On degeneracy of linear reconstruction from three views: linear line complex and applications

This paper investigates the linear degeneracies of projective structure estimation from line features across three views. We show that the rank of the linear system of equations for recovering the trilinear tensor of three views reduces to 23 (instead of 26) when the scene is a linear line complex [LLC] (a set of lines in space intersecting at a common line). The LLC situation is only linearly degenerate, and one can obtain a unique solution when the admissibility constraints of the tensor are accounted for. The line configuration described by an LLC, rather than being some obscure case, is in fact quite typical. It includes, as a particular example, the case of a camera moving down a hallway in an office environment or down an urban street. Furthermore, an LLC situation may occur as an artifact such as in direct estimation from spatio-temporal derivatives of image brightness. Therefore, an investigation into degeneracies and their remedy is important also in practice.