Motion fields are hardly ever ambiguous

Active gaze control improves optic flow-based segmentation and steering

An observer traversing an environment actively relocates gaze to fixate objects. Evidence suggests that gaze is frequently directed toward the center of an object considered as target but more likely toward the edges of an object that appears as an obstacle. We suggest that this difference in gaze might be motivated by specific patterns of optic flow that are generated by either fixating the center or edge of an object. To support our suggestion we derive an analytical model that shows: Tangentially fixating the outer surface of an obstacle leads to strong flow discontinuities that can be used for flow-based segmentation. Fixation of the target center while gaze and heading are locked without head-, body-, or eye-rotations gives rise to a symmetric expansion flow with its center at the point being approached, which facilitates steering toward a target. We conclude that gaze control incorporates ecological constraints to improve the robustness of steering and collision avoidance by actively generating flows appropriate to solve the task.

The projective geometry of ambiguous surfaces

The projective geometry underlying the ambiguous case of scene reconstruction from image correspondences is developed. The am biguous case arises when reconstruction yields two or more essentially different surfaces in space, each capable of giving rise to the image correspondences. Such surfaces naturally occur in complementary pairs. Ambiguous surfaces are examples of rectangular hyperboloids. Complementary ambiguous surfaces intersect in a space curve of degree four, which splits into two components, namely a twisted cubic (space curve of degree three), and a straight line. For each ambiguous surface compatible with a given set of image correspondences, a complementary surface compatible with the same image correspondences can always be found such that both the original surface and the twisted cubic contained in the intersection of the two surfaces are invariant under the same rotation through 180°. In consequence, each ambiguous surface is subject to a cubic polynomial constraint. This constraint is the basis of a new proof of the known result that there are, in general, exactly ten scene reconstructions compatible with five given image correspondences. Ambiguity also arises in reconstruction based on image velocities rather than on image correspondences. The two types of ambiguity have m any sim ilarities because image velocities are obtained from image correspondences as a limit, when the distances between corresponding points become small. It is shown that the amount of similarity is restricted, in that when passing from image correspondences to image velocities, some of the detailed geometry of the ambiguous case is lost.