What tasks can be performed with an uncalibrated stereo vision system?

This article studies the following question: `When is it possible to decide, on the basis of images of point-features observed by an imprecisely modeled two-camera stereo vision system, whether or not a prescribed robot positioning task has been precisely accomplished?' Results are shown for three camera model classes: injective cameras, weakly calibrated projective cameras, and uncalibrated projective cameras. In particular, given a weakly calibrated stereo pair, it is shown that a positioning task can be precisely accomplished if and only if the task specification is invariant to projective transformations. It is shown that injective and uncalibrated projective cameras can accomplish fewer tasks, but are still able to accomplish tasks involving point coincidences. The same formal framework is applied to the problem of determining the set of tasks which can be precisely accomplished with the well-known position-based control architecture. It is shown that, for any class of camera models, the set of tasks which can be precisely accomplished using a position-based control architecture is a subset of the complete set of tasks which can be decided on the set, but includes all positioning tasks based on point coincidences. Two ways of extending the idea of position-based control to accomplish more tasks are also presented.