 
Summary: Trajectory Triangulation: 3D Reconstruction
of Moving Points from a Monocular
Image Sequence
Shai Avidan, Member, IEEE, and Amnon Shashua, Member, IEEE
AbstractÐWe consider the problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera,
i.e., to reconstruct moving objects from lineofsight measurements only. The task is feasible only when some constraints are placed on
the shape of the trajectory of the moving point. We coin the family of such tasks as ªtrajectory triangulation.º We investigate the
solutions for points moving along a straightline and along conicsection trajectories. We show that if the point is moving along a
straight line, then the parameters of the line (and, hence, the 3D position of the point at each time instant) can be uniquely recovered,
and by linear methods, from at least five views. For the case of conicshaped trajectory, we show that generally nine views are
sufficient for a unique reconstruction of the moving point and fewer views when the conic is of a known type (like a circle in 3D
Euclidean space for which seven views are sufficient). The paradigm of trajectory triangulation, in general, pushes the envelope of
processing dynamic scenes forward. Thus static scenes become a particular case of a more general task of reconstructing scenes rich
with moving objects (where an object could be a single point).
Index TermsÐStructure from motion, multipleview geometry, dynamic scenes.
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1 INTRODUCTION
WE wish to remove the static scene assumption in
3Dfrom2D reconstruction by introducing a new
paradigm we call trajectory triangulation that pushes the
