Epipolar spaces and optimal sampling strategies

If precise calibration information is unavailable, as is often the case for active binocular vision systems, the determination of epipolar lines becomes untenable. Yet, even without instantaneous knowledge of the geometry, the search for corresponding points can be restricted to areas called epipolar spaces. For each point in one image, we define the corresponding epipolar space in the other image as the union of all associated epipolar lines over all possible system geometries. Epipolar spaces eliminate the need for calibration at the cost of an increased search region. One approach to mitigate this increase is the application of a space variant sampling or foveation strategy. While the application of such strategies to stereo vision tasks is not new, only rarely has a foveation scheme been specifically tailored for a stereo vision task. In this paper we derive a foundation of theorems that provide a means for obtaining optimal sampling schemes for a given set of epipolar spaces. An optimal sampling scheme is defined as a strategy that minimizes the average area per epipolar space.