Related papers: Beyond Gr\"obner Bases: Basis Selection for Minima…
Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements, i.e., solving minimal problems in a RANSAC framework. Minimal problems are usually formulated as…
Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e. solving minimal…
An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive…
Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e., solving minimal…
This paper introduces the first minimal solvers that jointly estimate lens distortion and affine rectification from repetitions of rigidly transformed coplanar local features. The proposed solvers incorporate lens distortion into the camera…
The efficiency of Gr\"obner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on…
Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(\alpha)$, a…
In this paper, we make a contribution to the computation of Gr\"obner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we…
Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the last years. However, the polynomial case has not been…
We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gr\"obner bases are a…
Symbolic computation for systems of differential equations is often computationally expensive. Many practical differential models have a form of polynomial or rational ODE system with specified outputs. A basic symbolic approach to analyze…
Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gr\"obner-basis or resultant-based methods due…
RANSAC is a popular technique for estimating model parameters in the presence of outliers. The best speed is achieved when the minimum possible number of points is used to estimate hypotheses for the model. Many useful problems can be…
Minimal problems in computer vision raise the demand of generating efficient automatic solvers for polynomial equation systems. Given a polynomial system repeated with different coefficient instances, the traditional Gr\"obner basis or…
We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…
This paper introduces the first minimal solvers that jointly estimate lens distortion and affine rectification from the image of rigidly-transformed coplanar features. The solvers work on scenes without straight lines and, in general, relax…
We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties…
We present a method for nonlinear parametric optimization based on algebraic geometry. The problem to be studied, which arises in optimal control, is to minimize a polynomial function with parameters subject to semialgebraic constraints.…
This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical…
We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner…