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Non-Single Viewpoint Catadioptric Cameras: Geometry and Analysis

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Abstract.

Conventional vision systems and algorithms assume the imaging system to have a single viewpoint. However, these imaging systems need not always maintain a single viewpoint. For instance, an incorrectly aligned catadioptric system could cause non-single viewpoints. Moreover, a lot of flexibility in imaging system design can be achieved by relaxing the need for imaging systems to have a single viewpoint. Thus, imaging systems with non-single viewpoints can be designed for specific imaging tasks, or image characteristics such as field of view and resolution. The viewpoint locus of such imaging systems is called a caustic.

In this paper, we present an in-depth analysis of caustics of catadioptric cameras with conic reflectors. We use a simple parametric model for both, the reflector and the imaging system, to derive an analytic solution for the caustic surface. This model completely describes the imaging system and provides a map from pixels in the image to their corresponding viewpoints and viewing direction. We use the model to analyze the imaging system's properties such as field of view, resolution and other geometric properties of the caustic itself. In addition, we present a simple technique to calibrate the class of conic catadioptric cameras and estimate their caustics from known camera motion. The analysis and results we present in this paper are general and can be applied to any catadioptric imaging system whose reflector has a parametric form.

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References

  • Arnold, V. 1978. Mathematical Methods of Classical Mechanics. Springer-Verlag.

  • Baker, S. and Nayar, S.K. 1998. A theory of catadioptric image formation. In Proc. ICCV, pp. 35–42.

  • Baker, S. and Nayar, S.K. 1999. A theory of single-viewpoint catadioptric image formation. IJCV, 35(2):1–22.

    Article  Google Scholar 

  • Bogner, S. 1995. Introduction to panoramic imaging. In IEEE SMC Conference, vol. 54, pp. 3100–3106.

    Google Scholar 

  • Bolles, R.C., Konolige, K.G., and Fischler, M.A. 1997. Extra set of eyes. In DARPA-IUW, pp. 41–44.

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

  • Bruce, J.W., Giblin, P.J., and Gibson, C.G. 1981. On caustics of plane curves. American Mathematical Monthly, 88:651–667.

    MathSciNet  Google Scholar 

  • Burkhard, D.G. and Shealy, D.L. 1973. Flux density for ray propoagation in gemoetrical optics. Journal of the Optical Society of America, 63(3):299–304.

    Google Scholar 

  • Chahl, J. and Srinivasan, M. 1997. Reflective surfaces for panoramic imaging. Applied Optics, 36(31):8275–8285.

    Google Scholar 

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

  • Derrien, S. and Konolige, K. 2000. Aproximating a single viewpoint in panoramic imaging devices. In International Conference on Robotics and Automation, pp. 3932–3939.

  • Gachter, S., Pajdla, T., and Micusik, B. 2001. Mirror design for an omnidirectional camera with space variant imager. In Overviews of the Workshops on Omnidirectional Vision, pp. 99–105.

  • Gaspar, J., Decco, C., Okamoto, J. Jr., and Santos-Victor, J. 2002. Constant resolution omnidirectional cameras. In Proc. OMNIVIS, p. 27.

  • Geyer, C. and Daniilidis, K. 1999. Catadioptric camera calibration. In Proc. ICCV, pp. 398–404.

  • Gluckman, J. and Nayar, S.K. 1999. Planar catadioptric stereo: Geometry and calibration. In Proc. CVPR, pp. I:22–28.

    Google Scholar 

  • Grossberg, M. and Nayar, S. 2001. A general imaging model and a method for finding its parameters. In Proc. ICCV, pp. 108–115.

  • Hamilton, W.R. 1828. Theory of systems of rays. Transactions of the Royal Irish Academy, 15:69–174.

    Google Scholar 

  • Hicks, A. 2002. Differential methods in catadioptric sensor design with applications to panoramic imaging. Technical report, Drexel University, Computer Science.

  • Hicks, R. and Bajcsy, R. 2000. Catadioptric sensors that approximate wide-angle perspective projections. In Proc. CVPR, pp. I:545–551.

    Google Scholar 

  • Hicks, R. and Perline, R. 2002. Equi-areal catadioptric sensors. In Proc. OMNIVIS, p. 13.

  • Hong, J.W., Tan, X., Pinette, B., Weiss, R., and Riseman, E.M. 1990. Image-based navigation using 360 views. In DARPA-IUW, pp. 782–791.

  • Jensen, H.W. 1996. In http://graphics.stanford.edu/henrik/images/caustics.html

  • Kang, S.B. 2000. Catadioptric self-calibration. In Proc. CVPR, pp. I:201–207.

    Google Scholar 

  • Murphy, J. 1997. Application of panoramic imaging to a teleoperated lunar rover. In IEEE SMC Conference, vol. 36, pp. 3117–3121.

    Google Scholar 

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

  • Nayar, S.K. 1997. Catadioptric omnidirectional cameras. In Proc. CVPR, pp. 482–488.

  • Nayar, S.K. and Karmarkar, A.D. 2000. 360 × 360 Mosaics. In Proc. CVPR, pp. I:388–395.

    Google Scholar 

  • Pajdla, T. 2001. Epipolar geometry of some non-classical cameras. In Computer Vision Winter Workshop, pp. 223–233.

  • Pajdla, T. 2002. Stereo with oblique cameras. Trans. IJCV, 47(1–3):161–170.

    MATH  Google Scholar 

  • Peleg, S., Pritch, Y., and Ben-Ezraet, M. 2000. Cameras for stereo panoramic imaging. In Proc. CVPR, pp. I:208–214.

    Google Scholar 

  • Peri, V.N. and Nayar, S.K. 1997. Generation of perspective and panoramic video from omnidirectional video. DARPA-IUW, pp. I:243–245.

    Google Scholar 

  • Rees, D. 1970. Panoramic television viewing system. United States Patent No.3,505,465.

  • Seitz, S. 2001. The space of all stereo images. In Proc. ICCV, pp. I:26–33.

  • Srinivasan, M. 2003. New class of mirrors for wide-angle imaging. In Proc. OMNIVIS.

  • Swaminathan, R., Grossberg, M., and Nayar, S. 2003. A perspective on distortions. In Proc. CVPR, pp. II: 594–601.

    Google Scholar 

  • Swaminathan, R., Grossberg, M.D., and Nayar, S.K. 2001. Caustics of catadioptric cameras. In Proc. ICCV, pp. II:2–9.

    Google Scholar 

  • Swaminathan, R., Grossberg, M.D., and Nayar, S.K. 2001. Non-single viewpoint catadioptric cameras: Geometry and analysis. Technical Report CUCS-004-01, Dept. of Computer Science, Columbia University.

  • Weinshall, D., Lee, M., Brodsky, T., Trajkovic, M., and Feldman, D. 2002. New view generation with a bi-centric camera. In Proc. ECCV (1), pp. 614–628.

  • Yagi, Y., Kawato, S., and Tsuji, S. 1994. Real-time omnidirectional image sensor (copis) for vision-guided navigation. 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 Proc. CVPR, pp. 160–165.

  • Yamazawa, K., Yagi, Y., and Yachida, M. 1993, Omnidirectional imaging with hyperboloidal projection. In Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1029–1034.

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Correspondence to Rahul Swaminathan.

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Swaminathan, R., Grossberg, M.D. & Nayar, S.K. Non-Single Viewpoint Catadioptric Cameras: Geometry and Analysis. Int J Comput Vision 66, 211–229 (2006). https://doi.org/10.1007/s11263-005-3220-1

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  • DOI: https://doi.org/10.1007/s11263-005-3220-1

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