Registration of 2D point sets by complex translation and rotation operations

Alignment of two sets containing two dimensional vectors (2D points) constitutes an important problem in medical imaging, remote sensing, and computer vision. We assume that the points in one set, called the transformed set, are constructed by translating and rotating the points in the other set, called the original set. The points in both sets are represented by complex numbers. In order to translate and then rotate a point, we add a complex constant and then multiply by a complex exponential respectively. We construct a cost function which tries to achieve the least-squares differences between a given transformed set and the set containing transformed points with respect to optimization parameters. We implement the Newton-Raphson optimization algorithm with polynomial line search in order to minimize this cost function. Simulation results with multiple datasets demonstrate that the proposed method aligns two sets efficiently and reliably.