Computing stable resultant-based minimal solvers by hiding a variable

TL;DR

This paper introduces a stable and efficient sparse resultant-based method for solving polynomial systems in computer vision, improving stability over existing methods by hiding a variable, especially in critical configurations.

Contribution

The paper proposes a novel sparse resultant approach that hides a variable, enhancing stability and efficiency of minimal solvers in computer vision applications.

Findings

The new method is more stable than Gr"obner bases-based solvers.

It avoids large matrix inversions, increasing numerical stability.

Demonstrated improvements on several computer vision problems.

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-style framework. Minimal problems often result in complex systems of polynomial equations. The existing state-of-the-art methods for solving such systems are either based on Gr\"obner bases and the action matrix method, which have been extensively studied and optimized in the recent years or recently proposed approach based on a sparse resultant computation using an extra variable. In this paper, we study an interesting alternative sparse resultant-based method for solving sparse systems of polynomial equations by hiding one variable. This approach results in a larger eigenvalue problem than the action matrix and extra variable sparse resultant-based methods; however, it does not need to compute an inverse or elimination of large matrices that may be numerically unstable. The proposed approach includes several improvements to the standard sparse resultant algorithms, which significantly improves the efficiency and stability of the hidden variable resultant-based solvers as we demonstrate on several interesting computer vision problems. We show that for the studied problems, our sparse resultant based approach leads to more stable solvers than the state-of-the-art Gr\"obner bases-based solvers as well as existing sparse resultant-based solvers, especially in close to critical configurations. Our new method can be fully automated and incorporated into existing tools for the automatic generation of efficient minimal solvers.