Abstract
Numerous geometric problems in computer vision involve the solution of systems of polynomial equations. This is particularly true for so called minimal problems, but also for finding stationary points for overdetermined problems. The state-of-the-art is based on the use of numerical linear algebra on the large but sparse coefficient matrix that represents the original equations multiplied with a set of monomials. The key observation in this paper is that the speed and numerical stability of the solver depends heavily on (i) what multiplication monomials are used and (ii) the set of so called permissible monomials from which numerical linear algebra routines choose the basis of a certain quotient ring. In the paper we show that optimizing with respect to these two factors can give both significant improvements to numerical stability as compared to the state of the art, as well as highly compact solvers, while still retaining numerical stability. The methods are validated on several minimal problems that have previously been shown to be challenging with improvement over the current state of the art.
Chapter PDF
Similar content being viewed by others
References
Agarwal, S., Chandraker, M.K., Kahl, F., Kriegman, D.J., Belongie, S.: Practical Global Optimization for Multiview Geometry. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006, Part I. LNCS, vol. 3951, pp. 592–605. Springer, Heidelberg (2006)
Brown, M., Hartley, R., Nister, D.: Minimal solutions for panoramic stitching. In: Proc. Conf. on Computer Vision and Pattern Recognition (CVPR 2007), Minneapolis (June 2007)
Bujnak, M., Kukelova, Z., Pajdla, T.: A general solution to the p4p problem for camera with unknown focal length. In: Proc. Conf. Computer Vision and Pattern Recognition, Anchorage, USA (2008)
Bujnak, M., Kukelova, Z., Pajdla, T.: Making Minimal Solvers Fast. In: Proc. Conf. Computer Vision and Pattern Recognition (2012)
Byröd, M., Brown, M., Åström, K.: Minimal solutions for panoramic stitching with radial distortion. In: Proc. British Machine Vision Conference, London, United Kingdom (2009)
Byröd, M., Josephson, K., Åström, K.: A Column-Pivoting Based Strategy for Monomial Ordering in Numerical Gröbner Basis Calculations. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part IV. LNCS, vol. 5305, pp. 130–143. Springer, Heidelberg (2008)
Byröd, M., Josephson, K., Åström, K.: Fast and stable polynomial equation solving and its application to computer vision. Int. Journal of Computer Vision 84(3), 237–255 (2009)
Byröd, M., Kukelova, Z., Josephson, K., Pajdla, T., Åström, K.: Fast and robust numerical solutions to minimal problems for cameras with radial distortion. In: Proc. Conf. Computer Vision and Pattern Recognition, Anchorage, USA (2008)
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer (1998)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer (2007)
Enqvist, O., Ask, E., Kahl, F., Åström, K.: Robust Fitting for Multiple View Geometry. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part I. LNCS, vol. 7572, pp. 738–751. Springer, Heidelberg (2012)
Enqvist, O., Josephson, K., Kahl, F.: Optimal correspondences from pairwise constraints. In: Proc. 12th Int. Conf. on Computer Vision, Kyoto, Japan (2009)
Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM 24(6), 381–395 (1981)
Hartley, R., Sturm, P.: Triangulation. Computer Vision and Image Understanding 68, 146–157 (1997)
Jin, H.: A three-point minimal solution for panoramic stitching with lens distortion. In: Computer Vision and Pattern Recognition, Anchorage, Alaska, USA, June 24-26 (2008)
Kahl, F.: Multiple view geometry and the l ∞ -norm. In: ICCV, pp. 1002–1009 (2005)
Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. In: Proc. 10th Int. Conf. on Computer Vision, Bejing, China, pp. 978–985 (2005)
Kukelova, Z., Bujnak, M., Pajdla, T.: Automatic Generator of Minimal Problem Solvers. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 302–315. Springer, Heidelberg (2008)
Kukelova, Z., Pajdla, T.: Two minimal problems for cameras with radial distortion. In: OMNIVIS (2007)
Naroditsky, O., Daniilidis, K.: Optimizing polynomial solvers for minimal geometry problems. In: ICCV, pp. 975–982 (2011)
Olsson, C., Enqvist, O., Kahl, F.: A polynomial-time bound for matching and registration with outliers. In: Proc. Conf. on Computer Vision and Pattern Recognition (2008)
Stewénius, H.: Gröbner Basis Methods for Minimal Problems in Computer Vision. PhD thesis, Lund University (April 2005)
Stewénius, H., Schaffalitzky, F., Nistér, D.: How hard is three-view triangulation really? In: Proc. Int. Conf. on Computer Vision, Beijing, China, pp. 686–693 (2005)
Torr, P., Zisserman, A.: Robust parameterization and computation of the trifocal tensor. Image and Vision Computing 15(8), 591–605 (1997)
Torr, P., Zisserman, A.: Robust computation and parametrization of multiple view relations. In: Proc. 6th Int. Conf. on Computer Vision, Mumbai, India, pp. 727–732 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kuang, Y., Åström, K. (2012). Numerically Stable Optimization of Polynomial Solvers for Minimal Problems. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds) Computer Vision – ECCV 2012. ECCV 2012. Lecture Notes in Computer Science, vol 7574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33712-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-33712-3_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33711-6
Online ISBN: 978-3-642-33712-3
eBook Packages: Computer ScienceComputer Science (R0)