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Many computer vision applications require robust and efficient estimation ofcamera geometry from a minimal number of input data measurements, i.e., solvingminimal problems in a RANSAC framework. Minimal problems are usually formulatedas complex systems of sparse polynomials. The systems usually areoverdetermined and consist of polynomials with algebraically constrainedcoefficients. Most state-of-the-art efficient polynomial solvers are based onthe action matrix method that has been automated and highly optimized in recentyears. On the other hand, the alternative theory of sparse resultants andNewton polytopes has been less successful for generating efficient solvers,primarily because the polytopes do not respect the constraints on thecoefficients. Therefore, in this paper, we propose a simple iterative scheme totest various subsets of the Newton polytopes and search for the most efficientsolver. Moreover, we propose to use an extra polynomial with a special form tofurther improve the solver efficiency via a Schur complement computation. Weshow that for some camera geometry problems our extra polynomial-based methodleads to smaller and more stable solvers than the state-of-the-art Grobnerbasis-based solvers. The proposed method can be fully automated andincorporated into existing tools for automatic generation of efficientpolynomial solvers. It provides a competitive alternative to popular Grobnerbasis-based methods for minimal problems in computer vision. We also study theconditions under which the minimal solvers generated by the state-of-the-artaction matrix-based methods and the proposed extra polynomial resultant-basedmethod, are equivalent. Specifically we consider a step-by-step comparisonbetween the approaches based on the action matrix and the sparse resultant,followed by a set of substitutions, which would lead to equivalent minimalsolvers.

Sparse resultant based minimal solvers in computer vision and their connection with the action matrix