A general framework for trajectory triangulation

The multiple view geometry of static scenes is now well understood. Recently attention was turned to dynamic scenes where scene points may move while the cameras move. The triangulation of linear trajectories is now well handled. The case of quadratic trajectories also received some attention. We present a complete generalization and address the problem of general trajectory triangulation of moving points from non-synchronized cameras. Two cases are considered: (i) the motion is captured in the images by tracking the moving point itself, (ii) the tangents of the motion only are extracted from the images. The first case is based on a new representation (to computer vision) of curves (trajectories) where a curve is represented by a family of hypersurfaces in the projective space P5. The second case is handled by considering the dual curve of the curve generated by the trajectory. In both cases these representations of curves allow: (i) the triangulation of the trajectory of a moving point from non-synchronized sequences, (ii) the recovery of more standard representation of the whole trajectory, (iii) the computations of the set of positions of the moving point at each time instant an image was made. Furthermore, theoretical considerations lead to a general theorem stipulating how many independent constraints a camera provides on the motion of the point. This number of constraint is a function of the camera motion. On the computation front, in both cases the triangulation leads to equations where the unknowns appear linearly. Therefore the problem reduces to estimate a high-dimensional parameter in presence of heteroscedastic noise. Several method are tested.