The Critical Sets of Lines for Camera Displacement Estimation: A MixedEuclidean-Projective and Constructive Approach

• Authors: Nassir Navab;Olivier D. Faugeras
• Venue: International Journal of Computer Vision
• Year: 1997

The problem of the recovery of the motion, and the structure from motionis relevant to many computer vision applications. Many algorithms have beenproposed to solve this problem. Some of these use line correspondences. Forobvious practical reasons, it is important to study the limitation of suchalgorithms. In this paper, we are concerned with the problem of recoveringthe relative displacements of a camera by using line matches in three views.In particular, we want to know whether there exist sets of 3D lines such thatno matter how many lines we observe there will always be several solutions tothe relative displacement estimation problem. Such sets of lines may becalled critical in the sense that they defeat the correspondingalgorithm. This question has been studied in detail in the case of pointmatches by early-century Austrian photogrammeters and, independently, in themid-seventies and early-eighties by computer vision scientists. The answerlies in the idea of a critical surface.The case of lines has been much less studied. Recently, Buchanan (1992a,1992b) provided a first analysis of the problem in which he gave a positiveanswer: there exist critical sets of lines and they are pretty big (∞² lines). In general these sets are algorithm dependent,for example the critical set of lines for the Liu-Huang algorithm introducedin (Buchanan, 1992a), but Buchanan has shown that there is a critical setthat defeats any algorithm. This paper is an attempt to build on hiswork and extend it in several directions. First, we cast his purelyprojective analysis in a more euclidean framework better suited toapplications and, currently, more familiar to most of the computer visioncommunity. Second, we clearly relate his critical set to those of previouslypublished algorithms, in particular (Liu and Huang, 1988a, 1988b). Third, weprovide an effective, i.e., computational, approach for describingthese critical sets in terms of simple geometric properties. This has allowedus to scrutinize the structure of the critical sets which we found to be bothintricate and beautiful.