Object-image correspondence for curves under projections

Joseph M. Burdis, Irina A. Kogan

Published 2012-02-06, updated 2013-03-15Version 2

We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The main advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem is then solved by differential signature construction based on Cartan's moving frame method. A similar approach can be used to decide whether a given finite set of ordered points on a plane is an image of a given finite set of ordered points in R^3. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.