Advertisement

International Journal of Computer Vision

, Volume 45, Issue 3, pp 223–243 | Cite as

Catadioptric Projective Geometry

  • Christopher Geyer
  • Kostas Daniilidis
Article

Abstract

Catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera. Central catadioptric sensors are the class of these devices having a single effective viewpoint. In this paper, we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications. We show that a central catadioptric projection is equivalent to a two-step mapping via the sphere. The second step is equivalent to a stereographic projection in the case of parabolic mirrors. Conventional lens-based perspective cameras are also central catadioptric devices with a virtual planar mirror and are, thus, covered by the unifying model. We prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images (conics). It turns out that planar and parabolic mirrors build a dual catadioptric projection pair. As a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system.

omnidirectional vision catadioptric systems conic sections duality stereographic projection calibration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baker, S. and Nayar, S. 1988. A theory of catadioptric image formation. In Proc. Int. Conf. on Computer Vision, Bombay, India, Jan. 3–5, 1998, pp. 35–42.Google Scholar
  2. Baker, S. and Nayar, S. 1999. A theory of single-viewpoint catadioptric image formation. International Journal of Computer Vision, 35:175–196.Google Scholar
  3. Benosman, R., Deforas, E., and Devars, J. 2000. A new catadioptric sensor for panoramic vision of mobile robots. In IEEE Workshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, pp. 112–118.Google Scholar
  4. Benosman, R. and Kang, S. 2000. Panoramic Vision. Springer-Verlag: New York, Berlin, Heidelberg.Google Scholar
  5. Bogner, S. 1995. An introduction to panospheric imaging. In Proc. IEEE Conf. Systems, Man, and Cybernetics, Vancouver, BC, pp. 3099–3116.Google Scholar
  6. Boult, T. 1998. Remote reality demonstration. In IEEE Conf. Computer Vision and Pattern Recognition, Santa Barbara, CA, June 23–25, 1998, pp. 966–967.Google Scholar
  7. Bruckstein, A. and Richardson, T. 2000. Omniview cameras with curved surface mirrors. In IEEE Workshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, originally published as Bell Labs Technical Memo, 1996, pp. 79–86.Google Scholar
  8. Chahl, J. and Srinivasan, M. 1997. Reflective surfaces for panoramic imaging. Applied Optics, 36:8275–8285.Google Scholar
  9. Daniilidis, K. (Ed.). 2000. IEEE Workshop on Omnidirectional Vision, Hilton Head Island, SC, June 12, 2000.Google Scholar
  10. Geyer, C. and Daniilidis, K. 1999. Catadioptric camera calibration. In Proc. Int. Conf. on Computer Vision, Kerkyra, Greece, Sept. 20–23, 1999, pp. 398–404.Google Scholar
  11. Greguss, P. 1985. The tube-peeper: A new concept in endoscopy Optics and Laser Technology, 32:41–45.Google Scholar
  12. Hartley, R. 2000. Chirality. International Journal of Computer Vision, 26:41–61.Google Scholar
  13. Hecht, E. and Zajac, A. 1997. Optics (3rd edn.). Addison-Wesley: Reading, MA.Google Scholar
  14. Hicks, R. and Bajcsy, R. 2000. Catadioptric sensors that approximate wide-angle perspective projections. In IEEE Conf. Computer Vision and Pattern Recognition, Hilton Head Island, SC, June 13–15, 2000, pp. 545–551.Google Scholar
  15. Hong, J., Tan, X., Weiss, R., and Riseman, E. 1991. Image-based homing. In IEEE Int. Conf. Robotics and Automation, pp. 620–625.Google Scholar
  16. Kang, S. 2000. Catadioptric self-calibration. In IEEE Conf. Computer Vision and Pattern Recognition, Hilton Head Island, SC, June 13–15, 2000, pp. I-201–207.Google Scholar
  17. Land, M. 1981. Optics and vision. in inverterbrates. In Handbook of Sensory Physiology, Autrum, H. (Ed.). Vol. VII/6B, Springer Verlag: Berlin, ch. 4, pp. 472–585.Google Scholar
  18. Laveau, S. and Faugeras, O. 1996. Oriented projective geometry in computer vision. In Proc. Fourth European Conference on Computer Vision, Cambridge, UK, April 14–18, 1996, B. Buxton (Ed.). Springer: Berlin, pp. 147–156. LNCS, 1064.Google Scholar
  19. Leonardis, A. and Jogan, M. 2000. Robust localization using eigenspace of spinning-images. In IEEE Workshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, pp. 37–46.Google Scholar
  20. Majumder, A., Gopi, M., Seales, B., and Fuchs, H. 1999. Immersive teleconferencing: A new algorithm to generate seamless panoramic video imagery. In Proceedings of the Seventh ACM International Conference on Multimedia, pp. 169–178.Google Scholar
  21. Nalwa, V. 1996. A true omnidirectional viewer. Technical report, Bell Labs, Holmdel, NJ.Google Scholar
  22. Nayar, S. 1997. Catadioptric omnidirectional camera. In IEEE Conf. Computer Vision and Pattern Recognition, Puerto Rico, June 17–19, 1997, pp. 482–488.Google Scholar
  23. Nayar, S. and Peri, V. 1999. Folded catadioptric cameras. In IEEE Conf. Computer Vision and Pattern Recognition, Fort Collins, CO, June 23–25, 1999, pp. 217–225.Google Scholar
  24. Needham, T. 1997. Visual Complex Analysis. Clarendon Press: Oxford.Google Scholar
  25. Nene, S. and Nayar, S. 1998. Stereo with mirrors. In Proc. Int. Conf. on Computer Vision, Bombay, India, Jan. 3–5, 1998, pp. 1087–1094.Google Scholar
  26. Onoe, Y., Yamazawa, K., Takemura, H., and Yokoya, N. 1998. Telepresence by real-time view-dependent image generation from omnidirectional video streams. Computer Vision and Image Understanding, 71:588–592.Google Scholar
  27. Pajdla, T., Werner, T., and Hlavac, V. 1998. Oriented projective reconstruction. In Proc. Austrian Association for Pattern Recognition.Google Scholar
  28. Penrose, R. and Rindler, W. (Eds). 1984. Spinors and Space-Time. Cambridge University Press: Cambridge, UK.Google Scholar
  29. Rees, D.W. 1971. Panoramic television viewing system. United States Patent No. 3, 505, 465, April 1970.Google Scholar
  30. Shah, S. and Aggarwal, J. 1996. Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation. Pattern Recognition, 29:1775–1788.Google Scholar
  31. Shum, H.-Y. and Szeliski, R. 2000. Systems and experiment paper: Construction of panoramic image mosaics with global and local alignment. International Journal of ComputerVision, 36:101–130.Google Scholar
  32. Stolfi, J. 1991. Oriented Projective Geometry. Academic Press: Boston.Google Scholar
  33. Sturm, P. 2000. A method for 3D-reconstruction of piecewise planar objects from single panoramic images. In IEEEWorkshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, pp. 119–126.Google Scholar
  34. Svoboda, T., Pajdla, T., and Hlavac, V. 1998. Epipolar geometry for panoramic cameras. In Proc. 5th European Conference on Computer Vision, pp. 218–231.Google Scholar
  35. Swaminathan, R. and Nayar, S. 2000. Non-metric calibration of wide-angle lenses and polycameras. In IEEE Conf. Computer Vision and Pattern Recognition, Hilton Head Island, SC, June 13–15, 2000, pp. II-413–419.Google Scholar
  36. Taylor, C. 2000. Video plus. In IEEE Workshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, pp. 3–10.Google Scholar
  37. Toomer, G. 1976. Diocles On Burning Mirrors. Sources in the History of Mathematics and the Physical Sciences. Springer-Verlag: Berlin.Google Scholar
  38. Winters, N., Gaspar, J., Lacey, G., and Santos-Victor, J. 2000. Omnidirectional vision for navigation In IEEE Workshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, pp. 21–28.Google Scholar
  39. Yagi, Y. 1999. Omnidirectional sensing and its application. IEICE Trans. Inform. and Systems, 3:568–579.Google Scholar
  40. Yagi, Y., Kawato, S., and Tsuji, S. 1994. Real-time omnidirectional image sensor (COPIS) for vision-guided navigation. Trans. on Robotics and Automation, 10:11–22.Google Scholar
  41. Zheng, J. and Tsuji, S. 1992. Panoramic representation for route recognition by a mobile robot. International Journal of Computer Vision, 9:55–76.Google Scholar
  42. Zhu, Z., Rajasekar, K., Riseman, E., and Hanson, A. 2000. Panoramic virtual stereo vision of cooperative mobile robots for localizing 3D moving objects. In IEEE Workshop on Omnidirectional Vision, Hilton Head, SC, June 12, 2000, pp. 29–36.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Christopher Geyer
    • 1
  • Kostas Daniilidis
    • 1
  1. 1.GRASP LaboratoryUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations