# On the Existence of Epipolar Matrices

arXiv: https://arxiv.org/abs/1510.01401 · HTML: https://mlsanity.com/paper/1510.01401/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2015.

## Built on by (most cited first, 6 of 6)

- [Learning to Solve Hard Minimal Problems](https://mlsanity.com/paper/2112.03424/) (2021)
- [Galois/monodromy groups for decomposing minimal problems in 3D reconstruction](https://mlsanity.com/paper/2105.04460/) (2021)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [Optimizing Elimination Templates by Greedy Parameter Search](https://mlsanity.com/paper/2203.14901/) (2022)
- [An Atlas for the Pinhole Camera](https://mlsanity.com/paper/2206.13468/) (2022)
- [PLMP -- Point-Line Minimal Problems for Projective SfM](https://mlsanity.com/paper/2503.04351/) (2025)

## Abstract (arXiv)

This paper considers the foundational question of the existence of a fundamental (resp. essential) matrix given m point correspondences in two views. We present a complete answer for the existence of fundamental matrices for any value of m. Using examples we disprove the widely held beliefs that fundamental matrices always exist whenever m ≤ 7. At the same time, we prove that they exist unconditionally when m ≤ 5. Under a mild genericity condition, we show that an essential matrix always exists when m ≤ 4. We also characterize the six and seven point configurations in two views for which all matrices satisfying the epipolar constraint have rank at most one.