This Article 
 Bibliographic References 
 Add to: 
Nonlinear Estimation of the Fundamental Matrix with Minimal Parameters
March 2004 (vol. 26 no. 3)
pp. 426-432
Peter Sturm, IEEE Computer Society

Abstract—The purpose of this paper is to give a very simple method for nonlinearly estimating the fundamental matrix using the minimum number of seven parameters. Instead of minimally parameterizing it, we rather update what we call its orthonormal representation, which is based on its singular value decomposition. We show how this method can be used for efficient bundle adjustment of point features seen in two views. Experiments on simulated and real data show that this implementation performs better than others in terms of computational cost, i.e., convergence is faster, although methods based on minimal parameters are more likely to fall into local minima than methods based on redundant parameters.

[1] K.B. Atkinson, ed., Close Range Photogrammetry and Machine Vision. Whittles Publishing, 1996.
[2] A. Bartoli, On the Non-Linear Optimization of Projective Motion Using Minimal Parameters Proc. Seventh European Conf. Computer Vision, vol. 2, pp. 340-354, May 2002.
[3] A. Bartoli and P. Sturm, Three New Algorithms for Projective Bundle Adjustment with Minimum Parameters Research Report 4236, INRIA, Grenoble, France, Aug. 2001.
[4] A. Bartoli, P. Sturm, and R. Horaud, Projective Structure and Motion from Two Views of a Piecewise Planar Scene Proc. Eighth Int'l Conf. Computer Vision, vol. 1, pp. 593-598, July 2001.
[5] R.I. Hartley, “In Defense of the 8-Point Algorithm,” Proc. Fifth Int'l Conf. Computer Vision, pp. 1,064-1,070, June 1995.
[6] R. Hartley, “Projective Reconstruction and Invariants from Multiple Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 10, pp. 1036-1041, Oct. 1994.
[7] R.I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision. Cambridge Univ. Press, June 2000.
[8] T. Huang and O. Faugeras,“Some properties of the E matrix in two-view motion estimation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 12, pp. 1,310-1,312, Dec. 1989.
[9] K. Kanatani and D.D. Morris, Gauges and Gauge Transformations for Uncertainty Description of Geometric Structure with Indeterminacy IEEE Trans. Information Theory, vol. 47, no. 5, July 2001.
[10] K. Levenberg, A Method for the Solution of Certain Non-Linear Problems in Least Squares Quarterly of Applied Math., pp. 164-168, 1944.
[11] H.C. Longuet-Higgins, A Computer Program for Reconstructing a Scene from Two Projections Nature, vol. 293, pp. 133-135, Sept. 1981.
[12] Q.T. Luong and O. Faugeras, The Fundamental Matrix: Theory, Algorithms and Stability Analysis Int'l J. Computer Vision, vol. 17, no. 1, pp. 43-76, 1996.
[13] Q.T. Luong and T. Vieville, Canonic Representations for the Geometries of Multiple Projective Views Computer Vision and Image Understanding, vol. 64, no. 2, pp. 193-229, 1996.
[14] D.W. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters J. SIAM, vol. 11, no. 2, pp. 431-441, June 1963.
[15] P.F. McLauchlan, Gauge Invariance in Projective 3D Reconstruction Proc. Multi-View Workshop, 1999.
[16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C The Art of Scientific Computing, second ed. Cambridge Univ. Press, 1992.
[17] G.A.F. Seber and C.J. Wild, Non-Linear Regression. John Wiley&Sons, 1989.
[18] J. Stuelpnagel, On the Parametrization of the Three-Dimensional Rotation Group SIAM Rev., vol. 6, no. 4, pp. 422-430, Oct. 1964.
[19] P. Torr and A. Zisserman, MLESAC: A New Robust Estimator with Application to Estimating Image Geometry Computer Vision and Image Understanding, vol. 78, no. 1, 2000.
[20] P.H.S. Torr, A. Zisserman, and S.J. Maybank, Robust Detection of Degenerate Configurations while Estimating the Fundamental Matrix Computer Vision and Image Understanding, vol. 71, no. 3, pp. 312-333, Sept. 1998.
[21] B. Triggs, P.F. McLauchlan, R.I. Hartley, and A. Fitzgibbon, Bundle Ajustment A Modern Synthesis Proc. Int'l Workshop Vision Algorithms: Theory and Practice, B. Triggs, A. Zisserman, and R. Szeliski, eds., 2000.
[22] R.Y. Tsai and T.S. Huang, Uniqueness and Estimation of Three-Dimensional Motion Parameters of Rigid Objects with Curved Surfaces IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, no. 1, pp. 13-27, Jan. 1984.
[23] Z. Zhang, Determining the Epipolar Geometry and Its Uncertainty: A Review Int'l J. Computer Vision, vol. 27, no. 2, pp. 161-195, Mar. 1998.
[24] Z. Zhang and C. Loop, Estimating the Fundamental Matrix by Transforming Image Points in Projective Space Computer Vision and Image Understanding, vol. 82, no. 2, pp. 174-180, May 2001.

Index Terms:
Structure-from-motion, bundle adjustment, minimal parameterization, fundamental matrix.
Adrien Bartoli, Peter Sturm, "Nonlinear Estimation of the Fundamental Matrix with Minimal Parameters," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 3, pp. 426-432, March 2004, doi:10.1109/TPAMI.2004.1262342
Usage of this product signifies your acceptance of the Terms of Use.