Determining the Singularities for the Observation of Three Image Lines

The determination of the singularity cases in the observation of image features is a complicated problem, which is still open, apart from image points. For the first time, we provide the singularity cases in the observation of three image lines. We show that a concept named the “hidden robot,” which was formerly used for understanding the singularities of a vision-based controller dedicated to parallel robots and, more recently, which proved to be efficient for finding the singularity cases in the observation of image points, can be used for interpreting the singularities in the observation of three image lines. The hidden robot concept considerably simplifies the analysis by using geometric interpretations of the mapping degeneracy and tools provided by the mechanical engineering community. We prove that in the most complicated case where three general lines in space are observed, singularities appear when the origin of the observed object frame is either on a quadric or a cubic surface whose location in space depends on the configuration of the observed lines. In simpler cases where at least two lines belong to the same plane, these two surfaces can degenerate into simpler geometrical loci (e.g., planes, cylinders, and lines).