Computing stable resultant-based minimal solvers by hiding a variable
This addresses the need for robust camera geometry estimation in computer vision applications, offering an incremental improvement in solver stability.
The paper tackles the problem of solving minimal polynomial systems in computer vision by proposing a sparse resultant-based method that hides a variable, resulting in more stable solvers than state-of-the-art Gröbner bases and existing sparse resultant methods, especially in critical configurations.
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 Gröbner 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 Gröbner 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.