Bounds on time-to-collision and rotational component from first-order derivatives of image flow

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 time-to-collision of the observer and the object and (ii) the maximum and minimum angular velocity 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 object.) 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 surface orientation of the object. In particular, the result that the minimum time-to-collision could 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.