pose estimation problems.

Previous works mainly focus on solving the least-squares problem. Although large progress has been made, achieving the optimal solution and real-time performance is still a challenge. Some algorithms [@olsson2009branch; @briales2017convex] are capable of finding the global minimum, however, their running time makes them not suitable for real-time applications. Recently, Wientapper [@wientapper2018universal] provided an efficient algorithm, but their algorithm can not achieve the global optimality in theory, and has the risk of no solution. Furthermore, previous works generally assume that the correct solution is the global minimizer. However, we find that the correct solution may come from a local minimizer in certain configurations.

Minimal solution is important to eliminate outliers in the RANSAC framework [@fischler1981random]. As the pose has 6 degrees of freedom (DoF), any combinations of 3D correspondences providing 6 independent constraints form a minimal configuration. Previous algorithms are proposed to solve these minimal configurations case by case [@chen1990pose; @ramalingam2010p2pi; @ramalingam2013theory]. The contributions of this paper are as follows:

First, we prove that there exist ambiguous configurations for any number of plane and line correspondences, which leads to multiple solutions. When a configuration approximates ambiguity, the correct solution of a problem may come from a local minimum. 