Rank Classification of Linear Line Structures from Images by Trifocal Tensor Determinability

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? The observed set of lines in space is called critical if there are multiple projectively nonequivalent configurations of the camera positions that can picture the same image triplet of the lines. We tackle the problem from the perspective of trifocal tensor, a quantity that captures the relative pose of the cameras in relation to the captured views. We show that the rank of a matrix that leads to the estimation of the tensor is reduced to 7, 11, 15 if the observed lines come from a line pencil, a line bundle, and a line field, respectively, which are line families belonging to linear line space; and 12, 19, 23 if the lines come from a general linear ruled surface, a general linear line congruence, and a general linear line complex, which are subclasses of linear line structures. We show that the above line structures, with the exception of linear line congruence and linear line complex, ought to be critical line structures. All of these 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.