Image Divergence and Deformation from Closed Curves

This paper describes a novel method to measure the differential invariants of the image velocity field from the integral of normal image velocities around image contours. This is equivalent to measuring the temporal changes in the area of a closed contour. This avoids having to recover a dense image velocity field and taking partial derivatives. It also does not require point or line correspondences. Moreover integration provides some immunity to image measurement noise. It is shown how an active observer making small, deliberate motions can use the estimates of the divergence and deformation of the image velocity field to determine the object surface orientation and time to contact. The results of real-time experiments are presented in which arbitrary image shapes are tracked using B-spline snakes and the invariants are computed efficiently as closed-form functions of the B-spline control points. This information is used to guide a robot manipulator in obstacle collision avoidance, object manipulation and navigation.