The critical sets of lines for camera displacement estimation: A mixed Euclidean-projective and constructive approach

The authors consider the problem of recovering the relative displacements of a camera by using line matches in three views. In particular, they try to find whether there exist sets of 3-D lines such that irrespective of how many lines are observed there will always be several solutions to the relative displacement estimation problem. Recently, T. Buchanan (1992) provided a first analysis of the problem in which he gave a positive answer. Here, an attempt is made to build on his work and extend it in several directions. First, his purely projective analysis is cast in a more Euclidean framework better suited to applications. Second, his critical set is related to those of previously published algorithms. Third, the algebraic equations are introduced and an effective, i.e., computational, approach is provided for describing these critical sets in terms of simple geometric properties. This has allowed analysis of the structure of the critical sets