This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Catadioptric Camera Calibration Using Geometric Invariants
October 2004 (vol. 26 no. 10)
pp. 1260-1271
Central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint. In this paper, we propose a novel method for the calibration of central catadioptric cameras using geometric invariants. Lines and spheres in space are all projected into conics in the catadioptric image plane. We prove that the projection of a line can provide three invariants whereas the projection of a sphere can only provide two. From these invariants, constraint equations for the intrinsic parameters of catadioptric camera are derived. Therefore, there are two kinds of variants of this novel method. The first one uses projections of lines and the second one uses projections of spheres. In general, the projections of two lines or three spheres are sufficient to achieve catadioptric camera calibration. One important conclusion in this paper is that the method based on projections of spheres is more robust and has higher accuracy than that based on projections of lines. The performances of our method are demonstrated by both the results of simulations and experiments with real images.

[1] D. Aliaga, "Accurate Catadioptric Calibration for Real-Time Pose Estimation in Room-Size Environments," Proc. IEEE Int'l Conf. Computer Vision (ICCV 01), IEEE CS Press, 2001, pp. 127-134.
[2] S. Baker and S. Nayar, A Theory of Single-Viewpoint Catadioptric Image Formation Int'l J. Computer Vision, vol. 35, no. 2, pp. 175-196, 1999.
[3] J. Barreto and H. Araújo, Geometric Properties of Central Catadioptric Line Images Proc. Seventh European Conf. Computer Vision, pp. 237-251, 2002.
[4] J. Barreto and H. Araújo, Direct Least Square Fitting of Paracatadioptric Line Images Proc. IEEE. Conf. Computer Vision and Pattern Recognition Workshop Omnidirectional Vision and Camera Networks, vol. VII, p. 78, 2003.
[5] S. Bogner, Introduction to Panoramic Imaging Proc. IEEE Int'l Conf. Systems Man and Cybernetics, pp. 3100-3106, 1995.
[6] D. Brown, Close Range Camera Calibration Photogrammetric Eng., vol. 37, no. 8, pp. 855-866, 1971.
[7] N. Daucher, M. Dhome, and J. Lapresté, Camera Calibration from Spheres Images Proc. Third European Conf. Computer Vision, pp. 449-454, 1994.
[8] A. Fitzgibbon and R. Fisher, A Buyer's Guide to Conic Fitting Proc. Sixth British Machine Vision Conf., pp. 513-522, 1995.
[9] A. Fitzgibbon, M. Pilu, and R.B. Fisher, “Direct Least Square Fitting of Ellipses,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 5, pp. 476-480, May 1999.
[10] C. Geyer and K. Daniilidis, “Catadioptric Camera Calibration,” Proc. Int'l Conf. Computer Vision, pp. 398-404, 1999.
[11] C. Geyer and K. Daniilidis, Catadioptric Projective Geometry Int'l J. Computer Vision, vol. 45, no. 3, pp. 223-243, 2001.
[12] C. Geyer and K. Daniilidis, Structure and Motion from Uncalibrated Catadioptric Views Proc. IEEE. Conf. Computer Vision and Pattern Recognition, vol. I, pp. 279-286, 2001.
[13] C. Geyer and K. Daniilidis, Paracatadioptric Camera Calibration IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 687-695, May 2002.
[14] M.D. Grossberg and S.K. Nayar, A General Imaging Model and a Method for Finding Its Parameters Proc. IEEE Int'l Conf. Computer Vision, vol. 2, pp. 108-115, July 2001.
[15] J. Hong, X. Tan, R. Weiss, and E. Riseman, Image-Based Homing Proc. IEEE. Conf. Robotics and Automation, pp. 620-625, 1991.
[16] S. Kang, “Catadioptric self-calibration,” IEEE Conf. Computer Vision and Pattern Recognition, pp. I-201-207, June 2000.
[17] V. Nalwa, A True Omnidirectional Viewer technical report, Bell Labs, Holmdel, 1996.
[18] S. Nene and S. Nayar, Stereo with Mirrors Proc. Sixth Int'l Conf. Computer Vision, pp. 1087-1094, 1998.
[19] T. Pajdla, T. Svoboda, and V. Hlavac, Epipolar Geometry of Central Panoramic Cameras Panoramic Vision: Sensors, Theory, and Applications, R. Benosman and S.B. Kang, eds., pp. 85-114, 2001.
[20] M. Penna, Camera Calibration: A Quick and Easy Way to Determine the Scale Factor IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 12, pp. 1240-1245, Dec. 1991.
[21] L. Quan, Conic Reconstruction and Correspondence from Two Views IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 2, pp. 151-160, Feb. 1996.
[22] J. Semple and G. Kneebone, Algebraic Projective Geometry. Oxford Science, 1952.
[23] G. Stein, Internal Camera Calibration Using Rotation and Geometric Shapes master's thesis, Artificial Intelligence Lab, MIT, 1993.
[24] T. Svoboda, T. Padjla, and V. Hlavac, Epipolar Geometry for Panoramic Cameras Int'l J. Computer Vision, vol. 49, no. 1, pp. 23-37, 2002.
[25] R. Swaminathan and S.K. Nayar, Nonmetric Calibration of Wide-Angle Lenses and Polycameras IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 10, Oct. 2000.
[26] R. Swaminathan, M.D. Grossberg, and S.K. Nayar, Caustics of Catadioptric Cameras Proc. IEEE Int'l Conf. Computer Vision, vol. 2, pp. 2-9, July 2001.
[27] Y. Yagi and S. Kawato, Panorama Scene Analysis with Conic Projection Proc. IEEE Workshop Intelligent Robots and Systems, pp. 181-187, 1990.
[28] Z. Zhang, Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting Image and Vision Computing, vol. 15, no. 1, pp. 59-76, 1997.

Index Terms:
Camera calibration, catadioptric camera, geometric invariant, omnidirectional vision, panoramic vision.
Citation:
Xianghua Ying, Zhanyi Hu, "Catadioptric Camera Calibration Using Geometric Invariants," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 10, pp. 1260-1271, Oct. 2004, doi:10.1109/TPAMI.2004.79
Usage of this product signifies your acceptance of the Terms of Use.