Bounds on translational and angular velocity components from flrst order derivatives of image flow

Abstract

A moving rigid object produces a moving image on the retina of an observer. It is shown that only the first order spatial derivatives of image motion are sufficient to determine (i) the maximum and minimum velocities of the object towards the observer, and (ii) the maximum and minimum angular velocities of the object along the direction of view. The second or higher order derivatives whose estimation is expensive and unreliable are not necessary. (The second order derivatives are necessary to determine the actual motion of the objectj many researchers have worked on this problem.) These results are interpreted in the image domain in terms of three differential invariants of the image flow field: divergence, curl, and shear magnitude. In the world domain, the above results are interpreted in terms of the motion and local sllrface orientation of the object. In particular, the result that the maximum velocity of approach of an object can be determined from only the first order derivatives has a fundamental significance to both biological and machine vision systems. It implies that an organism (or a robot) can quickly respond to avoid collision with a moving object from only coarse information. This capability exists irrespective of the shape or motion of the object. The only restriction is that motion should be rigid.