Observer Curve and Object Detection from the Optic Flow

A collision is an event where the robot path intersects with an object in the environment. Collisions can be desired if the object is a goal, or undesired if the object is an obstacle. We call the place of intersection a collision point. Prediction of collision points relies on a continuity assumption of the robot motion such as constant velocity. The robot is equipped with monocular vision to sense its environment. Motion of the robot results in motion of the environment in the sensory domain. The optic ow equals the projection of the environment motion on the image plane. We show that under the continuity assumption described above, the collision points can be computed from the optic ow without deriving a model of the environment. We show this for two types of observer motion. First we consider a translating observer and show that the projections of the collision points are located on the curve through the focus of expansion and in the direction of its temporal derivatives. Second we consider a mobile robot. We derive the collision points by introducing an invariant, the curvature scaled depth. This invariant couples the rotational velocity of the robot to its translational velocity and is closely related to the curvature of the mobile robot's path. We show that the spatial derivatives of the curvature scaled depth give the object surface orientation.