Critical line-set configurations to epipolar geometry determination and application to image line transfer

We address the issue of what linear line structures in space constitute critical configuration to three generally positioned cameras. By critical configuration we mean a set of lines in space whose image projections to the cameras do not allow unique determination of the cameras’ relative positions. The work is focused on the domain of a particular but common line structure group – the linear line structures – which include linear ruled surface, linear line congruence, and linear line complex, and more specifically line pencil, point star, and ruled plane. We tackle the issue by looking into the rank of a matrix that is related to the estimation of the trifocal tensor – a quantity that captures the relative geometry of the cameras. Our result is a summary of the following: which of the linear line structure families are critical structures, how critical each of them is, and how many lines need be present minimally in each of the families for the full information of the particular family to be revealed in their image projections. Real image results are presented to illustrate the findings. The findings are important to the validity and stability of algorithms related to structure from motion and projective reconstruction using line correspondences. This article is an extended version of the report (Zhao and Chung, 2008) that received IAPR’s Piero Zamperoni Best Paper Award in 2008.