# Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices

arXiv: https://arxiv.org/abs/2602.23450 · HTML: https://mlsanity.com/paper/2602.23450/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 27 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 13 of 13)

- [Learning to Solve Hard Minimal Problems](https://mlsanity.com/paper/2112.03424/) (2021)
- [Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings](https://mlsanity.com/paper/1904.02663/) (2019)
- [Minimal Problems for the Calibrated Trifocal Variety](https://mlsanity.com/paper/1611.05947/) (2016)
- [GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices](https://mlsanity.com/paper/1812.00426/) (2018)
- [Galois/monodromy groups for decomposing minimal problems in 3D reconstruction](https://mlsanity.com/paper/2105.04460/) (2021)
- [A New Rank Constraint on Multi-view Fundamental Matrices, and its Application to Camera Location Recovery](https://mlsanity.com/paper/1702.03023/) (2017)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [Averaging Essential and Fundamental Matrices in Collinear Camera Settings](https://mlsanity.com/paper/1912.00254/) (2019)
- [The Quadrifocal Variety](https://mlsanity.com/paper/1501.01266/) (2015)
- [On Some Properties of Calibrated Trifocal Tensors](https://mlsanity.com/paper/1601.01467/) (2016)
- [Minimal Perspective Autocalibration](https://mlsanity.com/paper/2405.05605/) (2024)
- [Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor](https://mlsanity.com/paper/2409.09313/) (2024)
- [Necessary and Sufficient Polynomial Constraints on Compatible Triplets of Essential Matrices](https://mlsanity.com/paper/1912.11987/) (2019)

## Abstract (arXiv)

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 Braatelund 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 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.