Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision

This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible.

[1]  Robert Pless,et al.  Using many cameras as one , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[2]  Henrik Stewenius,et al.  Gröbner Basis Methods for Minimal Problems in Computer Vision , 2005 .

[3]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[4]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[5]  Beatrice Gralton,et al.  Washington DC - USA , 2008 .

[6]  David Nistér,et al.  An efficient solution to the five-point relative pose problem , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  Richard I. Hartley,et al.  Minimal Solutions for Panoramic Stitching , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[8]  Gene H. Golub,et al.  Matrix computations , 1983 .

[9]  Thomas S. Huang,et al.  Algebraic methods for image processing and computer vision , 1996, IEEE Trans. Image Process..

[10]  Frederik Schaffalitzky,et al.  A minimal solution for relative pose with unknown focal length , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[11]  Alexandru Tupan,et al.  Triangulation , 1997, Comput. Vis. Image Underst..

[12]  Ilkka Karasalo,et al.  A criterion for truncation of theQR-decomposition algorithm for the singular linear least squares problem , 1974 .

[13]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..

[14]  Christopher Geyer,et al.  A Nine-point Algorithm for Estimating Para-Catadioptric Fundamental Matrices , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[15]  Andrew Zisserman,et al.  Robust computation and parametrization of multiple view relations , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[16]  P. McAree,et al.  Using Sturm sequences to bracket real roots of polynomial equations , 1990 .

[17]  F. Kahl.  Multiple View Geometry and the -norm , 2005 .

[18]  Martin Byröd,et al.  Image-Based Localization Using Hybrid Feature Correspondences , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[19]  David A. Cox,et al.  Solving Polynomial Equations: Foundations, Algorithms, and Applications (Algorithms and Computation in Mathematics) , 2005 .

[20]  Martin Byröd,et al.  A Column-Pivoting Based Strategy for Monomial Ordering in Numerical Gröbner Basis Calculations , 2008, ECCV.

[21]  Didier Henrion,et al.  Globally Optimal Estimates for Geometric Reconstruction Problems , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[22]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[23]  Zuzana Kukelova,et al.  A minimal solution to the autocalibration of radial distortion , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[24]  Martin Byröd,et al.  Improving Numerical Accuracy of Gröbner Basis Polynomial Equation Solvers , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[25]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[26]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[27]  Martin Byröd,et al.  Fast Optimal Three View Triangulation , 2007, ACCV.

[28]  David Nister,et al.  Recent developments on direct relative orientation , 2006 .

[29]  Richard I. Hartley,et al.  L-8Minimization in Geometric Reconstruction Problems , 2004, CVPR.

[30]  Kostas Daniilidis,et al.  Properties of the Catadioptric Fundamental Matrix , 2002, ECCV.

[31]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[32]  Zuzana Kukelova,et al.  A general solution to the P4P problem for camera with unknown focal length , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[33]  E. H. Thompson.  A RATIONAL ALGEBRAIC FORMULATION OF THE PROBLEM OF RELATIVE ORIENTATION , 2006 .

[34]  Frederik Schaffalitzky,et al.  How hard is 3-view triangulation really? , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[35]  R. Hartley,et al.  L/sub /spl infin// minimization in geometric reconstruction problems , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[36]  David J. Kriegman,et al.  Practical Global Optimization for Multiview Geometry , 2006, International Journal of Computer Vision.

[37]  Karl Johan Åström,et al.  Solutions to Minimal Generalized Relative Pose Problems , 2005 .

[38]  R. Hartley.  Triangulation, Computer Vision and Image Understanding , 1997 .

[39]  Andrew Zisserman,et al.  Robust parameterization and computation of the trifocal tensor , 1997, Image Vis. Comput..

[40]  Zuzana Kukelova,et al.  Fast and robust numerical solutions to minimal problems for cameras with radial distortion , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[41]  J. Philip.  A non-iterative algorithm for determining all essential matrices corresponding to five point pairs , 1996 .

[42]  Robert C. Bolles,et al.  Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography , 1981, CACM.

[43]  Fredrik Kahl,et al.  Multiple view geometry and the L/sub /spl infin//-norm , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[44]  Jan Verschelde,et al.  Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation , 1999, TOMS.

[45]  Zuzana Kukelova,et al.  Two Minimal Problems for Cameras with Radial Distortion , 2007, 2007 IEEE 11th International Conference on Computer Vision.