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 model ed two-camera stereo vision system, whether or not a prescribed robot posit ioning task has been precisely accomplished?" Results are shown for three c amera model classes: injective cameras, weakly calibrated projective camera s, and uncalibrated projective cameras. In particular, given a weakly calib rated stereo pair, it is shown that a positioning task can be precisely acc omplished if and only if the task specification is invariant to projective transformations. It is shown that injective and uncalibrated projective cam eras can accomplish fewer tasks, but are still able to accomplish tasks inv olving 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-b ased control architecture. It is shown that, for any class of camera models , the set of tasks which can be precisely accomplished using a position-bas ed 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 co incidences. Two ways of extending the idea of position-based control to acc omplish more tasks are also presented.