5D Motion Subspaces for Planar Motions

In practice, rigid objects often move on a plane. The object then rotates around a fixed axis and translates in a plane orthogonal to this axis. For a concrete example, think of a car moving on a street. Given multiple static affine cameras which observe such a rigidly moving object and track feature points located on this object, what can be said about the resulting feature point trajectories in the camera views? Are there any useful algebraic constraints hidden in the data? Is a 3D reconstruction of the scene possible even if there are no feature point correspondences between the different cameras? And if so, how many points are sufficient? Does a closed-form solution to this shape from motion reconstruction problem exist?

This paper addresses these questions and thereby introduces the concept of 5 dimensional planar motion subspaces: the trajectory of a feature point seen by any camera is restricted to lie in a 5D subspace. The constraints provided by these motion subspaces enable a closed-form solution for the reconstruction. The solution is based on multilinear analysis, matrix and tensor factorizations. As a key insight, the paper shows that already two points are sufficient to derive a closed-form solution. Hence, even two cameras where each of them is just tracking one single point can be handled. Promising results of a real data sequence act as a proof of concept of the presented insights.