Directions of motion fields are hardly ever ambiguous

If instead of the full motion field, we consider only the direction of the motion field due to a rigid motion, what can we say about the three-dimensional motion information contained in it? This paper provides a geometric analysis of this question based solely on the constraint that the depth of the surfaces in view is positive. The motivation behind this analysis is to provide a theoretical foundation for image constraints employing only the sign of flow in various directions and justify their utilization for addressing 3D dynamic vision problems. It is shown that, considering as the imaging surface the whole sphere, independently of the scene in view, two different rigid motions cannot give rise to the same directional motion field. If we restrict the image to half of a sphere (or an infinitely large image plane) two different rigid motions with instantaneous translational and rotational velocities (t(1), omega(1)) and (t(2), omega(2)) cannot give rise to the same directional motion field unless the plane through t(1) and t(2) is perpendicular to the plane through omega(1) and omega(2) (i.e., (t(1) x t(2)) . (omega(1) x omega(2)) = 0). In addition, in order to give practical significance to these uniqueness results for the case of a limited field of view, we also characterize the locations on the image where the motion vectors due to the different motions must have different directions. If (omega(1) x omega(2)) . (t(1) x t(2)) = 0 and certain additional constraints are met, then the two rigid motions could produce motion fields with the same direction. For this to happen the depth of each corresponding surface has to be within a certain range, defined by a second and a third order surface. Similar more restrictive constraints are obtained for the case of multiple motions. Consequently, directions of motion fields are hardly ever ambiguous. A byproduct of the analysis is that full motion fields are never ambiguous with a half sphere as the imaging surface.