Directions of motion fields are hardly ever ambiguous
Recent literature [7, 10, 11, 9, 13, 17] provides a number of results regarding uniqueness aspects of motion fields and exact image displacements due to 3-D rigid motion. Here instead of the full motion field we consider only the direction of the motion field due to a rigid motion and ask 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 fact that the depth of the surfaces in view is positive (i.e. that the surface in view is in front of the camera). With this analysis we thus offer a theoretical foundation for image constraints employing only the sign of flow in various directions and provide a solid basis for their utilization in addressing 3D dynamic vision problems.
For two different rigid motions (with instantaneous translational and rotational velocities (t1, Ω1) and (t2, Ω2)) to yield the same direction of the flow, the surfaces in view must satisfy certain inequality and equality constraints, called critical surface constraints. A complete description of image areas where the constraints cannot be satisfied is derived and it is shown that if the imaging surface is a whole sphere, any two motions with different translation and rotation axes can be distinguished using only the direction of the flow. In the case where the imaging surface is a hemisphere or a plane, it is shown that two motions could give rise to the same direction of the flow if (t1×t2)·(Ω1×Ω2)=0 and several additional constraints are satisfied. For this to occur, the surfaces in view must satisfy all the critical surface constraints; thus at some points only a single depth value is allowed. Similar results are obtained for the case of multiple motions. Consequently, directions of motion fields are hardly ever ambiguous.
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