Directions of Motion Fields are Hardly Ever Ambiguous

  • Authors:

  • Tomas Brodsky;Cornelia Fermüller;Yiannis Aloimonos

  • Affiliations:

  • Computer Vision Laboratory, Center for Automation Research, Computer Science Department, and Institute for Advanced Computer Studies, University of Maryland, College Park MD 20742-3275, USA. E-mai ...;Computer Vision Laboratory, Center for Automation Research, Computer Science Department, and Institute for Advanced Computer Studies, University of Maryland, College Park MD 20742-3275, USA. E-mai ...;Computer Vision Laboratory, Center for Automation Research, Computer Science Department, and Institute for Advanced Computer Studies, University of Maryland, College Park MD 20742-3275, USA. E-mai ...

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 1998

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Abstract

If instead of the full motion field, we consider only the directionof the motion field due to a rigid motion, what can we say about thethree-dimensional motion information contained in it? This paper provides ageometric analysis of this question based solely on the constraint that thedepth of the surfaces in view is positive. The motivation behind thisanalysis is to provide a theoretical foundation for image constraintsemploying only the sign of flow in various directions and justify theirutilization 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 giverise to the same directional motion field. If we restrict the image to halfof a sphere (or an infinitely large image plane) two different rigid motionswith instantaneous translational and rotational velocities(t_1ω_1) and(t_2,ω_2) cannot give rise to the samedirectional motion field unless the plane through t_1 andt_2 is perpendicular to the plane throughω_1 and ω_2 (i.e., (t_1× t_2) · (ω_1 ×ω_2) = 0. In addition, in order to give practicalsignificance to these uniqueness results for the case of a limited field ofview, we also characterize the locations on the image where the motionvectors due to the different motions must have different directions.If (ω_1 × ω_2) ·(t_1 × t_2) = 0 and certainadditional constraints are met, then the two rigid motions could producemotion fields with the same direction. For this to happen the depth of eachcorresponding surface has to be within a certain range, defined by a secondand a third order surface. Similar more restrictive constraints are obtainedfor the case of multiple motions. Consequently, directions of motion fieldsare hardly ever ambiguous. A byproduct of the analysis is that full motionfields are never ambiguous with a half sphere as the imaging surface.