Rank classification of linear line structure in determining trifocal tensor

Abstract

The problem we address is: given line correspondences over three views, what is the condition of the line correspondences for the spatial relation of the three associated camera positions to be uniquely recoverable? We tackle the problem from the perspective of trifocal tensor, a quantity that captures the relative positions of the cameras in relation to the three views. We show that the rank of the matrix that leads to the estimation of the tensor reduces to 7, 11, 15 respectively for line pencil, point star, and ruled plane, which are structures that belong to linear line space; and 12, 19, 23 for general ruled surface, general linear congruence, and general linear line complex. These critical structures are quite typical in reality, and thus the findings are important to the validity and stability of practically all algorithms related to structure from motion and projective reconstruction using line correspondences. © 2008 Springer Berlin Heidelberg.