Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

Abstract

Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically, we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.