Numerically Stable Optimization of Polynomial Solvers for Minimal Problems Yubin Kuang and Kalle ˚ Astr¨ om Centre for Mathematical Sciences Lund University, Sweden {yubin,kalle}@maths.lth.se Abstract. Numerous geometric problems in computer vision involve the solu- tion of systems of polynomial equations. This is particularly true for so called minimal problems, but also for finding stationary points for overdetermined prob- lems. 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 multi- plied 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 stabil- ity. 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. 1 Introduction Many problems in computer vision can be rephrased as several polynomials in several unknowns. This is particularly true for so called minimal structure and motion prob- lems. Solutions to minimal structure and motion problems are essential for RANSAC algorithms to find inliers in noisy data [13, 24, 25]. For such applications one needs to efficiently solve a large number of minimal structure and motion problems in order to find the best set of inliers. Once enough inliers have been found it is common to use non-linear optimization e.g. to find the best fit to all data in a least squares sense. Here fast and numerical stable solutions for the polynomials systems of the minimal prob- lems are crucial for generate initial parameters for the non-linear optimization. Another area of recent interest is global optimization used e.g. for optimal triangulation, resec- tioning and fundamental matrix estimation. While global optimization is promising, it is a difficult pursuit and various techniques has been tried, e.g. branch and bound [1], L ∞ -norm methods [16] and methods using linear matrix inequalities (LMIs) [17]. An alternative way to find the global optimum is to calculate stationary points directly, usually by solving some polynomial equation system [14,23]. So far, this has been an approach of limited applicability since calculation of stationary points is numerically difficult for larger problems. A third area are methods for finding the optimal set of inliers [11, 12, 21], which are based on solving certain geometrical problems, which involve solving systems of polynomial equations. A. Fitzgibbon et al. (Eds.): ECCV 2012, Part III, LNCS 7574, pp. 100–113, 2012. c  Springer-Verlag Berlin Heidelberg 2012