Ambiguous Configurations for 3-View Projective Reconstruction

The critical configurations for projective reconstruction from three views are discussed. 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 cameras uniquely, up to projective transformation. For two views, the classification of critical configurations is well known the configuration is critical if and only if the points and camera centres all lie on a ruled quadric. For three views the critical configurations have not been identified previously. In this paper it is shown that for any placement of three given cameras there always exists a critical set consisting of a fourth-degree curve – any number of points on the curve form a critical set for the three cameras. Dual to this result, for a set of seven points there exists a fourth-degree curve such that a configuration of any number of cameras placed on this curve is critical for the set of points. Other critical configurations exist in cases where the points all lie in a plane, or one of the cameras lies on a twisted cubic.