Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices

TL;DR

This paper characterizes the algebraic structure of compatible triples of fundamental and essential matrices in computer vision, providing new constraints and advancing understanding of their geometric compatibility varieties.

Contribution

It computes the multidegree and vanishing ideal of the variety of compatible matrix triples, introducing new quartic constraints that improve prior algebraic descriptions.

Findings

Introduces a simple set of quartic constraints for compatible fundamental matrices.

Shows these constraints help locally define the variety of compatible essential matrices.

Advances algebraic understanding of compatibility varieties in computer vision.

Abstract

We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Br{\aa}telund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples.