Beyond Gr\"obner Bases: Basis Selection for Minimal Solvers

TL;DR

This paper improves polynomial solvers for computer vision by efficiently selecting monomial bases beyond traditional Gr"obner bases, leading to faster and more effective minimal solvers in RANSAC frameworks.

Contribution

It introduces a novel enumeration method for monomial bases beyond Gr"obner bases and a basis sampling scheme to enhance solver efficiency.

Findings

More efficient solvers achieved in many cases

Enumeration of all relevant monomial bases is feasible

Sampling scheme improves solver performance

Abstract

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gr\"obner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gr\"obner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems.