Critical Configurations for -view Projective Reconstruction

In this paper we give a characterization of critical configurations for projective reconstruction with any number of points and views. A set of cameras and points is said to be critical if the projected image points are insufficient to determine the placement of the points and the cameras uniquely, up to a projective transformation. For two views, the critical configurations are well-known. In this paper it is shown that a configuration of cameras and points all lying on the intersection of two distinct ruled quadrics is critical. In distinction to the two-view case, which in general allows two alternative solutions, there is a family of ambiguous reconstructions for the -view case. As a partial converse, it is shown that for any critical configuration, all the points lie on the intersection of two ruled quadrics.