Effective curve registration using a novel solution method for overdetermined systems of polynomial equations

We propose a new method for registering a cloud of points in 2D onto a planar curve. This method does not require the knowledge of an initial guess for the position of the point cloud and proceeds without having to order, smooth out or otherwise process the points of the query point cloud in any way. The method consists in representing the planar curve by an algebraic curve, and in fitting the algebraic curve to the points of the point cloud by solving a corresponding over-constrained system of polynomial equations. The solution of this system is obtained using a new solution method for polynomial systems of equations, which we introduce in this paper. This solution method, which can be seen as an extension of the pseudo-inverse approach to solving linear systems of equations, naturally handles over-contrained systems of equations in a robust fashion.