Algebraic methods in 3‐d motion estimation from two‐view point correspondences

We consider the problem of determining motion (3‐D rotation and translation) of rigid objects from their images taken at two time instants. We assume that the locations of the perspective projection on the image plane of n points from the surface of the rigid body are known at two time instants. For n = 5, we show that there are at most ten possible motion values (in rotation and translation) and give many examples. For n ≥ 6, we show that the solution is generally unique. We derive a variety of necessary and sufficient conditions a solution must satisfy, show their equivalence, and use algebraic geometry to derive the bound on the number of solutions. A homotopy method is then used to compute all the solutions. Several examples are worked out and our computational experience is summarized.