Correspondence of projected 3-D points and lines using a continuous GRASP

Abstract

The field of computer vision has experienced rapid growth over the past 50 years. Many computer vision problems have been solved using theory and ideas from algebraic projective geometry. In this paper, we look at a previously unsolved problem from object recognition, namely object recognition when the correspondences between the object and image data are not known a priori. We formulate this problem as a mixed-integer non-linear optimization problem in terms of the unknown projection relating the object and image, as well as the unknown assignments of object points and lines to those in the image. The global optimum of this problem recovers the relationship between the object points and lines with those in the image. When certain assumptions are enforced on the allowable projections mapping the object into the image, a proof is provided which permits one to solve the optimization problem via a simple decomposition. We illustrate this decomposition approach on some example scenarios.