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 single 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 the last two decades. 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.

中文翻译：

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基本矩阵相容三元组的充分必要多项式约束

基本矩阵包含两个校准相机的相对旋转和平移参数。基本矩阵的众所周知的代数表征，即任意矩阵（二阶）变为基本矩阵的充分必要条件，由一个单一的三阶矩阵方程组成。基于这个方程，在过去的二十年里，已经提出了许多针对不同相对姿态估计问题的有效算法解决方案。在三个视图中，描述三个校准相机几何的可能方法来自考虑基本矩阵的兼容三元组。兼容性意味着三元组与校准相机的某种配置的对应关系。本文的主要目标是给出基本矩阵的相容三元组的代数表征。具体来说，我们对实数二阶基本矩阵的三元组提出了必要且充分的多项式约束，以确保其兼容性。约束以六个三次矩阵方程、一个四次和一个六次标量方程的形式给出。即使在具有共线中心的相机的情况下，所提出的约束的一个重要优点是它们的充分性。约束的应用可能包括三个或更多视图中的相对相机姿态估计、运动增量结构的基本矩阵的平均、多视图相机自动校准等。我们对实数二阶基本矩阵的三元组提出了必要且充分的多项式约束，以确保其兼容性。约束以六个三次矩阵方程、一个四次和一个六次标量方程的形式给出。即使在具有共线中心的相机的情况下，所提出的约束的一个重要优点是它们的充分性。约束的应用可能包括三个或更多视图中的相对相机姿态估计、运动增量结构的基本矩阵的平均、多视图相机自动校准等。我们对实数二阶基本矩阵的三元组提出了必要且充分的多项式约束，以确保其兼容性。约束以六个三次矩阵方程、一个四次和一个六次标量方程的形式给出。即使在具有共线中心的相机的情况下，所提出的约束的一个重要优点是它们的充分性。约束的应用可能包括三个或更多视图中的相对相机姿态估计、运动增量结构的基本矩阵的平均、多视图相机自动校准等。