A Column-Pivoting Based Strategy for Monomial Ordering in Numerical Gröbner Basis Calculations

Abstract

Abstract. This paper presents a new fast approach to improving stability in polynomial equation solving. 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. An interesting approach to stabilising the computations is to study basis selection for the quotient space C[x]/I. In this paper, the exact matrix computations involved in the solution procedure are clarified and using this knowledge we propose a new fast basis selection scheme based on QR-factorization with column pivoting. We also propose an adaptive scheme for truncation of the Gröbner basis to further improve stability. The new basis selection strategy is studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate a fourfold increase in speed and nearly as good over-all precision as the previous SVD-based method. Moreover, we get typically get similar or better reduction of the largest errors.

Pose estimation with radial distortion and unknown focal length

“…Gröbner basis methods have successfully been used to solve several systems of polynomial equations derived from computer vision problem in recent years, e.g. [4,5,12,19,27]. The advantages of using Gröbner basis methods is that if the structure of the system is the same for a large number of problem some calculations can be done symbolically in advance, which leads to an efficient method to solve the systems of polynomial equations.…”

“…With this the action matrix can be constructed and the eigenvectors of the transposed action matrix will hold all solutions to the system, see [5] for details on how to construct the action matrix with the method chosen in this paper. Matlab code for the solver used in this paper is available online at http://www.maths.lth.se/vision/ downloads.…”

3D LIDAR–camera intrinsic and extrinsic calibration: Identifiability and analytical least-squares-based initialization

“…This is not possible, however, when working with polynomials with floating-point coefficients. Therefore we employ the method proposed by Byröd et al (2008) to compute M ϕ . We first note that some of the monomials of ϕ( x) ·x B remain in x B , while some others do not.…”

Optimal estimation of vanishing points in a Manhattan world

“…This is not possible, however, when working with polynomials with floating-point coefficients. Therefore we employ the method proposed in [5] to compute M ϕ . We first note that some of the monomials of ϕ(x) · x B remain in x B , while some others do not.…”

“…Further details on this subject and also on possible numerical instabilities and their remedies cannot be provided here due to space considerations. We refer the interested reader to [5,16,22].…”