Critical motions and ambiguous Euclidean reconstructions in auto-calibration

The motions that lead to ambiguous Euclidean reconstructions in auto-calibration are investigated. Several auto-calibration constraints are considered: vanishing skew, known aspect ratio and internally calibrated cameras except for unknown focal lengths. We give a complete description of such critical motions in terms of algebraic manifolds and, in many cases, an explicit, geometric description for any number of cameras. For example, in the case of internally calibrated cameras except for unknown focal lengths, the only motions for which an affine reconstruction is ambiguous are either (i) rotations around (at most) two fired camera centres, or (ii) a planar motion on a conic with the optical axis tangent to the conic, or (iii) translation along the optical axis with arbitrary rotations around the optical axis. Moreover some practically important cases are also discussed