Abstract

This paper is a study of the motion field generated by moving 3D curves which are observed by a camera. We first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend. We then go ahead to study the motion field of a general curve. We first study the general case of a curve moving nonrigidly and introduce the notion of isometric motion. In order to do this, we introduce the notion of spatio-temporal surface and study its differential properties up to the second order. We show that, contrarily to what is commonly believed, the full motion field of the curve (i.e the component tangent to the curve) cannot be recovered from this surface. We also give the equations that characterize the spatio-temporal surface completely up to a rigid transformation. Those equations are the expressions of the first and second fundamental forms and the Gauss and Codazzi-Mainardi equations. We then...

Citations