Necessary and Sufficient Polynomial Constraints on Compatible Triplets of Essential Matrices

TL;DR

This paper introduces algebraic polynomial constraints that precisely characterize compatible triplets of essential matrices, aiding in multi-view camera pose estimation and structure from motion.

Contribution

It provides the first necessary and sufficient polynomial conditions for compatibility of triplets of essential matrices, including cases with collinear camera centers.

Findings

Six cubic matrix equations define compatibility.

One quartic and one sextic scalar equations are also part of the constraints.

Constraints are sufficient even for collinear camera centers.

Abstract

The essential matrix incorporates relative rotation and translation parameters of two calibrated cameras. The well-known algebraic characterization of essential matrices, i.e. necessary and sufficient conditions under which an arbitrary matrix (of rank two) becomes essential, consists of a unique matrix equation of degree three. Based on this equation, a number of efficient algorithmic solutions to different relative pose estimation problems have been proposed. In three views, a possible way to describe the geometry of three calibrated cameras comes from considering compatible triplets of essential matrices. The compatibility is meant the correspondence of a triplet to a certain configuration of calibrated cameras. The main goal of this paper is to give an algebraic characterization of compatible triplets of essential matrices. Specifically, we propose necessary and sufficient polynomial constraints on a triplet of real rank-two essential matrices that ensure its compatibility. The constraints are given in the form of six cubic matrix equations, one quartic and one sextic scalar equations. An important advantage of the proposed constraints is their sufficiency even in the case of cameras with collinear centers. The applications of the constraints may include relative camera pose estimation in three and more views, averaging of essential matrices for incremental structure from motion, multiview camera auto-calibration, etc.