A sparse resultant based method for efficient minimal solvers

TL;DR

This paper introduces a sparse resultant-based algebraic method for solving polynomial systems in computer vision, offering improved efficiency and stability over traditional Gr"obner basis solvers for minimal problems.

Contribution

The authors propose a novel sparse resultant approach that converts polynomial constraints into eigenvalue problems, enhancing solver efficiency and stability in computer vision applications.

Findings

The new method produces solvers comparable in size and accuracy to Gr"obner basis methods.

In some cases, the sparse resultant approach yields smaller, more stable solvers.

The approach can be fully automated and integrated into existing solver generation tools.

Abstract

Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e. solving minimal problems in a RANSAC framework. Minimal problems often result in complex systems of polynomial equations. Many state-of-the-art efficient polynomial solvers to these problems are based on Gr\"obner bases and the action-matrix method that has been automatized and highly optimized in recent years. In this paper we study an alternative algebraic method for solving systems of polynomial equations, i.e., the sparse resultant-based method and propose a novel approach to convert the resultant constraint to an eigenvalue problem. This technique can significantly improve the efficiency and stability of existing resultant-based solvers. We applied our new resultant-based method to a large variety of computer vision problems and show that for most of the considered problems, the new method leads to solvers that are the same size as the the best available Gr\"obner basis solvers and of similar accuracy. For some problems the new sparse-resultant based method leads to even smaller and more stable solvers than the state-of-the-art Gr\"obner basis solvers. Our new method can be fully automatized and incorporated into existing tools for automatic generation of efficient polynomial solvers and as such it represents a competitive alternative to popular Gr\"obner basis methods for minimal problems in computer vision.