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Issue No.07 - July (2012 vol.34)
pp: 1381-1393
Zuzana Kukelova , Czech Technical University in Prague, Prague
Martin Bujnak , Czech Technical University in Prague, Prague
Tomas Pajdla , Czech Technical University in Prague, Prague
ABSTRACT
We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.
INDEX TERMS
Structure from motion, relative camera pose, minimal problems, polynomial eigenvalue problems.
CITATION
Zuzana Kukelova, Martin Bujnak, Tomas Pajdla, "Polynomial Eigenvalue Solutions to Minimal Problems in Computer Vision", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.34, no. 7, pp. 1381-1393, July 2012, doi:10.1109/TPAMI.2011.230
REFERENCES
[1] Z. Bai, J. Demmel, J. Dongorra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems. SIAM, 2000.
[2] M. Bujnak, Z. Kukelova, and T. Pajdla, "A General Solution to the P4P Problem for Camera with Unknown Focal Length," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2008.
[3] M. Bujnak, Z. Kukelova, and T. Pajdla, "3D Reconstruction from Image Collections with a Single Known Focal Length," Proc. 12th IEEE Int'l Conf. Computer Vision, 2009.
[4] M. Bujnak, Z. Kukelova, and T. Pajdla, "New Efficient Solution to the Absolute Pose Problem for Camera with Unknown Focal Length and Radial Distortion," Proc. 10th Asian Conf. Computer Vision, 2010.
[5] M. Byröd, K. Josephson, and K. Åström, "Improving Numerical Accuracy of Gröbner Basis Polynomial Equation Solver," Proc. 11th IEEE Int'l Conf. Computer Vision, 2007.
[6] D. Cox, J. Little, and D. O'Shea, Using Algebraic Geometry. Springer-Verlag, 2005.
[7] D.K. Faddeev and W.N. Faddeeva, Computational Methods of Linear Algebra. Freeman, 1963.
[8] O. Faugeras and S. Maybank, "Motion from Point Matches: Multiplicity of Solutions," Int'l J. Computer Vision, vol. 4, no. 3, pp. 225-246, 1990.
[9] 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.
[10] A. Fitzgibbon, "Simultaneous Linear Estimation of Multiple View Geometry and Lens Distortion," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2001.
[11] C. Geyer and H. Stewenius, "A Nine-Point Algorithm for Estimating Para-Catadioptric Fundamental Matrices," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2007.
[12] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision. Cambridge Univ. Press, 2003.
[13] K. Josephson, M. Byröd, and K. Aström, "Pose Estimation with Radial Distortion and Unknown Focal Length," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2009.
[14] E. Kruppa, "Zur Ermittlung eines Objektes aus Zwei Perspektiven mit Innerer Orientierung," Sitz.-Ber. Akad.Wiss., Wien, Math. Naturw. Kl., Abt. IIa., vol. 122, pp. 1939-1948, 1918.
[15] Z. Kukelova and T. Pajdla, "A Minimal Solution to the Autocalibration of Radial Distortion," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2007.
[16] Z. Kukelova, M. Bujnak, and T. Pajdla, "Automatic Generator of Minimal Problem Solvers," Proc. 10th European Conf. Computer Vision, 2008.
[17] Z. Kukelova, M. Bujnak, and T. Pajdla, "Polynomial Eigenvalue Solutions to the 5-Pt and 6-Pt Relative Pose Problems," Proc. 19th British Machine Vision Conf., Sept. 2008.
[18] Z. Kukelova, M. Byröd, K. Josephson, T. Pajdla, and K. Åström, "Fast and Robust Numerical Solutions to Minimal Problems for Cameras with Radial Distortion," Computer Vision and Image Understanding, vol. 114, no. 2, pp. 234-244, Feb. 2010.
[19] Z. Kukelova and T. Pajdla, "A Minimal Solution to Radial Distortion Autocalibration," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 33, no. 12, pp. 2410-2422, Dec. 2011.
[20] H. Li and R. Hartley, "A Non-Iterative Method for Correcting Lens Distortion from Nine-Point Correspondences," Proc. OmniVision, 2005.
[21] H. Li, "A Simple Solution to the Six-Point Two-View Focal-Length Problem," Proc. European Conf. Computer Vision, pp. 200-213, 2006.
[22] H. Li and R. Hartley, "Five-Point Motion Estimation Made Easy," Proc. 18th Int'l Conf. Pattern Recognition, pp. 630-633, 2006.
[23] D. Hook and P. McAree, "Using Sturm Sequences to Bracket Real Roots of Polynomial Equations," Graphic Gems I, pp. 416-423, Academic Press, 1990.
[24] X. Li, C. Wu, C. Zach, S. Lazebnik, and J. Frahm, "Modeling and Recognition of Landmark Image Collections Using Iconic Scene Graphs," Proc. 10th European Conf. Computer Vision, 2008.
[25] D. Manocha and J.F. Canny, "Multipolynomial Resultant Algorithms," J. Symbolic Computation, vol. 15, no. 2, pp. 99-122, 1993.
[26] D. Manocha, "Solving Systems of Polynomial Equations," IEEE Computer Graphics and Applications, vol. 14, no. 2, pp. 46-55, Mar. 1994.
[27] D. Martinec and T. Pajdla, "Robust Rotation and Translation Estimation in Multiview Reconstruction," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2007.
[28] B. Micusik and T. Pajdla, "Estimation of Omnidirectional Camera Model from Epipolar Geometry," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 485-490, 2003.
[29] D. Nister, "An Efficient Solution to the Five-Point Relative Pose," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 6, pp. 756-770, June 2004.
[30] D. Nister, R. Hartley, and H. Stewenius, "Using Galois Theory to Prove that Structure from Motion Algorithms Are Optimal," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2007.
[31] J. Philip, "A Non-Iterative Algorithm for Determining All Essential Matrices Corresponding to Five Point Pairs," Photogrammetric Record, vol. 15, no. 88, pp. 589-599, 1996.
[32] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing. Cambridge Univ. Press, Feb. 2002.
[33] N. Snavely, S.M. Seitz, and R.S. Szeliski, "Photo Tourism: Exploring Image Collections in 3D," Proc. Siggraph, pp. 835-846, 2006.
[34] N. Snavely, S. Seitz, and R. Szeliski, "Skeletal Graphs for Efficient Structure from Motion," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2008.
[35] H.J. Stetter, Numerical Polynomial Algebra. SIAM, 2004.
[36] H. Stewénius, D. Nister, F. Kahl, and F. Schaffalitzky, "A Minimal Solution for Relative Pose with Unknown Focal Length," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 789-794, 2005.
[37] H. Stewenius, D. Nister, M. Oskarsson, and K. Astrom, "Solutions to Minimal Generalized Relative Pose Problems," Proc. Workshop Omnidirectional Vision, 2005.
[38] H. Stewenius, C. Engels, and D. Nister, "Recent Developments on Direct Relative Orientation," ISPRS J. Photogrammetry and Remote Sensing, vol. 60, pp. 284-294, 2006.
[39] M. Urbanek, R. Horaud, and P. Sturm, "Combining Off- and On-Line Calibration of a Digital Camera," Proc. Third Int'l Conf. 3-D Digital Imaging and Modeling, 2001.
[40] A. Wallack, I.Z. Emiris, and D. Manocha, "MARS: A Maple/Matlab/C Resultant-Based Solver," Proc. Int'l Symp. Symbolic and Algebraic Computation, pp. 244-251, 1998.
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