Galois/monodromy groups for decomposing minimal problems in 3D reconstruction

TL;DR

This paper explores the use of Galois/monodromy groups to analyze and decompose polynomial problems in 3D reconstruction, leading to more efficient solvers and new insights into symmetries and problem degrees.

Contribution

It introduces a framework applying Galois/monodromy groups to classical and novel 3D reconstruction problems, revealing symmetries and reducing problem degrees.

Findings

Identified new symmetries in absolute pose problems with mixed features

Reduced the degree of a 3-image homography estimation problem from 64 to 16

Provided new algebraic constraints on compatible homographies

Abstract

We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three images. For this problem of degree 64, we can reduce the degree to 16; the latter better reflecting the intrinsic difficulty of algebraically solving the problem. As a byproduct, we obtain new constraints on compatible homographies, which may be of independent interest.