Using dynamics to recover Euclidian 3-dimensional structure from 2-dimensional perspective projections

In this paper we consider the problem of recovering the 3-dimensional Euclidian structure of a rigid object from multi-frame point correspondence data in a sequence of 2-D images obtained under perspective projection. The main idea is to recast the problem as the identification of an LTI system based on partial data. The main result of the paper shows that, under mild conditions, the lowest order system whose projections interpolate the 2-D data, yields (up to a single scaling constant) the correct 3 dimensional Euclidean coordinates of the points. Finally, we show that the problem of finding this system (and hence the associated 3-D data) can be recast into a rank minimization form that can be efficiently solved using convex relaxations. In contrast, existing approaches to the problem, based on iterative matrix factorizations can recover structure only up to a projective transformation that does not preserve the Euclidian geometry of the object.