Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices

2026-02-26

AI Summary

This work addresses the incomplete characterization of compatibility conditions for fundamental and essential matrices in three-view geometry, which in prior studies often relied on restrictive assumptions. By leveraging tools from multiview projective geometry and computational algebraic geometry, the paper provides the first complete description of the vanishing ideal of compatible triples of fundamental matrices, rigorously establishing their underlying algebraic structure. This result is further extended to the case of essential matrices. The central contribution is the discovery of a concise set of quartic algebraic constraints that locally define the compatibility variety, substantially improving upon existing incomplete conditions. These findings offer a solid theoretical foundation for multi-view geometry with implications for robust structure-from-motion and visual reconstruction pipelines.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Multi-instance/Multi-view Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• fundamental matrix
• essential matrix
• compatible triples
• multiprojective geometry
• vanishing ideal

Innovation

• fundamental matrix
• essential matrix
• vanishing ideal
• quartic constraints
• multiprojective geometry