Monocular pose determination from lines: critical sets and maximum number of solutions

In this paper we consider a subpart of the following problem. We assume that we have a set of known three-dimensional lines that we observe with a camera with an unknown pose and orientation. The problem is to recover the position and orientation of the camera from the observed image lines assuming that the correspondence has been established between the 2D and the 3D lines. Numerical methods have already been proposed for solving this problem but the question of the uniqueness of the solution has not yet been addressed. We show that there exist infinite sets of three-dimensional lines such that no matter how many lines we observe in these sets, the solution to the orientation or pose determination problem is not unique. We also give the maximum number of possible solutions. These results are important because they clearly define the domain of validity of algorithms which solve the orientation or pose determination problem.