Approximate Point-to-Surface Registration with a Single Characteristic Point

We present approximation algorithms for point-to-surface registration problems which have applications in medical navigation systems. One of the central tasks of such a system is to determine a “good” mapping (the registration transformation or registration for short) of the coordinate system of the operation theatre onto the coordinate system of a 3D model M of a patient, generated from CRor MRT scans. The registration φ is computed by matching a 3D point set P measured on the skin of the patient to the 3D model M. It is chosen from a class R of admissible transformations (e.g., rigid motions) so that it approximately minimizes a suitable error function e (such as the directed Hausdorff or mean squared error distance) between φ(P) and M, i.e., φ = argminφ′∈R e(φ′(P),M). A common technique to support the registration process is to determine either automatically or manually so-called characteristic points or landmarks, which are corresponding points on the model and in the point set. Since corresponding characteristic points are supposed to be mapped onto (or close to) each other, this reduces the number of degrees of freedom of the matching problem. We provide approximation algorithms which compute a rigid motion registration in the most difficult setting of only a single characteristic point.