The analysis of multiple views of the same scene has received a lot of attention recently. Many algebraic relations have been discovered between primitives in multiple views. Most relate either points or lines across views. Scene independent multiview constraints, like the Fundamental matrix, the Trilinear Tensor and the Quadrilinear Tensor that relate two, three, and four views respectively [4,5,7,8,12,13], encapsulate the viewing parameters such as the pose and the internal parameters of the cameras producing these views. View-independent constraints on configurations of points is an active area of research currently. The relationships between projections of points moving with uniform velocity presented recently [11] fall under this category.
Multiple view situations in Computer Vision have been analyzed with two objectives: to derive scene-independent constraints relating multiple views and to derive view-independent constraints relating the multiple scene points. The second approach overlaps with the Structure From Motion (SFM), which attempts to compute the 3D information of a set of world points undergoing a rigid motion from multiple observations using a single camera. Non-rigid motion is difficult to analyze in this scheme. The case of multiple objects moving with different velocities can be considered very close to the case of non-rigid motion. The algebraic relations between the projections of multiple, linearly moving objects in a scene [11] was shown to be view-independent, under the assumption that the points move with uniform velocity. The constraints relating multiple moving objects can be classified into time-dependent and time-independent relationships, similar to the scene and view dependence mentioned above. The two view constraints on points moving with a constant velocity is another noteworthy contribution in this direction [2].
In this paper, we study the algebraic relationships between moving objects imaged from different points of view. We derive time-dependent and time-independent relationships between the velocities and accelerations of the affine projections of moving objects in Section 2. We show in Section 3 that by observing an object motion over time using stationary cameras, we can arrive at time-independent constraints on the motion. Some applications of these are outlined in Section 4. Section 5 discusses how these relationships can be extended for the general perspective projections. We present a comprehensive analysis of the multiview relations in Section 6. Section 7 presents a few concluding remarks.