Critical configurations of lines to geometry determination of three cameras

We address the question of, what structure of a set of lines in space constitutes a critical configuration to three generally positioned cameras. By critical configuration, it is meant a set of lines in space whose image projections to the cameras do not allow unique determination of the cameraspsila relative positions in space. We approach the question by looking into the trifocal tensor of the cameraspsila a quantity tightly related to the camera geometry. We focus on structures of lines in space that are common in reality - the linear structures - and examine which of them would not allow the tensor to be uniquely determined. Linear structures of lines include linear ruled surface, linear line congruence, and linear line complex, and more specifically line pencil, point star, and ruled plane. We offer a summary of by how much such families of line set are shy of the general enough structure for a unique determination of the tensor. We also point out how many lines need be visible minimally in each of them, for the full information of the associated structure to be revealed in the image data. We present synthetic and real image results to confirm the findings. The findings are important to the validity and stability of algorithms related to structure from motion and projective reconstruction using line correspondences.