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A Theory of Single-Viewpoint Catadioptric Image Formation

Abstract

Conventional video cameras have limited fields of view which make them restrictive for certain applications in computational vision. A catadioptric sensor uses a combination of lenses and mirrors placed in a carefully arranged configuration to capture a much wider field of view. One important design goal for catadioptric sensors is choosing the shapes of the mirrors in a way that ensures that the complete catadioptric system has a single effective viewpoint. The reason a single viewpoint is so desirable is that it is a requirement for the generation of pure perspective images from the sensed images. In this paper, we derive the complete class of single-lens single-mirror catadioptric sensors that have a single viewpoint. We describe all of the solutions in detail, including the degenerate ones, with reference to many of the catadioptric systems that have been proposed in the literature. In addition, we derive a simple expression for the spatial resolution of a catadioptric sensor in terms of the resolution of the cameras used to construct it. Moreover, we include detailed analysis of the defocus blur caused by the use of a curved mirror in a catadioptric sensor.

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References

  • Adelson, E.H. and Bergen, J.R. 1991. The plenoptic function and elements of early vision. In Computational Models of Visual Processing, chap. 1, Landy and Movshon (Eds.), MIT Press.

  • Baker, S. and Nayar, S.K. 1998. A theory of catadioptric image formation. In Proceedings of the 6th Internation Conference on Computer Vision, Bombay, India, IEEE Computer Society, pp. 35–42.

  • Bogner, S. 1995. Introduction to panoramic imaging. In Proceedings of the IEEE SMC Conference, pp. 3100–3106.

  • Born, M. and Wolf, E. 1965. Principles of Optics. Permagon Press.

  • Chahl, J.S. and Srinivassan, M.V. 1997. Reflective surfaces for panoramic imaging. Applied Optics, 36(31):8275_8285.

    Google Scholar 

  • Charles, J.R., Reeves, R., and Schur, C. 1987. How to build and use an all-sky camera. Astronomy Magazine, April.

  • Drucker, D. and Locke, P. 1996. A natural classification of curves and surfaces with reflection properties. Mathematics Magazine, 69(4):249–256.

    Google Scholar 

  • Gortler, S.J., Grzeszczuk, R., Szeliski, R., and Cohen, M. 1996. The lumigraph. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, pp. 43–54.

  • Goshtasby, A. and Gruver, W.A. 1993. Design of a single-lens stereo camera system. Pattern Recognition, 26(6):923–937.

    Google Scholar 

  • Hecht, E. and Zajac, A. Optics. Addison-Wesley.

  • Hong, J. 1991. Image based homing. In Proceedings of the IEEE International Conference on Robotics and Automation.

  • Inaba, M., Hara, T., and Inoue, H. 1993. A stereo viewer based on a single camera with view-control mechanism. In Proceedings of the International Conference on Robots and Systems.

  • Murphy, J.R. 1995. Application of panoramic imaging to a teleoperated lunar rover. In Proceedings of the IEEE SMC Conferences, pp. 3117–3121.

  • Nalwa, V.S. 1996. A true omnidirectional viewer. Technical Report, Bell Laboratories, Holmdel, NJ 07733, USA.

    Google Scholar 

  • Nayar, S.K. 1988. Sphereo: Recovering depth using a single camera and two specular spheres. In Proceedings of SPIE: Optics, Illumination, and Image Sensing for Machine Vision II.

  • Nayar, S.K. 1997a. Catadioptric omnidirectional camera. In Proceedings of the 1997 Conference on Computer Vision and Pattern Recognition, pp. 482–488.

  • Nayar, S.K. 1997b. Omnidirectional video camera. In Proceedings of the 1997 DARPA Image Understanding Workshop.

  • Nayar, S.K. and Baker, S. 1997. Catadioptric image formation. In Proceedings of the 1997 DARPA Image UnderstandingWorkshop, New Orleans, Louisiana, pp. 1431–1437.

  • Nene, S.A. and Nayar, S.K. 1998. Stereo with mirrors. In Proceedings of the 6th Internation Conference on Computer Vision, Bombay, India, IEEE Computer Society.

  • Peri, V. and Nayar, S.K. 1997. Generation of perspective and panoramic video from omnidirectional video. In Proceedings of the 1997 DARPA Image Understanding Workshop, New Orleans. Rees, D.W. 1970. Panoramic television viewing system. United States Patent No. 3,505,465.

  • Yagi, Y. and Kawato, S. 1990. Panoramic scene analysis with conic projection. In Proceedings of the International Conference on Robots and Systems.

  • Yagi, Y., Kawato, S., and Tsuji, S. 1994. Real-time omnidirectional image sensor (COPIS) for vision-guided navigation. IEEE Transactions on Robotics and Automation, 10(1):11–22.

    Google Scholar 

  • Yagi, Y. and Yachida, M. 1991. Real-time generation of environmental map and obstacle avoidance using omnidirectional image sensor with conic mirror. In Proceedings of the 1991 Conference on Computer Vision and Pattern Recognition, pp. 160–165.

  • Yamazawa, K., Yagi, Y., and Yachida, M. 1993. Omnidirectional imaging with hyperboloidal projection. In Proceedings of the International Conference on Robots and Systems.

  • Yamazawa, K., Yagi, Y., and Yachida, M. 1995. Obstacle avoidance with omnidirectional image sensor HyperOmni Vision. In Proceedings of the IEEE International Conference on Robotics and Automation, pp. 1062–1067.

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Baker, S., Nayar, S.K. A Theory of Single-Viewpoint Catadioptric Image Formation. International Journal of Computer Vision 35, 175–196 (1999). https://doi.org/10.1023/A:1008128724364

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  • DOI: https://doi.org/10.1023/A:1008128724364

  • image formation
  • sensor design
  • sensor resolution
  • defocus blur
  • omnidirectional imaging
  • panoramic imaging