On the Exact Linearization of Structure From Motion

Abstract

The estimation of structure from motion has been a central task of computational vision over the last decade. As it is very well known, the problem is nonlinear due to the perspective nature of the measurements. One may ask at this point: does there exist a clever choice of coordinates which simplifies the estimation task? In particular, since "linearity" is a coordinate-dependent notion, is there a choice of coordinates such that the problem of estimating structure from motion becomes linear? In this paper we prove that the answer to the above question is no. An immediate consequence is that all choices of coordinates representations are structurally equivalent, in the sense that, at the current state of understanding of nonlinear estimation, none of them has an advantage based on geometric properties; instead, the difference between them is based purely on computational (numerical) ground. A further consequence of our result is the legitimation of the use of local linearization-based techniques (such as the Extended Kalman Filter) for estimating structure from known motion.