Necessary and Sufficient Polynomial Constraints on Compatible Triplets of Essential Matrices
This provides a foundational algebraic tool for multi-view geometry problems like camera pose estimation and structure from motion, though it is incremental as it builds on existing essential matrix theory.
The paper tackles the problem of algebraically characterizing compatible triplets of essential matrices for three calibrated cameras, proposing necessary and sufficient polynomial constraints in the form of six cubic matrix equations, one quartic, and one sextic scalar equations.
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.