. 13.

(a), (b) Maximum values, minimum values, mean values, standard deviations of the distances among the horizontally (vertically) adjoining points of the chessboard. (c) Fitting errors of the plane to the 3D data of corner points.

Fig. 14.
Fig. 14.

(a) Image taken by the central catadioptric camera, and (b) 3D scene representation of the central catadioptric-perspective camera system in this experiment.

Tables (3)

Tables Icon

Table 1. Reconstruction Results with Different Positions of Chessboard Image Pairs

Tables Icon

Table 2. Reconstruction Results with Different Positions of Chessboard Image by Ref. [16]

Tables Icon

Table 3. Reconstruction Results with Different Positions of Chessboard Image by Our Method

Equations (21)

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

λ[upvp1]=[kxpβpu0p0kypv0p001][XpYpZp],
[kxpβpu0p0kypv0p001]
(Z+c)2a2X2+Y2b2=1,
(Zm+2c)[uovo1]=[kxoβou0o0kyov0o001][YmXmZm+2c],
[kxoβou0o0kyov0o001]
{[YmiXmiZmi]=λi[kxoβou0o0kyov0o001]1[uoivoi1][002c](Zmi+c)2a2Xmi2+Ymi2b2=1,
R1=[cos(ang1)sin(ang1)0sin(ang1)cos(ang1)0001].
βi=atanZmiXmi2+Ymi2.
R2=[1000cos(ang2)sin(ang2)0sin(ang2)cos(ang2)].
Kv=[100010001],
λ[uvvv1]=Kv[Rv|Tv][XpYpZp1],
Kv1[uvvv1]T=[Rv|Tv][XpYpZp1]Tλ.
[XmiYmiZmi]T=R2R1[XmiYmiZmi]T,
[uvivvi1]T=Kv[XmiYmiZmi]TZmi=[XmiYmiZmi]TZmi.
Rm2c=R11R21Rm2cTm2c=R11R21Tm2c.
Rp2m=Rp2cRm2c1Tp2m=Tp2cRp2cRm2c1Tm2c.
E1=i=1Nj=148mpers,ijm¯pers(Kpers,Rp2c,i,Tp2c,iM¯j)2,
E2=i=1Nj=148mcata,ijm¯cata(Kcata,a,b,c,Rp2m,Tp2m,Rp2c,i,Tp2c,iM¯j)2,
[kxpβpu0p0kypv0p001]=[3318.300616.8703330.37372.28001].
[kxoβou0o0kyov0o001]=[2410.60963.3502405.0810.11001].
Rp2m=[0.992820.10410.0588680.0736840.144750.986720.0941990.983980.15138]Tp2m=[160.57422.7235.19]T(mm).

Metrics