Abstract

A family of catadioptric imaging systems has been developed that can achieve omnidirectional viewing with a single planar imager while still being able to recover perspective images, provided that they satisfy the single-viewpoint (SVP) constraint. It has been shown that the only mirror shapes that can have SVP when paired with a sole focusing planar imager camera are the surfaces of revolution of conic section curves. However, the special case of such a surface, the cone-shaped mirror itself, has not been deemed a viable SVP mirror shape. We present a comprehensive imaging theory of the cone mirror in its SVP configuration. We show that the SVP, cone mirror catadioptric system not only is practical but also has unique advantages for certain applications. The detailed theory explains why and how a practical SVP cone configuration can be set up, the merits and weaknesses of such systems, and how one can remedy the weaknesses to create a workable imaging system. We also derive the tolerance formula for estimating effects of alignment errors. A prototype has been constructed, and experimental results validate our theory.

© 2006 Optical Society of America

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  1. S. Baker and S. K. Nayar, "A theory of single-viewpoint catadioptric image formation," Int. J. Comput. Vis. 35, 175-196 (1999).
    [CrossRef]
  2. K. Yamazawa, Y. Yagi, and M. Yachida, "Ommdirectional imaging with hyperbolodial projection," in Proceedings of the 1993 IEEE/Robotics Society of Japan International Conference on Intelligent Robots and Systems (IEEE, 1993) Vol. 2, pp. 1029-1034.
    [CrossRef]
  3. T. Svoboda, T. Pajdla, and V. Hlavác, "Epipolar geometry for panoramic cameras," in Proceedings of the 5th European Conference on Computer Vision (IEEE, 1998), pp. 218-232.
  4. S.-S. Lin and R. Bajcsy, "True single view point cone mirror omni-directional catadioptric system," in Proceedings of the 8th International Conference on Computer Vision (IEEE, 2001), Vol. II, pp. 102-107.
  5. C. Geyer and K. Daniilidis, "Paracatadioptric camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
    [CrossRef]
  6. S.-S. Lin, "Omni-directional 3D Stereo computer vision sensor using reflective cone mirror," Ph.D. thesis (Computer and Information Science Department, University of Pennsylvania, 2003).
  7. S.-S. Lin and R. Bajcsy, "Single-view-point omnidirectional catadioptric cone mirror imager," IEEE Trans. Pattern Anal. Mach. Intell. 28, 840-845 (2006).
    [CrossRef] [PubMed]
  8. Y. Yagi, S. Kawato, and S. Tsuji, "Real-time omnidirectional image sensor (COPIS) for vision-guided navigation," IEEE Trans. Rob. Autom. 10, 11-22 (1994).
    [CrossRef]
  9. S. Bogner, "Introduction to panoramic imaging," in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetices 1995 (IEEE, 1995), pp. 3100-3106.
  10. D. Southwell, B. Vandegriend, and A. Basu, "A conical mirror pipeline inspection system," in Proceedings of the 1996 IEEE International Conference on Robotics and Automation (IEEE, 1996), Vol. 4, pp. 3253-3258.
  11. J. S. Chahl and M. V. Srinvasan, "Reflective surfaces for panoramic imaging," Appl. Opt. 36, 8275-8285 (1997).
    [CrossRef]
  12. Y. Yagi, "Omnidirectional sensing and its applications," IEICE Trans. Inf. Syst. E82D, 568-579 (1999).
  13. Y. Yagi and M. Yachida, "Real-time omnidirectional image sensors," Int. J. Comput. Vis. , 58, 173-207 (2004).
    [CrossRef]
  14. V. S. Nalwa, "Panoramic viewing system with offset virtual optical centers," U.S. patent 6,219,090 (17 April 2000).
  15. H. Hua and N. Ahuja, "A high-resolution panoramic camera," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE2001), Vol. 1, pp. 960-967.
  16. R. Swaminathan, M. D. Grossberg, and S. K. Nayar, "A perspective on distortions," in Proceedings of the 2003 International Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. II, pp. 594-601.
  17. D. W. Rees, "Panoramic television viewing system," U.S. patent 3,505,465 (7 April 1970).
  18. R. A. Hicks and R. K. Perline, "Geometric distributions for catadioptric sensor design," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 584-589.
  19. R. A. Hicks and R. Bajcsy, "Reflective surfaces as computational sensors," Image Vis. Comput. 19, 773-777 (2001).
    [CrossRef]
  20. A. Krishna and N. Ahuja, "Panoramic image acquisition," in Proceedings of the 1996 International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 379-384.
  21. H. Nagahara, Y. Yagi, and M. Yachida, "Resolution improving method for a 3D environment modeling using omnidirectional image sensor," in Proceedings of IEEE International Conference on Robotics and Automation 2002 (IEEE, 2002), Vol. 1, pp. 900-907.
  22. H. Ishiguro, M. Yamamoto, and S. Tsuji, "Omnidirectional stereo," IEEE Trans. Pattern Anal. Mach. Intell. 14, 257-262 (1992).
    [CrossRef]
  23. D. W. Murray, "Recovering range using virtual multicamera stereo," Comput. Vis. Image Underst. 61, 285-291 (1995).
    [CrossRef]
  24. S. B. Kang and R. Szeliski, "3-d scene data recovery using omnidirectional multibaseline stereo," Int. J. Comput. Vis. 25, 167-183 (1997).
    [CrossRef]
  25. R. Benosman and J. Devars, "Panoramic stereovision sensor," in Proceedings of the International Conference on Pattern Recognition 1998 (International Association for Pattern Recognition, 1998), Vol. 1, pp. 767-769.
  26. S. Peleg, M. Ben-Ezra, and Y. Pritch, "Omnistereo: panoramic stereo imaging," IEEE Trans. Pattern Anal. Mach. Intell. 23, 279-290 (2001).
    [CrossRef]
  27. D. Southwell, A. Basu, M. Fiala, and J. Reyda, "Panoramic stereo in Proceedings of the 13th International Conference on Pattern Recognition (IEEE, 1996), Vol. 1, pp. 378-382.
    [CrossRef]
  28. S.-S. Lin and R. Bajcsy, "High resolution catadioptric omni-directional stereo sensor for robot vision," in IEEE International Conference on Robotics and Automation (IEEE, 2003), pp. 1694-1699.
  29. H. Ishiguro, "Development of low-cost compact omnidirectional vision sensors," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer-Verlag, 2001), pp. 21-38.
  30. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Permagon, 1984).
    [PubMed]
  31. W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, 2000).
  32. C. Carathéodory, "Sitzungsberichte der Beyer," Abh. Math.-Naturwiss. Kl., Akad. Wiss. Lit., Mainz 56, 1 (1926).
  33. ZEMAX Optical Design Program User's Guide Version 10.0 (Focus Software, Incorporated, Tucson, Arizona, USA, 2002).

2006

S.-S. Lin and R. Bajcsy, "Single-view-point omnidirectional catadioptric cone mirror imager," IEEE Trans. Pattern Anal. Mach. Intell. 28, 840-845 (2006).
[CrossRef] [PubMed]

2004

Y. Yagi and M. Yachida, "Real-time omnidirectional image sensors," Int. J. Comput. Vis. , 58, 173-207 (2004).
[CrossRef]

2002

C. Geyer and K. Daniilidis, "Paracatadioptric camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

2001

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

S. Peleg, M. Ben-Ezra, and Y. Pritch, "Omnistereo: panoramic stereo imaging," IEEE Trans. Pattern Anal. Mach. Intell. 23, 279-290 (2001).
[CrossRef]

1999

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

Y. Yagi, "Omnidirectional sensing and its applications," IEICE Trans. Inf. Syst. E82D, 568-579 (1999).

1997

S. B. Kang and R. Szeliski, "3-d scene data recovery using omnidirectional multibaseline stereo," Int. J. Comput. Vis. 25, 167-183 (1997).
[CrossRef]

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

1995

D. W. Murray, "Recovering range using virtual multicamera stereo," Comput. Vis. Image Underst. 61, 285-291 (1995).
[CrossRef]

1994

Y. Yagi, S. Kawato, and S. Tsuji, "Real-time omnidirectional image sensor (COPIS) for vision-guided navigation," IEEE Trans. Rob. Autom. 10, 11-22 (1994).
[CrossRef]

1992

H. Ishiguro, M. Yamamoto, and S. Tsuji, "Omnidirectional stereo," IEEE Trans. Pattern Anal. Mach. Intell. 14, 257-262 (1992).
[CrossRef]

1926

C. Carathéodory, "Sitzungsberichte der Beyer," Abh. Math.-Naturwiss. Kl., Akad. Wiss. Lit., Mainz 56, 1 (1926).

Ahuja, N.

A. Krishna and N. Ahuja, "Panoramic image acquisition," in Proceedings of the 1996 International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 379-384.

H. Hua and N. Ahuja, "A high-resolution panoramic camera," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE2001), Vol. 1, pp. 960-967.

Bajcsy, R.

S.-S. Lin and R. Bajcsy, "Single-view-point omnidirectional catadioptric cone mirror imager," IEEE Trans. Pattern Anal. Mach. Intell. 28, 840-845 (2006).
[CrossRef] [PubMed]

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

S.-S. Lin and R. Bajcsy, "High resolution catadioptric omni-directional stereo sensor for robot vision," in IEEE International Conference on Robotics and Automation (IEEE, 2003), pp. 1694-1699.

S.-S. Lin and R. Bajcsy, "True single view point cone mirror omni-directional catadioptric system," in Proceedings of the 8th International Conference on Computer Vision (IEEE, 2001), Vol. II, pp. 102-107.

Baker, S.

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

Basu, A.

D. Southwell, A. Basu, M. Fiala, and J. Reyda, "Panoramic stereo in Proceedings of the 13th International Conference on Pattern Recognition (IEEE, 1996), Vol. 1, pp. 378-382.
[CrossRef]

D. Southwell, B. Vandegriend, and A. Basu, "A conical mirror pipeline inspection system," in Proceedings of the 1996 IEEE International Conference on Robotics and Automation (IEEE, 1996), Vol. 4, pp. 3253-3258.

Ben-Ezra, M.

S. Peleg, M. Ben-Ezra, and Y. Pritch, "Omnistereo: panoramic stereo imaging," IEEE Trans. Pattern Anal. Mach. Intell. 23, 279-290 (2001).
[CrossRef]

Benosman, R.

R. Benosman and J. Devars, "Panoramic stereovision sensor," in Proceedings of the International Conference on Pattern Recognition 1998 (International Association for Pattern Recognition, 1998), Vol. 1, pp. 767-769.

Bogner, S.

S. Bogner, "Introduction to panoramic imaging," in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetices 1995 (IEEE, 1995), pp. 3100-3106.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Permagon, 1984).
[PubMed]

Carathéodory, C.

C. Carathéodory, "Sitzungsberichte der Beyer," Abh. Math.-Naturwiss. Kl., Akad. Wiss. Lit., Mainz 56, 1 (1926).

Chahl, J. S.

Daniilidis, K.

C. Geyer and K. Daniilidis, "Paracatadioptric camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

Devars, J.

R. Benosman and J. Devars, "Panoramic stereovision sensor," in Proceedings of the International Conference on Pattern Recognition 1998 (International Association for Pattern Recognition, 1998), Vol. 1, pp. 767-769.

Fiala, M.

D. Southwell, A. Basu, M. Fiala, and J. Reyda, "Panoramic stereo in Proceedings of the 13th International Conference on Pattern Recognition (IEEE, 1996), Vol. 1, pp. 378-382.
[CrossRef]

Geyer, C.

C. Geyer and K. Daniilidis, "Paracatadioptric camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

Grossberg, M. D.

R. Swaminathan, M. D. Grossberg, and S. K. Nayar, "A perspective on distortions," in Proceedings of the 2003 International Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. II, pp. 594-601.

Hicks, R. A.

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

R. A. Hicks and R. K. Perline, "Geometric distributions for catadioptric sensor design," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 584-589.

Hlavác, V.

T. Svoboda, T. Pajdla, and V. Hlavác, "Epipolar geometry for panoramic cameras," in Proceedings of the 5th European Conference on Computer Vision (IEEE, 1998), pp. 218-232.

Hua, H.

H. Hua and N. Ahuja, "A high-resolution panoramic camera," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE2001), Vol. 1, pp. 960-967.

Ishiguro, H.

H. Ishiguro, M. Yamamoto, and S. Tsuji, "Omnidirectional stereo," IEEE Trans. Pattern Anal. Mach. Intell. 14, 257-262 (1992).
[CrossRef]

H. Ishiguro, "Development of low-cost compact omnidirectional vision sensors," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer-Verlag, 2001), pp. 21-38.

Kang, S. B.

S. B. Kang and R. Szeliski, "3-d scene data recovery using omnidirectional multibaseline stereo," Int. J. Comput. Vis. 25, 167-183 (1997).
[CrossRef]

Kawato, S.

Y. Yagi, S. Kawato, and S. Tsuji, "Real-time omnidirectional image sensor (COPIS) for vision-guided navigation," IEEE Trans. Rob. Autom. 10, 11-22 (1994).
[CrossRef]

Krishna, A.

A. Krishna and N. Ahuja, "Panoramic image acquisition," in Proceedings of the 1996 International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 379-384.

Lin, S.-S.

S.-S. Lin and R. Bajcsy, "Single-view-point omnidirectional catadioptric cone mirror imager," IEEE Trans. Pattern Anal. Mach. Intell. 28, 840-845 (2006).
[CrossRef] [PubMed]

S.-S. Lin and R. Bajcsy, "True single view point cone mirror omni-directional catadioptric system," in Proceedings of the 8th International Conference on Computer Vision (IEEE, 2001), Vol. II, pp. 102-107.

S.-S. Lin and R. Bajcsy, "High resolution catadioptric omni-directional stereo sensor for robot vision," in IEEE International Conference on Robotics and Automation (IEEE, 2003), pp. 1694-1699.

S.-S. Lin, "Omni-directional 3D Stereo computer vision sensor using reflective cone mirror," Ph.D. thesis (Computer and Information Science Department, University of Pennsylvania, 2003).

Murray, D. W.

D. W. Murray, "Recovering range using virtual multicamera stereo," Comput. Vis. Image Underst. 61, 285-291 (1995).
[CrossRef]

Nagahara, H.

H. Nagahara, Y. Yagi, and M. Yachida, "Resolution improving method for a 3D environment modeling using omnidirectional image sensor," in Proceedings of IEEE International Conference on Robotics and Automation 2002 (IEEE, 2002), Vol. 1, pp. 900-907.

Nalwa, V. S.

V. S. Nalwa, "Panoramic viewing system with offset virtual optical centers," U.S. patent 6,219,090 (17 April 2000).

Nayar, S. K.

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

R. Swaminathan, M. D. Grossberg, and S. K. Nayar, "A perspective on distortions," in Proceedings of the 2003 International Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. II, pp. 594-601.

Pajdla, T.

T. Svoboda, T. Pajdla, and V. Hlavác, "Epipolar geometry for panoramic cameras," in Proceedings of the 5th European Conference on Computer Vision (IEEE, 1998), pp. 218-232.

Peleg, S.

S. Peleg, M. Ben-Ezra, and Y. Pritch, "Omnistereo: panoramic stereo imaging," IEEE Trans. Pattern Anal. Mach. Intell. 23, 279-290 (2001).
[CrossRef]

Perline, R. K.

R. A. Hicks and R. K. Perline, "Geometric distributions for catadioptric sensor design," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 584-589.

Pritch, Y.

S. Peleg, M. Ben-Ezra, and Y. Pritch, "Omnistereo: panoramic stereo imaging," IEEE Trans. Pattern Anal. Mach. Intell. 23, 279-290 (2001).
[CrossRef]

Rees, D. W.

D. W. Rees, "Panoramic television viewing system," U.S. patent 3,505,465 (7 April 1970).

Reyda, J.

D. Southwell, A. Basu, M. Fiala, and J. Reyda, "Panoramic stereo in Proceedings of the 13th International Conference on Pattern Recognition (IEEE, 1996), Vol. 1, pp. 378-382.
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, 2000).

Southwell, D.

D. Southwell, B. Vandegriend, and A. Basu, "A conical mirror pipeline inspection system," in Proceedings of the 1996 IEEE International Conference on Robotics and Automation (IEEE, 1996), Vol. 4, pp. 3253-3258.

D. Southwell, A. Basu, M. Fiala, and J. Reyda, "Panoramic stereo in Proceedings of the 13th International Conference on Pattern Recognition (IEEE, 1996), Vol. 1, pp. 378-382.
[CrossRef]

Srinvasan, M. V.

Svoboda, T.

T. Svoboda, T. Pajdla, and V. Hlavác, "Epipolar geometry for panoramic cameras," in Proceedings of the 5th European Conference on Computer Vision (IEEE, 1998), pp. 218-232.

Swaminathan, R.

R. Swaminathan, M. D. Grossberg, and S. K. Nayar, "A perspective on distortions," in Proceedings of the 2003 International Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. II, pp. 594-601.

Szeliski, R.

S. B. Kang and R. Szeliski, "3-d scene data recovery using omnidirectional multibaseline stereo," Int. J. Comput. Vis. 25, 167-183 (1997).
[CrossRef]

Tsuji, S.

Y. Yagi, S. Kawato, and S. Tsuji, "Real-time omnidirectional image sensor (COPIS) for vision-guided navigation," IEEE Trans. Rob. Autom. 10, 11-22 (1994).
[CrossRef]

H. Ishiguro, M. Yamamoto, and S. Tsuji, "Omnidirectional stereo," IEEE Trans. Pattern Anal. Mach. Intell. 14, 257-262 (1992).
[CrossRef]

Vandegriend, B.

D. Southwell, B. Vandegriend, and A. Basu, "A conical mirror pipeline inspection system," in Proceedings of the 1996 IEEE International Conference on Robotics and Automation (IEEE, 1996), Vol. 4, pp. 3253-3258.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Permagon, 1984).
[PubMed]

Yachida, M.

Y. Yagi and M. Yachida, "Real-time omnidirectional image sensors," Int. J. Comput. Vis. , 58, 173-207 (2004).
[CrossRef]

H. Nagahara, Y. Yagi, and M. Yachida, "Resolution improving method for a 3D environment modeling using omnidirectional image sensor," in Proceedings of IEEE International Conference on Robotics and Automation 2002 (IEEE, 2002), Vol. 1, pp. 900-907.

K. Yamazawa, Y. Yagi, and M. Yachida, "Ommdirectional imaging with hyperbolodial projection," in Proceedings of the 1993 IEEE/Robotics Society of Japan International Conference on Intelligent Robots and Systems (IEEE, 1993) Vol. 2, pp. 1029-1034.
[CrossRef]

Yagi, Y.

Y. Yagi and M. Yachida, "Real-time omnidirectional image sensors," Int. J. Comput. Vis. , 58, 173-207 (2004).
[CrossRef]

Y. Yagi, "Omnidirectional sensing and its applications," IEICE Trans. Inf. Syst. E82D, 568-579 (1999).

Y. Yagi, S. Kawato, and S. Tsuji, "Real-time omnidirectional image sensor (COPIS) for vision-guided navigation," IEEE Trans. Rob. Autom. 10, 11-22 (1994).
[CrossRef]

K. Yamazawa, Y. Yagi, and M. Yachida, "Ommdirectional imaging with hyperbolodial projection," in Proceedings of the 1993 IEEE/Robotics Society of Japan International Conference on Intelligent Robots and Systems (IEEE, 1993) Vol. 2, pp. 1029-1034.
[CrossRef]

H. Nagahara, Y. Yagi, and M. Yachida, "Resolution improving method for a 3D environment modeling using omnidirectional image sensor," in Proceedings of IEEE International Conference on Robotics and Automation 2002 (IEEE, 2002), Vol. 1, pp. 900-907.

Yamamoto, M.

H. Ishiguro, M. Yamamoto, and S. Tsuji, "Omnidirectional stereo," IEEE Trans. Pattern Anal. Mach. Intell. 14, 257-262 (1992).
[CrossRef]

Yamazawa, K.

K. Yamazawa, Y. Yagi, and M. Yachida, "Ommdirectional imaging with hyperbolodial projection," in Proceedings of the 1993 IEEE/Robotics Society of Japan International Conference on Intelligent Robots and Systems (IEEE, 1993) Vol. 2, pp. 1029-1034.
[CrossRef]

Abh. Math.-Naturwiss. Kl., Akad. Wiss. Lit., Mainz

C. Carathéodory, "Sitzungsberichte der Beyer," Abh. Math.-Naturwiss. Kl., Akad. Wiss. Lit., Mainz 56, 1 (1926).

Appl. Opt.

Comput. Vis. Image Underst.

D. W. Murray, "Recovering range using virtual multicamera stereo," Comput. Vis. Image Underst. 61, 285-291 (1995).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. Peleg, M. Ben-Ezra, and Y. Pritch, "Omnistereo: panoramic stereo imaging," IEEE Trans. Pattern Anal. Mach. Intell. 23, 279-290 (2001).
[CrossRef]

H. Ishiguro, M. Yamamoto, and S. Tsuji, "Omnidirectional stereo," IEEE Trans. Pattern Anal. Mach. Intell. 14, 257-262 (1992).
[CrossRef]

C. Geyer and K. Daniilidis, "Paracatadioptric camera calibration," IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

S.-S. Lin and R. Bajcsy, "Single-view-point omnidirectional catadioptric cone mirror imager," IEEE Trans. Pattern Anal. Mach. Intell. 28, 840-845 (2006).
[CrossRef] [PubMed]

IEEE Trans. Rob. Autom.

Y. Yagi, S. Kawato, and S. Tsuji, "Real-time omnidirectional image sensor (COPIS) for vision-guided navigation," IEEE Trans. Rob. Autom. 10, 11-22 (1994).
[CrossRef]

IEICE Trans. Inf. Syst.

Y. Yagi, "Omnidirectional sensing and its applications," IEICE Trans. Inf. Syst. E82D, 568-579 (1999).

Image Vis. Comput.

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

Int. J. Comput. Vis.

S. B. Kang and R. Szeliski, "3-d scene data recovery using omnidirectional multibaseline stereo," Int. J. Comput. Vis. 25, 167-183 (1997).
[CrossRef]

Y. Yagi and M. Yachida, "Real-time omnidirectional image sensors," Int. J. Comput. Vis. , 58, 173-207 (2004).
[CrossRef]

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

Other

K. Yamazawa, Y. Yagi, and M. Yachida, "Ommdirectional imaging with hyperbolodial projection," in Proceedings of the 1993 IEEE/Robotics Society of Japan International Conference on Intelligent Robots and Systems (IEEE, 1993) Vol. 2, pp. 1029-1034.
[CrossRef]

T. Svoboda, T. Pajdla, and V. Hlavác, "Epipolar geometry for panoramic cameras," in Proceedings of the 5th European Conference on Computer Vision (IEEE, 1998), pp. 218-232.

S.-S. Lin and R. Bajcsy, "True single view point cone mirror omni-directional catadioptric system," in Proceedings of the 8th International Conference on Computer Vision (IEEE, 2001), Vol. II, pp. 102-107.

D. Southwell, A. Basu, M. Fiala, and J. Reyda, "Panoramic stereo in Proceedings of the 13th International Conference on Pattern Recognition (IEEE, 1996), Vol. 1, pp. 378-382.
[CrossRef]

S.-S. Lin and R. Bajcsy, "High resolution catadioptric omni-directional stereo sensor for robot vision," in IEEE International Conference on Robotics and Automation (IEEE, 2003), pp. 1694-1699.

H. Ishiguro, "Development of low-cost compact omnidirectional vision sensors," in Panoramic Vision: Sensors, Theory, and Applications, R.Benosman and S.B.Kang, eds. (Springer-Verlag, 2001), pp. 21-38.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Permagon, 1984).
[PubMed]

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, 2000).

V. S. Nalwa, "Panoramic viewing system with offset virtual optical centers," U.S. patent 6,219,090 (17 April 2000).

H. Hua and N. Ahuja, "A high-resolution panoramic camera," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE2001), Vol. 1, pp. 960-967.

R. Swaminathan, M. D. Grossberg, and S. K. Nayar, "A perspective on distortions," in Proceedings of the 2003 International Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. II, pp. 594-601.

D. W. Rees, "Panoramic television viewing system," U.S. patent 3,505,465 (7 April 1970).

R. A. Hicks and R. K. Perline, "Geometric distributions for catadioptric sensor design," in Proceedings of the 2001 International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 584-589.

R. Benosman and J. Devars, "Panoramic stereovision sensor," in Proceedings of the International Conference on Pattern Recognition 1998 (International Association for Pattern Recognition, 1998), Vol. 1, pp. 767-769.

A. Krishna and N. Ahuja, "Panoramic image acquisition," in Proceedings of the 1996 International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), pp. 379-384.

H. Nagahara, Y. Yagi, and M. Yachida, "Resolution improving method for a 3D environment modeling using omnidirectional image sensor," in Proceedings of IEEE International Conference on Robotics and Automation 2002 (IEEE, 2002), Vol. 1, pp. 900-907.

S. Bogner, "Introduction to panoramic imaging," in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetices 1995 (IEEE, 1995), pp. 3100-3106.

D. Southwell, B. Vandegriend, and A. Basu, "A conical mirror pipeline inspection system," in Proceedings of the 1996 IEEE International Conference on Robotics and Automation (IEEE, 1996), Vol. 4, pp. 3253-3258.

S.-S. Lin, "Omni-directional 3D Stereo computer vision sensor using reflective cone mirror," Ph.D. thesis (Computer and Information Science Department, University of Pennsylvania, 2003).

ZEMAX Optical Design Program User's Guide Version 10.0 (Focus Software, Incorporated, Tucson, Arizona, USA, 2002).

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Figures (21)

Fig. 1
Fig. 1

SVP cone mirror imaging model in the pinhole camera model explained by the concept of (a) virtual image, (b) direct ray tracing.

Fig. 2
Fig. 2

Meridional ray tracing illustration of two cone SVP configurations: SVP at (a) cone tip, (b) front focal point.

Fig. 3
Fig. 3

Multiple SVPs: projection centers and principal points of a focusing camera system in Gaussian optics model.

Fig. 4
Fig. 4

SVP cone angular FOV limits determined by the shape of the mirror and the FOV of lens camera.

Fig. 5
Fig. 5

Resolution (larger value better) variation of an example-SVP cone system in both radial (vertical, meridional) and circumferential (horizontal, sagittal) directions in relation to view angle (elevation). This is modeled after our prototype system with the lens FOV 34° and cone tip 107°; CCD chip is standard 640 × 480 so maximum r p = 240 pixel. This arrangement has omnidirectional FOV of elevation angle from 0 to 17°. The y axis value is the reciprocal of Eqs. (7, 8).

Fig. 6
Fig. 6

Deviation from SVP along optical axis (a) is robust, along the lateral direction (b) is robust.

Fig. 7
Fig. 7

(a) Meridional plane and (b) sagittal plane for world point W and the cone mirror system. (c) Reflection pattern at meridional plane and sagittal plane at the cone surface. (d) Overview. Images in (c) and (d) are actual ray trace result by ZEMAX.

Fig. 8
Fig. 8

Uncorrected system focusing plot.

Fig. 9
Fig. 9

Focusing diagrams of (a) meridional and (b) sagittal rays.

Fig. 10
Fig. 10

F/number change reduces best focus spot sizes.

Fig. 11
Fig. 11

Aperture stop placed away from the front lens improves image quality, (a) By placing the stop at the lens the rays are reflected near the tip of the cone where the surface curvature difference is greatest. (b) Placing the stop behind the front lens selects rays reflected at mirror surface points farther away from the tip. Note that the image positions do not change when stop position is changed. Both graphs have F/8.

Fig. 12
Fig. 12

The best focus that is diffraction limited and without vignetting is achieved at F/8 with stop 4.865 mm behind front lens for our test system.

Fig. 13
Fig. 13

Effects of changing mirror shape to the image blur size at image locations 0.8 3.3 mm from the center of the image plane. Setup is for front focus at the SVP.

Fig. 14
Fig. 14

Rms spot radius (image position 2.2 mm from the center of CCD) versus lens effective focal length (F/number fixed at F/8, aperture stop at lens, SVP at object side focus). Sagittal, meridional and best indicate focus settings.

Fig. 15
Fig. 15

SVP curved convex surface (hyperbolic and parabolic) mirror omnicam system (left) and non-SVP cone omnicam system (right) get better focused image when larger mirrors are used. This figure shows that the main reason for better images is the larger radius of curvature at the mirror patch that actually reflects the incoming rays. The larger curved-surface convex mirrors have larger radius of curvature everywhere. Cone mirrors are different. It is the shift of reflecting point caused by the camera moving farther away from the cone mirror that is causing the change of radius of curvature at the reflection point.

Fig. 16
Fig. 16

(a) Real lens ray tracing using Edmund Optics 6 mm lens, stock no. 54852. (b) Real lens ray tracing results adjusted to the optimum configuration. Here we use F/16 as a compromise before the diffraction effects grow too large.

Fig. 17
Fig. 17

LSF along tangential and sagittal directions.

Fig. 18
Fig. 18

(a) MTF curves (T=tangential, S=sagittal) of meridional focus at the rim of the image. (b) MTF curves of best focus at the rim of the image. (c) MTF curves of four image positions at the meridional focus. (d) MTF curves of four image positions at the best focus. (e) MTF of the normal lens (Edmund no. 54852).

Fig. 19
Fig. 19

(a) SVP parabolic mirror MTF at the image plane. (b) The same MTF adjusted for original object spatial frequency.

Fig. 20
Fig. 20

Robustness tests. (a) Precise SVP image. (b) Experiment setup. (c) Subtraction image with camera moved 0.025 in. back longitudinally. (d) Subtraction image taken with camera moved 0.025 in. laterally (toward the bottom side of the picture). (e) Subtraction result of an image digitally shifted 10 pixels down. Note: subtraction image means the image obtained by subtracting the precise SVP image from the image taken at the relocated camera positions. All subtraction images are histogram enhanced; otherwise they appear almost entirely black.

Fig. 21
Fig. 21

(a) Best focus, f = 6.5 mm , F/2, 1 800 s . (b) Best focus, f = 6.5 mm , F/10, 1 30 sec . (c) Unwarped from the omniview in (b) (azimuth, −160°; elevation, −6°; same camera parameters as the original). (d) Another unwarped view from (b) (azimuth, elevation same as in (c)).

Tables (1)

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Table 1 Image Displacement Caused by Camera Offset from the SVP Position a

Equations (33)

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1 s o + 1 s i = 1 f .
S o h y p e r f o c a l = f 2 ( A d ) = f D d .
r p = f p tan ( θ π 2 Φ ) ,
d Λ d ν = ( r d Θ d r ) ( cos Φ d Θ d Φ ) = ( csc Φ r d r d Φ ) ,
d r d Φ = f [ 1 + tan 2 ( θ c o n e _ t i p π 2 Φ ) ] = f + r 2 f .
d Λ d ν = f r [ 1 + ( r f ) 2 ] cos Φ .
360 ° ( 2 π r p ) = 180 ° π f p tan ( θ π 2 Φ ) .
180 ° π f [ 1 + ( r f ) 2 ] = 180 ° π f [ 1 + tan 2 ( θ π 2 Φ ) ] .
x f = H z
x f = H ( z + b )
( x x ) = ( f f ) ( ( b z ) + 1 ) .
x x = ( 1 { 1 [ ( b z ) + 1 ] } ) x
{ 1 1 [ ( b z ) + 1 ] } R max < 1 ,
z > b ( R max 1 ) b R max .
x f = H z
x f = ( H d ) z ,
x x = f d z .
z > f d s = f pix d ,
y a = x 2 4 a
y b = x 2 4 b ,
y = k x .
m Y = k m X Y = k X .
G ( x ) = U + A cos ( 2 π ν x ) ,
F ( x ) = LSF ( δ ) G ( x δ ) d δ LSF ( δ ) d δ .
F ( x ) = U + A T ( ν ) cos ( 2 π ν x ϕ ) = U + A T c ( ν ) cos ( 2 π ν x ϕ ) + A T s ( ν ) sin ( 2 π ν x ϕ ) ,
T ( ν ) = [ T c ( ν ) 2 + T s ( ν ) 2 ] 1 2
T c ( ν ) = LSF ( δ ) cos ( 2 π ν δ ) d δ LSF ( δ ) d δ ,
T s ( ν ) = LSF ( δ ) sin ( 2 π ν δ ) d δ LSF ( δ ) d δ ,
tan ϕ = T s ( ν ) T c ( ν ) .
M = Max Min Max + Min ,
M o = U + A ( U A ) U + A + U A = A U ,
M i = A U T ( ν ) = M o T ( ν ) .
MTE ( ν ) = T ( ν ) = M i M o .

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