# Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings

arXiv: https://arxiv.org/abs/1904.02663 · HTML: https://mlsanity.com/paper/1904.02663/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2019.

## Built on by (most cited first, 18 of 18)

- [Global Structure-from-Motion Revisited](https://mlsanity.com/paper/2407.20219/) (2024)
- [Deep Permutation Equivariant Structure from Motion](https://mlsanity.com/paper/2104.06703/) (2021)
- [On the Instability of Relative Pose Estimation and RANSAC's Role](https://mlsanity.com/paper/2112.14651/) (2021)
- [Learning Structure-from-Motion with Graph Attention Networks](https://mlsanity.com/paper/2308.15984/) (2023)
- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)
- [On-the-Fly SfM: What you capture is What you get](https://mlsanity.com/paper/2309.11883/) (2023)
- [Towards Initialization-free Calibrated Bundle Adjustment](https://mlsanity.com/paper/2506.23808/) (2025)
- [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)
- [RESfM: Robust Deep Equivariant Structure from Motion](https://mlsanity.com/paper/2404.14280/) (2024)
- [A Taxonomy of Structure from Motion Methods](https://mlsanity.com/paper/2505.15814/) (2025)
- [Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC](https://mlsanity.com/paper/2310.02719/) (2023)
- [Consensus Learning with Deep Sets for Essential Matrix Estimation](https://mlsanity.com/paper/2406.17414/) (2024)
- [PRaDA: Projective Radial Distortion Averaging](https://mlsanity.com/paper/2504.16499/) (2025)
- [On the Robustness of Multi-View Rotation Averaging](https://mlsanity.com/paper/2102.05454/) (2021)
- [QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition](https://mlsanity.com/paper/2602.22639/) (2026)
- [Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices](https://mlsanity.com/paper/2602.23450/) (2026)
- [Learning Global Camera Poses from Noisy View-Graphs for Structure from Motion](https://mlsanity.com/paper/2609.09491/) (2026)

## Abstract (arXiv)

Essential matrix averaging, i.e., the task of recovering camera locations and orientations in calibrated, multiview settings, is a first step in global approaches to Euclidean structure from motion. A common approach to essential matrix averaging is to separately solve for camera orientations and subsequently for camera positions. This paper presents a novel approach that solves simultaneously for both camera orientations and positions. We offer a complete characterization of the algebraic conditions that enable a unique Euclidean reconstruction of n cameras from a collection of (n2) essential matrices. We next use these conditions to formulate essential matrix averaging as a constrained optimization problem, allowing us to recover a consistent set of essential matrices given a (possibly partial) set of measured essential matrices computed independently for pairs of images. We finally use the recovered essential matrices to determine the global positions and orientations of the n cameras. We test our method on common SfM datasets, demonstrating high accuracy while maintaining efficiency and robustness, compared to existing methods.