Subscribe
Issue No.06 - June (2010 vol.32)
pp: 1044-1059
Jae-Hak Kim , Queen Mary University of London, London
Hongdong Li , The Australian National University, Canberra
Richard Hartley , The Australian National University, Canberra
ABSTRACT
We investigate the problem of estimating the ego-motion of a multicamera rig from two positions of the rig. We describe and compare two new algorithms for finding the 6 degrees of freedom (3 for rotation and 3 for translation) of the motion. One algorithm gives a linear solution and the other is a geometric algorithm that minimizes the maximum measurement error—the optimal L_\infty solution. They are described in the context of the General Camera Model (GCM), and we pay particular attention to multicamera systems in which the cameras have nonoverlapping or minimally overlapping field of view. Many nonlinear algorithms have been developed to solve the multicamera motion estimation problem. However, no linear solution or guaranteed optimal geometric solution has previously been proposed. We made two contributions: 1) a fast linear algebraic method using the GCM and 2) a guaranteed globally optimal algorithm based on the L_\infty geometric error using the branch-and-bound technique. In deriving the linear method using the GCM, we give a detailed analysis of degeneracy of camera configurations. In finding the globally optimal solution, we apply a rotation space search technique recently proposed by Hartley and Kahl. Our experiments conducted on both synthetic and real data have shown excellent results.
INDEX TERMS
Multicamera rigs, generalized camera, motion estimation, epipolar equation, branch and bound, linear programming.
CITATION
Jae-Hak Kim, Hongdong Li, Richard Hartley, "Motion Estimation for Nonoverlapping Multicamera Rigs: Linear Algebraic and {\rm L}_\infty Geometric Solutions", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 6, pp. 1044-1059, June 2010, doi:10.1109/TPAMI.2009.82
REFERENCES
 [1] 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. [2] P. Sturm, "Multi-View Geometry for General Camera Models," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 206-212, June 2005. [3] R. Pless, "Using Many Cameras as One," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 587-593, 2003. [4] E. Mouragnon, M. Lhuillier, M. Dhome, F. Dekeyster, and P. Sayd, "Generic and Real-Time Structure from Motion," Proc. British Machine Vision Conf., 2007. [5] M. Lhuillier, "Effective and Generic Structure from Motion Using Angular Error," Proc. Int'l Conf. Pattern Recognition, pp. 67-70, 2006. [6] R. Hartley and F. Schaffalitzky, "${L}_\infty$ Minimization in Geometric Reconstruction Problems," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 504-509, June 2004, http://dx.doi.org/10.1109CVPR.2004.140. [7] F. Kahl, "Multiple View Geometry and the $L_\infty$ -Norm," Proc. IEEE Int'l Conf. Computer Vision, pp. 1002-1009, 2005. [8] K. Sim and R. Hartley, "Recovering Camera Motion Using ${L}_\infty$ Minimization," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 1230-1237, 2006, http://dx.doi.org/10.1109CVPR.2006.247. [9] K. Sim and R. Hartley, "Removing Outliers Using the ${L}_\infty$ Norm," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 485-494, 2006, http://dx.doi.org/10.1109CVPR.2006.253. [10] R. Hartley and F. Kahl, "Global Optimization through Searching Rotation Space and Optimal Estimation of the Essential Matrix," Proc. IEEE Int'l Conf. Computer Vision, Oct. 2007, http://dx.doi.org/10.1109ICCV.2007.4408896 . [11] R. Hartley and F. Kahl, "Global Optimization through Rotation Space Search," Int'l J. Computer Vision, vol. 82, no. 1, pp. 64-79, Apr. 2009, [12] H. Li, "A Practical Algorithm for $L_{\infty }$ Triangulation with Outliers," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 1-8, June 2007. [13] J. Stolfi, Oriented Projective Geometry. Academic Press Professional, Inc., 1991. [14] H. Stewénius, D. Nistér, M. Oskarsson, and K. Åström, "Solutions to Minimal Generalized Relative Pose Problems," Proc. Workshop Omnidirectional Vision, Oct. 2005. [15] M. Byröd, K. Josephson, and K. Åström, "Improving Numerical Accuracy of Gröbner Basis Polynomial Equation Solvers," Proc. IEEE Int'l Conf. Computer Vision, 2007. [16] G. Schweighofer and A. Pinz, "Fast and Globally Convergent Structure and Motion Estimation for General Camera Models," Proc. British Machine Vision Conf., 2006. [17] J.-M. Frahm, K. Köser, and R. Koch, "Pose Estimation for Multi-Camera Systems," Proc. DAGM, 2004. [18] J.-H. Kim, R. Hartley, J.-M. Frahm, and M. Pollefeys, "Visual Odometry for Non-Overlapping Views Using Second-Order Cone Programming," Proc. Asian Conf. Computer Vision, vol. 2, pp. 353-362, Nov. 2007, http://dx.doi.org/10.1007978-3-540-76390-1_35 . [19] J.-H. Kim, H. Li, and R. Hartley, "Motion Estimation for Multi-Camera Systems Using Global Optimization," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2008, http://dx.doi.org/10.1109CVPR.2008.4587680 . [20] H. Li, R. Hartley, and J.-H. Kim, "Linear Approach to Motion Estimation Using Generalized Camera Models," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2008, http://dx.doi.org/10.1109CVPR.2008.4587545 . [21] R. Gupta and R.I. Hartley, "Linear Pushbroom Cameras," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 9, pp. 963-975, Sept. 1997, http://dx.doi.org/10.110934.615446. [22] R. Hartley, "Photogrammetric Techniques for Panoramic Cameras," Proc. SPIE Conf. Integrating Photogrammetric Techniques with Scene Analysis and Machine Vision, pp. 127-139, Apr. 1993, http://dx.doi.org/10.111712.155798. [23] R.I. Hartley and T. Saxena, "The Cubic Rational Polynomial Camera Model," Proc. Defense Advanced Research Projects Agency Image Understanding Workshop, pp. 649-653, 1997. [24] A. Zomet, D. Feldman, S. Peleg, and D. Weinshall, "Non-Perspective Imaging and Rendering with the Crossed-Slits Projection," technical report, Leibnitz Center, Hebrew Univ. of Jerusalem, http://citeseerx.ist.psu.edu/viewdocsummary? doi=10.1.1.19.5676 , 2002. [25] D. Feldman, T. Pajdla, and D. Weinshall, "On the Epipolar Geometry of the Crossed-Slits Projection," Proc. IEEE Int'l Conf. Computer Vision, pp. 988-995, 2003. [26] R.I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, second ed. Cambridge Univ. Press, 2004. [27] R.I. Hartley, "Projective Reconstruction and Invariants from Multiple Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 10, pp. 1036-1041, Oct. 1994, http://dx.doi.org/10.110934.329005. [28] H. Li and R. Hartley, "Five-Point Motion Estimation Made Easy," Proc. Int'l Conf. Pattern Recognition, pp. 630-633, Aug. 2006, http://dx.doi.org/10.1109ICPR.2006.579. [29] B. Clipp, J.-H. Kim, J.-M. Frahm, M. Pollefeys, and R. Hartley, "Robust 6DOF Motion Estimation for Non-Overlapping Multi-Camera Systems," Proc. Workshop Applications of Computer Vision, pp. 1-8, Jan. 2008, http://dx.doi.org/10.1109WACV.2008. 4544011 . [30] GNU Project, GNU Linear Programming Kit Version 4.9, http://www.gnu.org/softwareglpk/, 2009. [31] P.G.R. Inc., "Ladybug2 Camera," http:/www.ptgrey.com, 2006. [32] 2d3 Limited, "2d3 Boujou," http:/www.2d3.com, 2005. [33] M.A. Fischler and R.C. Bolles, "Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography," Comm. ACM, vol. 24, no. 6, pp. 381-395, 1981.