Motion field and optical flow: qualitative properties

It is shown that the motion field the 2-D vector field which is the perspective projection on the image plane of the 3-D velocity field of a moving scene, and the optical flow, defined as the estimate of the motion field which can be derived from the first-order variation of the image brightness pattern, are in general different, unless special conditions are satisfied. Therefore, dense optical flow is often ill-suited for computing structure from motion and for reconstructing the 3-D velocity field by algorithms which require a locally accurate estimate of the motion field. A different use of the optical flow is suggested. It is shown that the (smoothed) optical flow and the motion field can be interpreted as vector fields tangent to flows of planar dynamical systems. Stable qualitative properties of the motion field, which give useful informations about the 3-D velocity field and the 3-D structure of the scene, usually can be obtained from the optical flow. The idea is supported by results from the theory of structural stability of dynamical systems.>