The information in the direction of image flow

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 information regarding the three-dimensional motion? In this paper 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 (t₁, ω₁) and (t₂, ω₂) cannot give rise to the same directional motion field unless the plane through t₁ and t₂ is perpendicular to the plane through ω₁ and ω₂ (i.e., (t₁×t₂)·(ω₁ ×ω₂)=0). In addition, in order to give a 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 (ω₁×ω₂)·(t₁ ×t₂)=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