Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings

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.

Necessary and Sufficient Polynomial Constraints on Compatible Triplets of Essential Matrices

“…The symmetric 9 × 9 matrix E (as well as its analog for fundamental matrices) has been previously introduced in [19,12,11] where some of its spectral properties were investigated. Matrix E is called compatible if it is constructed from a compatible triplet.…”

“…Matrix E is called compatible if it is constructed from a compatible triplet. In [11], the authors propose necessary and sufficient conditions on the compatibility of matrix E (more precisely, of an n-view generalization of matrix E) in terms of its spectral or singular value decomposition. In particular, these conditions imply that the characteristic polynomial of a compatible matrix E has the form…”

“…Their rectification (averaging) is one of the possible approaches to overcome the incompatibility. In [11], the authors proposed a novel solution to the essential matrix averaging problem based on their own algebraic characterization of compatible sets of essential matrices. Although the method from [11] showed good results outperforming the existing state-of-the-art solutions both in accuracy and in speed, it is not applicable to the practically important case of cameras with collinear centers.…”

“…The compatibility means that a triplet must correspond to a certain configuration of three calibrated cameras. In the recent paper [11], the authors considered compatible n-view multiplets of essential matrices and proposed their algebraic characterization in terms of either the spec-tral decomposition or the singular value decomposition of a symmetric 3n × 3n matrix constructed from the multiplet.…”

A Hybrid Global Image Orientation Method for Simultaneously Estimating Global Rotations and Global Translations

“…Later, the authors extended their work. Exploring the algebraic characterizations of essential matrices, they introduced a method to simultaneously solve for rotation and translation of each image from essential matrices (Kasten et al, 2019b). The disadvantage is that this method cannot deal with projection centres that are all (nearly) collinear.…”