A Column-Pivoting Based Strategy for Monomial Ordering in Numerical Gröbner Basis Calculations (original) (raw)

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

International Journal of Computer Vision, 2009

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 Introduction Numerous geometric problems in computer vision involve the solution of systems of polynomial equations. This is particularly true for so called minimal structure and motion

Numerical computation of Gröbner bases

Proceedings of CASC2004 ( …, 2004

In this paper we deal with the problem of numerical computation of Gröbner bases of zero-dimensional polynomial systems. It is well known that the computation of a Gröbner basis cannot be generally executed in floating-point arithmetic by a standard approach. This, however, would be highly desirable for practical applications. We present an approach for computing Gröbner bases numerically. It is an elaboration of the idea of a stabilized Gröbner basis computation initially proposed by Hans Stetter. Our implementation of the algorithm based on the presented results is available online.

Beyond Grobner Bases: Basis Selection for Minimal Solvers

2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gröbner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gröbner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems.

Algebraic methods in computer vision

Many problems in computer vision require efficient solvers for solving systems of non-linear polynomial equations. For instance relative and absolute camera pose computations and estimation of the camera lens distortion are examples of problems that can be formulated as minimal problems, i.e. they can be solved from a minimal number of input data and lead to solving systems of polynomial equations with a finite number of solutions. Often, polynomial systems arising from minimal problems are not trivial and general algorithms for solving systems of polynomial equations are not efficient for them. Therefore, special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this thesis we review two general algebraic methods for solving systems of polynomial equations, the Gröbner basis and the resultant based methods, and suggest their modifications, which are suitable for many computer vision problems. The main difference between the modified methods and the general methods is that the modified methods use the structure of the system of polynomial equations representing a particular problem to design an efficient specific solver for this problem. These modified methods consist of two phases. In the first phase, preprocessing and computations common to all considered instances of the given problem are performed and an efficient specific solver is constructed. For a particular problem this phase needs to be performed only once. In the second phase, the specific solver is used to efficiently solve concrete instances of the particular problem. This efficient specific solver is not general and solves only systems of polynomial equations of one form. However, it is faster than a general solver and suitable for applications that appear in computer vision and robotics. Construction of efficient specific solvers can be easily automated and therefore used even by non-experts to solve technical problems leading to systems of polynomial equations. In this thesis we propose an automatic generator of such efficient specific solvers based on the modified Gröbner basis method. We demonstrate the usefulness of our approach by providing new, efficient and numerical stable solutions to several important relative pose problems, most of them previously unsolved. These problems include estimating relative pose and internal parameters of calibrated, partially calibrated (with unknown focal length), or completely uncalibrated perspective or radially distorted cameras observing general scenes or scenes with dominant plane. All these problems can be efficiently used in many applications such as camera localization, structure-from-motion, scene reconstruction, tracking and recognition. The quality of all presented solvers is demonstrated on synthetic and real data. I would like to express my thanks to my colleagues at CMP who I had the pleasure of working with, especially to Martin Bujňák for his ideas and collaborative efforts in a large part of my work. I am greatly indebted to my advisor Tomáš Pajdla for guiding me throughout my research. Their friendly support, patience and theoretical and practical help have been paramount to the successful completion of my PhD study. Furthermore, I would like to thank Martin Byröd, Klas Josephson and KalleÅström at the Mathematical Imaging Group in Lund for interesting discussions, useful comments and collaboration on two papers. Finally, I would like to thank my family and my friends for all their support that made it possible for me to finish this thesis.

Gröbner Bases and Polynomial Equations

2016

Let S = k[x1, x2, . . . , xn] denote a polynomial ring over a field k where x1, x2, . . . , xn are indeterminates. A Gröbner basis is a set of polynomials in S which has several remarkable properties which enable us to carry out standard operations on ideals, rings and modules in an algorithmic way. Every set of polynomials in S can be transformed into a Gröbner basis. This process generalises three important algorithms: (1) Gauss elimination method for solving a system of linear equations, (2) Euclid’s algorithm for finding the greatest common divisor and (3) The simplex method of linear programming. One of the goals of these two lectures is to explain how to reduce the problem of solving a system of polynomial equations to a problem of finding eigenvalues of commuting matrices. We will introduce term orders first on the set of monomials in S and define the concept of Gróbner basis of an ideal. Term orders on monomials in k[x1, x2, . . . , xn] The set of monomials in the polynomial...

A modified LLL Algorithm for Change of Ordering of Gröbner Basis

In this paper, a modified version of LLL algorithm, which is a an algorithm with output-sensitive complexity, is presented to convert a given Gröbner basis with respect to a specific order of a polynomial ideal I in arbitrary dimensions to a Gröbner basis of I with respect to another term order. Also a comparison with the FGLM conversion and Buchberger method is considered.

Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation

Lecture Notes in Computer Science, 2008

Computation of Gröbner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for a long time; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes large errors, in various cases. On the basis of this analysis, we propose a practical method for computing the Gröbner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a method of estimating the ill-conditionedness of the input system.

Solving polynomial equations for minimal problems in computer vision

2006

Many vision tasks require efficient solvers of systems of polynomial equations. Epipolar geometry and relative camera pose computation are tasks which can be formulated as minimal problems which lead to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this work we suggest improvements of current techniques for solving systems of polynomial equations suitable for some vision problems. We introduce two tricks. The first trick helps to reduce the number of variables and degrees of the equations. The second trick can be used to replace computationally complex construction of Gröbner basis by a simpler procedure. We demonstrate benefits of our technique by providing a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation. We provide an efficient and robust solver of this problem. The quality of the solver is demonstrated on synthetic and real data.

Numerical stability and stabilization of Groebner basis computation

Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02, 2002

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only approximately known. We report on some numerical experiments, that show that the intrinsic instability of the problem is high but not such as to make the problem unmanageable, and that there is space to improve the numerical stability of the algorithms. We suggest some algorithms to deal with the case of overdetermined and unstable systems.

Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases

… of the 2009 international symposium on …, 2009

In this paper, we propose an efficient method to solve polynomial systems whose equations are left invariant by the action of a finite group G. The idea is to simultaneously compute a truncated SAGBI-Gröbner bases (a generalisation of Gröbner bases to ideals of subalgebras of polynomial ring) and a Gröbner basis in the invariant ring K[σ1, . . . , σn] where σi is the i-th elementary symmetric polynomial.