The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.12 - Dec. (2012 vol.34)
pp: 2303-2314
R. Hartley , Sch. of Eng. (RSISE), Australian Nat. Univ., Canberra, ACT, Australia
Hongdong Li , Sch. of Eng. (RSISE), Australian Nat. Univ., Canberra, ACT, Australia
ABSTRACT
In this paper, we present an efficient new approach for solving two-view minimal-case problems in camera motion estimation, most notably the so-called five-point relative orientation problem and the six-point focal-length problem. Our approach is based on the hidden variable technique used in solving multivariate polynomial systems. The resulting algorithm is conceptually simple, which involves a relaxation which replaces monomials in all but one of the variables to reduce the problem to the solution of sets of linear equations, as well as solving a polynomial eigenvalue problem (polyeig). To efficiently find the polynomial eigenvalues, we make novel use of several numeric techniques, which include quotient-free Gaussian elimination, Levinson-Durbin iteration, and also a dedicated root-polishing procedure. We have tested the approach on different minimal cases and extensions, with satisfactory results obtained. Both the executables and source codes of the proposed algorithms are made freely downloadable.
INDEX TERMS
polynomials, eigenvalues and eigenfunctions, Gaussian processes, image sensors, iterative methods, motion estimation, dedicated root-polishing procedure, hidden variable approach, minimal-case camera motion estimation, five-point relative orientation problem, six-point focal-length problem, multivariate polynomial systems, monomials, linear equations, polynomial eigenvalue problem, quotient-free Gaussian elimination, Levinson-Durbin iteration, Polynomials, Cameras, Eigenvalues and eigenfunctions, Calibration, Motion estimation, Mathematical model, polynomial root finding, Camera calibration, camera motion estimation, epipolar geometry, minimal solver
CITATION
R. Hartley, Hongdong Li, "An Efficient Hidden Variable Approach to Minimal-Case Camera Motion Estimation", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.34, no. 12, pp. 2303-2314, Dec. 2012, doi:10.1109/TPAMI.2012.43
REFERENCES
[1] D. Nistér, "An Efficient Solution to the Five-Point Relative Pose Problem," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2003.
[2] H. Li and R. Hartley, "Five-Point Motion Estimation Made Easy," Proc. 18th Int'l Conf. Pattern Recognition, pp. 630-633, 2006.
[3] H. Li, "A Simple Solution to the Six-Point Two-View Focal-Length Problem," Proc. Ninth European Conf. Computer Vision, pp. 200-213, 2006.
[4] D. Nistér, "An Efficient Solution to the Five-Point Relative Pose Problem," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 6, pp. 756-777, June 2004.
[5] E.H. Bareiss, "Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination," Math. Computation, vol. 22, no. 103, pp. 565-578, 1968.
[6] H. Stewénius, C. Engels, and D. Nistér, "Recent Developments on Direct Relative Orientation," ISPRS J. Photogrammetry and Remote Sensing, vol. 60, pp. 284-294, June 2006.
[7] H. Stewénius, F. Kahl, D. Nistér, and F. Schaffalitzky, "A Minimal Solution for Relative Pose with Unknown Focal Length," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 789-794, June 2005.
[8] H. Stewénius and C. Engels, "An Efficient Minimal Solution for Infinitesimal Camera Motion," Proc. IEEE Conf. Computer Vision and Pattern Recognition, June 2006.
[9] Z. Kukelova, M. Bujnak, and T. Pajdla, "Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems," Proc. British Machine Vision Conf., 2008.
[10] R.I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, second ed. Cambridge Univ. Press, 2004.
[11] O.D. Faugeras and S.J. Maybank, "Motion from Point Matches: Multiplicity of Solutions," Proc. Workshop Visual Motion, 1989.
[12] D. Cox, J. Little, and D. O'Shea, Using Algebraic Geometry, second ed. Springer, 2005.
[13] D. Nistér, R. Hartley, and H. Stewenius, "Using Galois Theory to Prove Structure from Motion Algorithms Are Optimal," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2007.
[14] A.W. Fitzgibbon, "Simultaneous Linear Estimation of Multiple View Geometry and Lens Distortion," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, 2001.
[15] G. Cybenko, "The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations," SIAM J. Scientific and Statistical Computing, vol. 1, pp. 303-310, 1980.
[16] U. Helmke, K. Hüper, P.Y. Lee, and J.B. Moore, "Essential Matrix Estimation Using Gauss-Newton Iterations on a Manifold," Int'l J. Computer Vision, vol. 74, no. 2, pp. 117-136, 2007.
[17] R.I. Hartley, "Estimation of Relative Camera Positions for Uncalibrated Cameras," Proc. Second European Conf. Computer Vision, pp. 579-587, 1992.
47 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool