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Directions of Motion Fields are Hardly Ever Ambiguous

Authors
  • Brodsky, Tomas1
  • Fermüller, Cornelia1
  • Aloimonos, Yiannis1
  • 1 University of Maryland, Computer Vision Laboratory, Center for Automation Research, Computer Science Department, and Institute for Advanced Computer Studies, College Park MD, 20742-3275, USA
Type
Published Article
Journal
International Journal of Computer Vision
Publisher
Springer-Verlag
Publication Date
Jan 01, 1998
Volume
26
Issue
1
Pages
5–24
Identifiers
DOI: 10.1023/A:1007928406666
Source
Springer Nature
Keywords
License
Yellow

Abstract

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(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 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. 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.

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