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Structure from Motion with Wide Circular Field of View Cameras
July 2006 (vol. 28 no. 7)
pp. 1135-1149
This paper presents a method for fully automatic and robust estimation of two-view geometry, autocalibration, and 3D metric reconstruction from point correspondences in images taken by cameras with wide circular field of view. We focus on cameras which have more than 180^\circ field of view and for which the standard perspective camera model is not sufficient, e.g., the cameras equipped with circular fish-eye lenses Nikon FC-E8 (183^\circ), Sigma 8mm-f4-EX (180^\circ), or with curved conical mirrors. We assume a circular field of view and axially symmetric image projection to autocalibrate the cameras. Many wide field of view cameras can still be modeled by the central projection followed by a nonlinear image mapping. Examples are the above-mentioned fish-eye lenses and properly assembled catadioptric cameras with conical mirrors. We show that epipolar geometry of these cameras can be estimated from a small number of correspondences by solving a polynomial eigenvalue problem. This allows the use of efficient RANSAC robust estimation to find the image projection model, the epipolar geometry, and the selection of true point correspondences from tentative correspondences contaminated by mismatches. Real catadioptric cameras are often slightly noncentral. We show that the proposed autocalibration with approximate central models is usually good enough to get correct point correspondences which can be used with accurate noncentral models in a bundle adjustment to obtain accurate 3D scene reconstruction. Noncentral camera models are dealt with and results are shown for catadioptric cameras with parabolic and spherical mirrors.

[1] D.G. Aliaga, “Accurate Catadioptric Calibration for Real-Time Pose Estimation in Room-Size Environments,” Proc. IEEE Int'l Conf. Computer Vision, pp. 127-134, 2001.
[2] Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst eds. Philadelphia: SIAM, 2000.
[3] S. Baker and S.K. Nayar, “A Theory of Single-Viewpoint Catadioptric Image Formation,” Int'l J. Computer Vision, vol. 35, no. 2, pp. 175-196, 1999.
[4] H. Bakstein and T. Pajdla, “Panoramic Mosaicing with a $180^\circ$ Field of View Lens,” Proc. IEEE Workshop Omnidirectional Vision, pp. 60-67, 2002.
[5] J.P. Barreto and H. Araujo, “Issues on the Geometry of Central Catadioptric Image Formation,” Proc. Conf. Computer Vision and Pattern Recognition, pp. II:422-427, 2001.
[6] Panoramic Vision: Sensors, Theory, and Applications, R. Benosman and S.B. Kang, eds. New York: Springer Verlag, 2001.
[7] C. Bräuer-Burchardt and K. Voss, “A New Algorithm to Correct Fish-Eye- and Strong Wide-Angle-Lens-Distortion from Single Images,” Proc. IEEE Conf. Image Processing, pp. 225-228, 2001.
[8] T. Brodsky, C. Fermuller, and Y. Aloimonos, “Directions of Motion Fields Are Hardly Ever Ambiguous,” Proc. European Conf. Computer Vision, 1996.
[9] S. Derrien and K. Konolige, “Approximating a Single Viewpoint in Panoramic Imaging Devices,” Proc. IEEE Workshop Omnidirectional Vision, pp. 85-90, 2000.
[10] O. Faugeras, Q. Tuan Luong, and T. Papadopoulo, The Geometry of Multiple Images: The Laws that Govern the Formation of Multiple Images of a Scene and Some of Their Applications. Cambridge, Mass.: MIT Press, 2001.
[11] C. Fermuller and Y. Aloimonos, “Ambiguity in Structure from Motion: Sphere vs. Plane,” Int'l J. Computer Vision, vol. 28, pp. 137-154, 1998.
[12] A. Fitzgibbon, “Simultaneous Linear Estimation of Multiple View Geometry and Lens Distortion,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. I, pp. 125-132, 2001.
[13] M.M. Fleck, “Perspective Projection: The Wrong Imaging Model,” Technical Report TR 95-01, Dept. of Computer Science, Univ. of Iowa, 1995.
[14] S. Gächter, T. Pajdla, and B. Mičušík, “Mirror Design for an Omnidirectional Camera with a Space Variant Imager,” Proc. IEEE Workshop Omnidirectional Vision, pp. 99-105, 2001.
[15] C. Geyer and K. Daniilidis, “A Unifying Theory for Central Panoramic Systems and Practical Applications,” Proc. European Conf. Computer Vision, 2000.
[16] C. Geyer and K. Daniilidis, “Structure and Motion from Uncalibrated Catadioptric Views,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 279-286, 2001.
[17] C. Geyer and K. Daniilidis, “Para-Catadioptric Camera Calibration,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 687-695, May 2002.
[18] C. Geyer and K. Daniilidis, “Mirrors in Motion: Epipolar Geometry and Motion Estimation,” Proc. IEEE Int'l Conf. Computer Vision, pp. 766-773, 2003.
[19] C. Geyer and K. Daniilidis, “Omnidirectional Video,” The Visual Computer, vol. 19, no. 6, pp. 405-416, 2003.
[20] J. Gluckman and S.K. Nayar, “Ego-Motion and Omnidirectional Cameras,” Proc. Int'l Conf. Computer Vision, pp. 999-1005, 1998.
[21] 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, pp. 108-115, 2001.
[22] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision. Cambridge, U.K.: Cambridge Univ. Press, 2000.
[23] A. Hicks and R. Bajcsy, “Catadioptric Sensors that Approximate Wide-Angle Perspective Projections,” Proc. IEEE Workshop Omnidirectional Vision, pp. 97-103, 2000.
[24] http:/www.2d3.com, 2006.
[25] http:/www.remotereality.com, 2006.
[26] S.B. Kang, “Catadioptric Self-Calibration,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 201-207, 2000.
[27] J. Kumler and M. Bauer, “Fish-Eye Lens Designs and Their Relative Performance,” (SPIE) Proc.— Current Developments in Lens Design and Optical Systems Eng., vol. 4093, pp. 360-369, 2000.
[28] J. Matas, O. Chum, M. Urban, and T. Pajdla, “Robust Wide Baseline Stereo from Maximally Stable Extremal Regions,” Proc. British Machine Vision Conf., vol. 1, pp. 384-393, 2002.
[29] B. Mičušík, “Two View Geometry of Omnidirectional Cameras,” PhD thesis, Dept. of Cybernetics, Faculty of Electrical Eng., Czech Technical Univ., Prague, Czech Republic, 2004.
[30] B. Mičušík and T. Pajdla, “Estimation of Omnidirectional Camera Model from Epipolar Geometry,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 485-490, 2003.
[31] B. Mičušík and T. Pajdla, “Omnidirectional Camera Model and Epipolar Geometry Estimation by RANSAC with Bucketing,” Proc. Scandinavian Conf. Image Analysis, 2003.
[32] B. Mičušík and T. Pajdla, “Autocalibration and 3D Reconstruction with Noncentral Catadioptric Cameras,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2004.
[33] B. Mičušík and T. Pajdla, “Para-Catadioptric Camera Autocalibration from Epipolar Geometry,” Proc. Asian Conf. Computer Vision, vol. 2, pp. 748-753, 2004.
[34] D. Nister, “A Minimal Solution to the Generalised 3-Point Pose Problem,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 560-567, 2004.
[35] J. Oliensis, “Exact Two-Image Structure from Motion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 12, pp. 1618-1633, Dec. 2002.
[36] S.F. Ray, Applied Photographic Optics: Lenses and Optical Systems for Photography, Film, Video, Electronic and Digital Imaging, third ed. Oxford, U.K.: Focal Press, 2002.
[37] S. Shah and J.K. Aggarwal, “Intrinsic Parameter Calibration Procedure for a (High Distortion) Fish-Eye Lens Camera with Distortion Model and Accuracy Estimation,” Pattern Recognition, vol. 29, no. 11, pp. 1775-1788, 1996.
[38] O. Shakernia, R. Vidal, and S. Sastry, “Omnidirectional Vision-Based Egomotion Estimation from Backprojection Flows,” Proc. IEEE Workshop Omnidirectional Vision, 2003.
[39] D. Strelow, J. Mishler, D. Koes, and S. Singh, “Precise Omnidirectional Camera Calibration,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 689-694, 2001.
[40] P. Sturm, “A Method for 3D Reconstruction of Piecewise Planar Objects from Single Panoramic Images,” Proc. IEEE Workshop Omnidirectional Vision, pp. 119-126, 2000.
[41] P. Sturm, “Mixing Catadioptric and Perspective Cameras,” Proc. IEEE Workshop Omnidirectional Vision, pp. 60-67, 2002.
[42] P. Sturm and S. Ramalingam, “A Generic Concept for Camera Calibration,” Proc. European Conf. Computer Vision, pp. 1-13, 2004.
[43] T. Svoboda and T. Pajdla, “Epipolar Geometry for Central Catadioptric Cameras,” Int'l J. Computer Vision, vol. 49, no. 1, pp. 23-37, 2002.
[44] T. Svoboda, T. Pajdla, and V. Hlaváč, “Epipolar Geometry for Panoramic Cameras,” Proc. European Conf. Computer Vision, pp. 218-232, 1998.
[45] T. Svoboda, T. Pajdla, and V. Hlaváč, “Motion Estimation Using Central Panoramic Cameras,” Proc. IEEE Int'l Conf. Intelligent Vehicles, pp. 335-340, 1998.
[46] 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, 2001.
[47] 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, pp. 1172-1178, Oct. 2000.
[48] C.J. Taylor, “Videoplus: A Method for Capturing the Structure and Appearance of Immersive Environments,” IEEE Trans. Visualization and Computer Graphics, vol. 8, no. 2, pp. 171-182, Apr.-June 2003.
[49] S. Thirthala and M. Pollefeys, “The Radial Trifocal Tensor: A Tool for Calibrating the Radial Distortion of Wide-Angle Cameras,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 321-328, 2005.
[50] F. Tisseur and K. Meerbergen, “The Quadratic Eigenvalue Problem,” SIAM Rev., vol. 43, no. 2, pp. 235-286, 2001.
[51] Y. Xiong and K. Turkowski, “Creating Image-Based VR Using a Self-Calibrating Fisheye Lens,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 237-243, 1997.
[52] X. Ying and Z. Hu, “ We Consider Central Catadioptric Cameras and Fisheye Cameras within a Unified Imaging Model?” Proc. European Conf. Computer Vision, 2004.

Index Terms:
Omnidirectional vision, fish-eye lens, catadioptric camera, autocalibration.
Citation:
Branislav Micu??, Tom? Pajdla, "Structure from Motion with Wide Circular Field of View Cameras," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 28, no. 7, pp. 1135-1149, July 2006, doi:10.1109/TPAMI.2006.151
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