  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 complex systems of sparse polynomials. 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 and
Newton polytopes has been less successful for generating efficient solvers,
primarily because the polytopes do not respect the constraints on the
coefficients. Therefore, in this paper, 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 a Schur complement computation. We
show that for some camera geometry problems our extra polynomial-based method
leads to smaller and more stable solvers than the state-of-the-art Grobner
basis-based solvers. The proposed method can be fully automated and
incorporated into existing tools for automatic generation of efficient
polynomial solvers. It provides a competitive alternative to popular Grobner
basis-based methods for minimal problems in computer vision. We also 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.
