Critical motions for auto-calibration when some intrinsic parameters can vary

Auto-calibration is the recovery of the full camera geometry and Euclidean scene structure from several images of an unknown 3D scene, using rigidity constraints and partial knowledge of the camera intrinsic parameters. It fa ils for certain special classes of camera motion. This paper derives necess ary and sufficient conditions for unique auto-calibration, for several prac tically important cases where some of the intrinsic parameters are known (e .g. skew, aspect ratio) and others can vary (e.g. focal length). We introdu ce a novel subgroup condition on the camera calibration matrix, which helps to systematize this sort of auto-calibration problem. We show that for sub group constraints, criticality is independent of the exact Values of the in trinsic parameters and depends only on the camera motion. We study such cri tical motions for arbitrary numbers of images under the following constrain ts: vanishing skew, known aspect ratio and full internal calibration module unknown focal lengths. We give explicit, geometric descriptions for most o f the singular cases. For example, in the case of unknown focal lengths, th e only critical motions are: (i) arbitrary rotations about the optical axis and translations, (ii) arbitrary rotations about at most two centres, (iii ) forward-looking motions along an ellipse and/or a corresponding hyperbola in an orthogonal plane. Some practically important special cases are also analyzed in more detail.