Abstract. We describe a method for determining a ne and metric calibration of a camera with unchanging internal parameters undergoing planar motion. It is shown that a ne calibration is recovered uniquely, and metric calibration up to a two fold ambiguity. The novel aspects of this work are: rst, relating the distinguished objects of 3D Euclidean geometry to xed entities in the image second, showing that these xed entities can be computed uniquely via the trifocal tensor between image triplets third, a robust and automatic implementation of the method. Results are included of a ne and metric calibration and structure recovery using images of real scenes.

Three-dimensional (3D) projective structure, that is structure modulo a projectivity of 3D space, can be recovered from its projection in multiple perspective images. The images might be acquired, for example, by a moving monocular camera or a stereo rig. This projective structure can be upgraded to Euclidean structure by identifying two entities, the plane at infinity and the absolute conic.Auto-calibration methods use constraints induced by the rigid motion of the camera to determine the Euclidean structure (or equivalently the camera calibration). Often these motion constraints are supplemented by known values of the camera's internal parameters or scene constraints in order to resolve ambiguities or stabilize the algorithms.It is shown in this paper that in certain common situations this supplementary information may not resolve the ambiguity. This is illustrated for the particular ambiguity arising for motions with a single direction of the rotation axis. Four types of constraint are analysed, and the conditions under which the ambiguity is not resolved are given. The constraint cases are: perpendicular image axes (the zeroskew constraint); specified image aspect ratio; specified image principal point; and perpendicularity of scene features.

A number of recent papers have demonstrated that camera "selfcalibration" can be accomplished purely from image measurements, without requiring special calibration objects or known camera motion. We describe a method, based on self-calibration, for obtaining (scaled) Euclidean structure from multiple uncalibrated perspective images using only point matches between views.The method is in two stages. First, using an uncalibrated camera, structure is recovered up to an affine ambiguity from two views. Second, from one or more further views of this affine structure the camera intrinsic parameters are determined, and the structure ambiguity reduced to scaled Euclidean. The technique is independent of how the affine structure is obtained. We analyse its limitations and degeneracies.Results are given for images of real scenes. An application is described for active vision, where a Euclidean reconstruction is obtained during normal operation with an initially uncalibrated camera. Finally, it is demonstrated that Euclidean reconstruction can be obtained from a single perspective image of a repeated structure.