Numerically stable optimization of polynomial solvers for minimal problems

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. © 2012 Springer-Verlag.