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

## TL;DR

This paper introduces a novel algebraic approach for simultaneous estimation of camera positions and orientations in multiview settings, improving accuracy and efficiency in essential matrix averaging for structure from motion.

## Contribution

It provides a complete algebraic characterization of conditions for unique Euclidean reconstruction and formulates essential matrix averaging as a constrained optimization problem.

## Key findings

- High accuracy in SfM datasets
- Efficient and robust performance
- Outperforms existing methods

## Abstract

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 $(^n_2)$ 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.